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On Easiest Functions for Mutation Operators in Bio-Inspired Optimisation
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Understanding which function classes are easy and which are hard for a given algorithm is a fundamental question for the analysis and design of bio-inspired search heuristics. A natural starting point is to consider the easiest and hardest functions for an algorithm. For the (1+1) EA using standard bit mutation (SBM) it is well known that OneMax is an easiest function with unique optimum while Trap is a hardest. In this paper we extend the analysis of easiest function classes to the contiguous somatic hypermutation (CHM) operator used in artificial immune systems. We define a function MinBlocks and prove that it is an easiest function for the (1+1) EA using CHM, presenting both a runtime and a fixed budget analysis. Since MinBlocks is, up to a factor of 2, a hardest function for standard bit mutations, we consider the effects of combining both operators into a hybrid algorithm. We rigorously prove that by combining the advantages of k operators, several hybrid algorithmic schemes have optimal asymptotic performance on the easiest functions for each individual operator. In particular, the hybrid algorithms using CHM and SBM have optimal asymptotic performance on both OneMax and MinBlocks. We then investigate easiest functions for hybrid schemes and show that an easiest function for a hybrid algorithm is not just a trivial weighted combination of the respective easiest functions for each operator.
KeywordsRunning time analysis Theory Hybridisation Evolutionary algorithms Artificial immune systems
1 Introduction
Over the past years many bio-inspired search heuristics such as evolutionary algorithms, swarm intelligence algorithms, or artificial immune systems, have been developed and successfully applied to various optimisation problems. These heuristics have different strengths and weaknesses in coping with different fitness landscapes. Determining which heuristic is the best choice for a given problem is a fundamental and important question.
We use rigorous theoretical analyses to contribute to a better understanding of bio-inspired algorithms. A natural step is to investigate which functions are easy and which are hard for a given algorithm as this yields fundamental insights into their working principles, particularly with respect to their strengths and weaknesses.
Knowing how well an algorithm performs on its easiest and hardest functions, with regards to the expected optimisation time, provides general insights, as the expected optimisation time of every function will be no less than that of an easiest function and no more than that of a hardest function. With respect to applications, such research may also provide guidelines for the design of benchmarks for experimental studies [10].
For a simple elitist (1+1) evolutionary algorithm (EA) using only standard bit mutations (SBM) Doerr, Johannsen, and Winzen showed that OneMax, a function simply counting the number of ones in a bit string, is the easiest problem among all functions with a single global optimum [8]. This statement generalises to the class of all evolutionary algorithms that only use standard bit mutation for variation [33] as well as higher mutation rates and stochastic dominance [37]. On the other hand, He et al. [10] showed that the highly deceptive function Trap is the hardest function for the (1+1) EA.
In this work we consider typical mutation operators in artificial immune systems where no such results are available. Mutation operators in artificial immune systems [6] usually come with much larger mutation rates than mutation operators in evolutionary algorithms. One such example is the somatic contiguous hypermutation (CHM) operator from the B-cell algorithm (BCA) [23], where large contiguous blocks of a bit string are flipped simultaneously. Previous theoretical work on comparing SBM and CHM has contributed to the understanding of their benefits and drawbacks in the context of its runtime on different problems [14, 17, 18], the expected solution quality after a pre-defined number of steps [21], and in dynamic environments [22]. It is easy to see that hardest functions for CHM with respect to expected optimisation times are those where the algorithm gets trapped with positive probability such that the optimisation time is not finite, e. g., if there exists a second best search point for which all direct mutations to the optimum involve non-contiguous bits [17]. However, it is an open problem and the next step forward to determine what kind of functions are easiest for this type of mutation operator.
The BCA uses standard bit mutations alongside CHM and thus can be considered as a hybrid algorithm combining two mutation operators. More specifically, it uses a population of search points and creates \(\lambda \) clones for each of them. It then applies standard bit mutation to a randomly selected clone for each parent search point and subsequently applies CHM to all clones. The interplay between these two operators has been investigated for the vertex cover problem [14].
Hybridisations such as hyper-heuristics [30] and memetic algorithms [27] have become very popular over recent years. But despite their practical success their theoretical analysis is still in its infancy, noteworthy examples being the work in [1, 24, 32, 34]. However, nothing is known about easiest or hardest functions for such algorithms.
The goals of this paper are twofold. We first want to understand what functions are easy for CHM and investigate the performance of CHM and SBM on these functions. Afterwards we consider hybridisations of SBM and CHM and easiest functions for such algorithms. We first use the method by He et al. [10], explained in Sect. 3, to derive an easiest function for CHM which we call MinBlocks. In Sect. 4 we present an analysis of the optimisation time as well as a fixed budget analysis for CHM on MinBlocks. We then show that SBM alone is not able to optimise the constructed function and that a hybridisation of SBM and CHM can have significant advantages (Sect. 5.1). Finally, we investigate properties of easiest functions for such hybrid algorithms (Sect. 5.3).
This journal paper extends a preliminary conference paper [4] in several ways. Firstly, in Sect. 3 it is discussed how the framework for determining easiest and hardest functions is not restricted to (1+1)-style algorithms, but is general enough to apply for a much larger class of algorithms including (1+\(\lambda \)) EAs [13] and the recently popular (1+(\(\lambda ,\lambda \))) GA using mutation and crossover [7]. Secondly, the analysis of the advantages of hybridisation in Sect. 5.1 has been considerably extended. Rather than just giving an example of a hybrid algorithm using one out of two operators at each step with constant probability, the analysis has been generalised to allow different hybridisation schemes and an arbitrary number k of operators. Finally, in Sect. 5.2 an experimental analysis is presented to shed light on the performance of the algorithms for fitness functions that depend both on the number of ones (i.e., OneMax) and the number of blocks (MinBlocks) in the bit string.
2 Preliminaries
We will refer to the (1+1) \(A\) that uses SBM as the (1+1) EA, the most widely studied evolutionary algorithm.
Note that Algorithm 1 uses strict selection, i. e. only strict improvements are accepted. For most of our theoretical results we also consider a variant of Algorithm 1 with non-strict selection, where the acceptance condition "\(f(y) > f(x)\)" is replaced by "\(f(y) \ge f(x)\)".
3 Easiest and Hardest Functions
Let T(A, f, x) denote the expected number of function evaluations for the (1+1) \(A\) to find an optimal solution for the first time when starting at x (expected hitting time). In the following we only consider algorithms and functions that lead to a finite expected hitting time.
Definition 1
[10, Definition 1] Given a (1+1) \(A\) for maximising a class of fitness functions with the same optima (denoted by \(\mathcal {F}\)), a function f in the class is said to be an easiest function for the (1+1) \(A\) if \(T(A,f,x) \le T(A, g,x)\) for every \(g \in \mathcal {F}\) and every \(x \in S\). A function f in the class is said to be a hardest function for the (1+1) \(A\) if \(T(A,f,x) \ge T(A, g,x)\) for every \(g \in \mathcal {F}\) and every \(x \in S\).
The above definition of easiest and hardest functions is based on a point-by-point comparison of the runtime of the EA on two fitness functions. The criteria stated in the following Lemmas 4 and 5 for determining whether a fitness function is an easiest or hardest function for a (1+1) \(A\) were originally given in [10]. The main proof idea in [10] is to apply additive drift theorems [11], taking the expected hitting time as drift function (distance to the target state). We state these drift theorems as Theorem 2, referring to the presentation of Lehre and Witt [25], who provided a self-contained proof.
Theorem 2
- (i)
If \({\text {E}\left( X_t - X_{t+1} \mid X_0, \dots , X_t; X_t > 0\right) } \ge \delta _u\) then \({\text {E}\left( T \mid X_0\right) } \le X_0/\delta _u\).
- (ii)
If \({\text {E}\left( X_t - X_{t+1} \mid X_0, \dots , X_t\right) } \le \delta _\ell \) then \({\text {E}\left( T \mid X_0\right) } \ge X_0/\delta _\ell \).
Before presenting those criteria, we state and prove the following helper lemma. It is stated as Lemma 3 in [10] without proof.
Lemma 3
If the expected time T(A, f, x) is used as the drift function, then the expected drift1 is \(\varDelta _f(x)=1\) for all non-optimal search points x.
First, we have the following criterion of determining whether a fitness function is an easiest function for a (1+1) \(A\). The two lemmas below extend Theorem 1 and Theorem 2 in [10], respectively, as they are applicable to both strict elitist selection (that is, the parent x is replaced by the child y if \(f(y) > f(x)\)) and non-strict elitist selection (that is, the parent x is replaced by the child y if \(f(y) \ge f(x)\)). The framework in [10] was restricted to strict selection.
Lemma 4
for any two points x and y, if \(T(A,f,x) < T(A,f,y)\), then \(f(x) > f(y)\),
Let g(x) be any fitness function with the same optima as f(x). Choose the runtime T(A, f, x) as the drift function: \(d(x)= T(A,f,x)\).
When maximising f(x), according to Lemma 3, for any non-optimal point x the drift \(\varDelta _{f}(x) =1.\) In the following we prove that the drift \(\varDelta _{g}(x) \le \varDelta _{f}(x)=1\).
- Negative drift:
- let x and y be two points such that \(d(x) < d(y)\). According to the monotonically decreasing condition, \(f(x) > f(y)\). For f(x), since \(f(x) > f(y)\), y is never accepted and then \( \varDelta ^-_{f} (x)= 0. \) But for g(x), the negative drift is not positive. Thus we have$$\begin{aligned} \varDelta ^-_{g} (x) \le 0=\varDelta ^-_{f} (x). \end{aligned}$$(2)
- Positive drift:
let x and y be two points such that \(d(x) > d(y)\). If y is an optimum, then naturally \(f(x) < f(y)\). If y is not an optimum, then according to the monotonically decreasing condition, \(f(x) < f(y)\).
Let \(P^{[m]}(x,y)\) denote the probability of mutating x into y (which is independent of f or g).For f(x), since \(f(x)<f(y)\), y is always accepted and then \( P_{f}(x,y)=P^{[m]}(x,y). \) But for g(x), since g(x) might be larger, smaller than or equal to g(y), \(P_{g}(x,y)\le P^{[m]}(x,y)\). Hence$$\begin{aligned} \varDelta ^+_{g}(x) =\;&\sum _{y: d(x)> d (y)} P_{g}( x, y) ( d (x) -d(y)) \\ \le \;&\sum _{y:d (x) >d (y)} P_{f}( x, y) ( d (x) -d(y))=\varDelta ^+_{f}(x). \end{aligned}$$
Considering both negative drift and positive drift, we get \( {\varDelta _{g} (x) \le \varDelta _{f}(x)=1} \). Since \(\varDelta _{g} (x) \le 1\) for all x, according to Theorem 2, the expected runtime is \(T(A,g,x) \ge d(x)=T(A,f,x)\) and the theorem statement is derived. \(\square \)
In a similar way, we have the following criterion of determining whether a fitness function is a hardest function for a (1+1) \(A\), assuming that all expected optimisation times are finite2. The monotonically decreasing condition in the above lemma is replaced by the monotonically increasing condition.
Lemma 5
for any two non-optimal points x and y, if \(T(A,f,x)< T(A,f,y)\), then \(f(x) < f(y)\),
Let g(x) be any fitness function with the same optima as f(x). Choose the runtime T(A, f, x) as the drift function: \(d(x)= T(A,f,x)\).
- Positive drift:
- let x and y be two points such that \(d(x) > d(y)\). Let \(P^{[m]}(x,y)\) denote the probability of mutating x into y (which is independent of f or g). For f(x), if y is an optimum point, then the probability that x is mutated into y and is accepted, \(P_{f}(x,y)\), is equal to \(P^{[m]}(x,y)\). Similarly for g(x), \(P_{g}(x,y)=P^{[m]}(x,y)\) if y is an optimum point. If y is not an optimum point, according to the monotonically increasing condition, \(f(x) > f(y)\). For f(x), since \(f(x) > f(y)\), the probability \(P_{f}(x,y)=0\). But for g(x), since g(x) might be larger, smaller than or equal to g(y), \(P_{g}(x,y)\ge 0\). Thus we have \(P_{g}(x,y)\ge P_{f}(x,y)\) for all (x, y) such that \(d(x)>d(y)\). Therefore, the positive drifts for g and f satisfy$$\begin{aligned} \varDelta ^+_{g}(x) =\;&\sum _{y: d(x)> d (y)} P_{g}( x, y) ( d (x) -d(y)) \\ \ge \;&\sum _{y:d (x) >d (y)} P_{f}( x, y) ( d (x) -d(y))=\varDelta ^+_{f}(x). \end{aligned}$$
- Negative drift:
let x and y be two points such that \(d(x) < d(y)\). Then according to the monotonically increasing condition, \(f(x) < f(y)\).For f(x), since \(f(x)<f(y)\), y is always accepted and then \( P_{f}(x,y)=P^{[m]}(x,y). \) But for g(x), since g(x) might be larger, smaller than or equal to g(y), \(P_{g}(x,y)\le P^{[m]}(x,y)\). Hence$$\begin{aligned} \varDelta ^-_{g}(x) =\;&\sum _{y: d(x)< d (y)} P_{g}( x, y) ( d (x) -d(y)) \\ \ge \;&\sum _{y:d (x) <d (y)} P_{f}( x, y) ( d (x) -d(y))=\varDelta ^-_{f}(x). \end{aligned}$$
Considering both negative drift and positive drift, we get \( {\varDelta _{g} (x) \ge \varDelta _{f}(x)=1} \). Since \(\varDelta _{g} (x) \ge 1\) for all x, according to Theorem 2, the expected runtime is \(T(A,g,x) \le d(x)=T(A,f,x)\) and the theorem statement is derived. \(\square \)
The framework due to He et al. [10] is restricted to search heuristics that base their search on a single point in the search space and it is not obvious how to expand this to population-based algorithms. However, it is worth mentioning that the framework is not restricted to (1+1)-style algorithms. It can also be applied to algorithms that employ elitist selection but have a more complicated way of deciding on the next search point to use as next parameter. Such algorithms can be described as (1+1)-style algorithms with a much more complex mutation operator or, perhaps more adequately, by exactly the same kind of Markov chain as that of a (1+1)-style algorithm, i. e., by a Markov chain of the same size \(2^n \times 2^n\). Algorithms where this is true include the (1+\(\lambda \)) EA (see, e.g., [13]) that creates \(\lambda \) offspring by means of mutation, independently and identically distributed, and a best one replaces the current population if its fitness is not worse. In this sense the framework can deal with population-based heuristics (given that the population is restricted to a larger offspring population, not a larger parent population). It can also be applied beyond mutation-based algorithms to genetic algorithms with crossover if the restriction of a population of size only 1 is met. An example for such an algorithm is the so-called (1+(\(\lambda \), \(\lambda \))) GA [7], the first realistic evolutionary algorithm to provably beat the \({\varOmega \left( n\log n\right) }\) lower bound on \(\text {OneMax} \). We see that the framework is much more general and useful than it may appear at first sight.
We have already mentioned that we consider functions where an algorithm does not have finite expected optimisation time to be harder than those with finite expected optimisation times. It is well known that CHM (Algorithm 3) can be trapped in local optima when the parameter r is set to \(r=1\) [2] (see [17, p. 521] for a concrete example demonstrating this effect). Setting \(r=1\), however, reveals properties of the hypermutation operator in the clearest way and this is the reason we stick to this choice (compare [17]). This implies that analysing hardest functions for (1+1) CHM does not make much sense because it is easy to find functions where there is a positive probability that the algorithm gets stuck in a local optimum so that, consequently, the expected optimisation time is not finite. One could consider different measures of hardness for this situation, e. g., considering the conditional expected optimisation time given that a global optimum is found or, alternatively, considering the probability not to find a global optimum. This, however, is beyond the scope of this article.
4 Contiguous Hypermutations on an Easiest Function with a Unique Global Optimum
We are now ready to derive an easiest function with a unique global optimum for contiguous hypermutations and analyse the performance of the (1+1) CHM on this function.
4.1 Notation and Definition
We use \(x = x[0] x[1] \cdots x[n-1] \in \{0, 1\}^n\) as notation for bit strings of length n. For \(a, b \in \{0, 1, \dots , n-1\}\) we denote by \(x[a \dots b]\) the concatenation of x[a], \(x[(a+1) \bmod n]\), \(x[(a+2) \bmod n]\), ..., \(x[(a+i) \bmod n]\) where i is the smallest number from \(\{0, 1, \dots , n-1\}\) with \((a+i) \bmod n = b\). We denote by \(\left| x[a \dots b]\right| \) the number of bits in \(x[a \dots b]\), i. e., its length. We say that \(x \in \{0, 1\}^n\) contains a 1-block from a to b if \(x[a \dots b] = 1^{\left| x[a \dots b]\right| }\) and \(x[(a-1)\bmod n] = x[(b+1)\bmod n] = 0\) hold. Analogously we may speak of a 0-block from a to b. Note that bit strings need to contain at least one 0-bit and at least one 1-bit to contain a 0-block or a 1-block. It is easy to see that each \(x \in \{0, 1\}^n \setminus \{0^n, 1^n\}\) contains an equal number of 0-blocks and 1-blocks.
Definition 6
We define an easiest function with unique global optimum for contiguous hypermutations by defining a partition \(L_0 \mathrel {\dot{\cup }} L_1\mathrel {\dot{\cup }} L_2 \mathrel {\dot{\cup }} \cdots \mathrel {\dot{\cup }} L_l = \{0, 1\}^n\) and assigning fitness values accordingly. We call the function \(\textsc {MinBlocks} \) and define \(\textsc {MinBlocks} (x) = l - i\) for \(x \in L_i\). We define \(l = \left\lfloor n/2\right\rfloor +1\) and level sets \(L_0 = \{ 1^n \}\), \(L_1 = \{ 0^n \}\) and \(L_i = \large \{ x \in \{0, 1\}^n \mid x\) contains \(i-1\) different 1-blocks\(\large \}\) for each \(i \in \{2, 3, \dots , l\}\).
The function has a unique global optimum \(1^n\) with function value \(l = \left\lfloor n/2\right\rfloor +1\). The second best bit string is \(0^n\) with function value \(l-1\). All other bit strings contain an equal number of 0-blocks and 1-blocks. A bit string with j 1-blocks is in level set \(L_{j+1}\) and thus, its function value equals \(l-(j+1)=l-j-1\).
We defer the proof that MinBlocks is indeed an easiest function for (1+1) CHM to the next section (Theorem 8) in order to make use of arguments from the analysis of the expected optimisation time performed there.
4.2 Expected Optimisation Time
We analyse the expected optimisation time of the (1+1) CHM on MinBlocks, i. e., the expected number of function evaluations executed until the global optimum is reached [12]. To facilitate our analysis, we start with an analysis of the expected optimisation time starting from a bit string from a particular level set which will in turn allow us to prove that MinBlocks is an easiest function for (1+1) CHM. We will then continue with the overall upper and lower bounds for the optimisation time.
Lemma 7
The statement about \({\text {E}\left( T_0\right) }\) is trivial. For \({\text {E}\left( T_1\right) }\) it suffices to observe that any hypermutation which chooses as mutation length n leads from \(0^n\) to the unique global optimum \(1^n\). Such a mutation has probability \(1/(n+1)\) which implies \({\text {E}\left( T_1\right) } = n+1\). For \({\text {E}\left( T_2\right) }\) we observe that for each \(x \in L_2\) there are two mutations which lead to \(L_0 \cup L_1\), one leading to \(L_0\) and the other leading to \(L_1\). Since \(L_0\) and \(L_1\) are reached with equal probability 1 / 2 we have \({\text {E}\left( T_2\right) } = n(n+1)/2 + {\text {E}\left( T_1\right) }/2 = (n+1)^2/2\). For \({\text {E}\left( T_i\right) }\) with \(i>2\) we observe that only mutations that reduce the number of 1-blocks can lead to some \(L_j\) with \(j<i\). It is easy to see that, the number of 1-blocks can only be reduced by 1 in one contiguous hypermutation. In order to achieve that a mutation must start at the first bit of a block (either a 0-block or a 1-block) and end at the last bit of a block (either a 0-block or a 1-block) but this block cannot be the one just before the block containing the first flipped bit (otherwise the length of the mutation is n, the bit string is inverted and the number of 1-blocks remains unchanged). Thus, if there are j blocks in the bit string, the number of such mutations equals \(j(j-1)\). For \(x \in L_i\) the number of 1-blocks equals \(i-1\) and therefore the number of blocks equals \(2i-2\) so that there are \((2i-2)(2i-3)\) such mutations. Thus, the expected time to leave \(L_i\) equals \(n(n+1)/((2i-2)(2i-3))\) and \({\text {E}\left( T_i\right) } = n(n+1)/((2i-2)(2i-3))+ {\text {E}\left( T_{i-1}\right) }\) follows. \(\square \)
The expected runtimes provided in Lemma 7 allow us to verify whether MinBlocks satisfies the criteria set by Lemma 4. The following theorem establishes MinBlocks as an easiest function for (1+1) CHM.
Theorem 8
MinBlocks is an easiest function for the (1+1) CHM with strict or non-strict selection among all functions with a unique optimum.
The theorem follows from Lemma 7, the definition of MinBlocks, and Lemma 4. According to Definition 6, for all \(i\in \{1, 2, \ldots ,\left\lfloor n/2\right\rfloor \}\) the fitness value of solutions in subset \(L_i\) is strictly less than the fitness value of solutions in \(L_{i-1}\). Therefore, a solution x has a better MinBlocks value than a solution y, if and only if x and y belong to two distinct subsets \(L_i\) and \(L_j\) respectively such that \(i<j\). Note that in Lemma 7 the expected runtime of the (1+1) CHM initialised with a solution from \(L_i\), \({\text {E}\left( T_i\right) }\), satisfies \({\text {E}\left( T_i\right) }>{\text {E}\left( T_{i-1}\right) }\) for all i and thus \(i<j\) implies \({\text {E}\left( T_i\right) }<{\text {E}\left( T_j\right) }\). Therefore, \(\textsc {MinBlocks} (x)>\textsc {MinBlocks} (y)\) if and only if the expected runtimes of (1+1) CHM starting from solutions x and y satisfy \(T(A,f,x)={\text {E}\left( T_i\right) } < {\text {E}\left( T_{j}\right) }=T(A,f,y)\). According to Lemma 4, the above two way implication makes MinBlocks an easiest function for the (1+1) CHM.
We continue our analysis of the expected optimisation time. Note that the following bound asymptotically matches the lower bound of \({\varOmega \left( n^2\right) }\) for contiguous hypermutations and functions with a unique global optimum proven by Jansen and Zarges [17].
Theorem 9
Let T denote the expected optimisation time of the (1+1) CHM with strict or non-strict selection on MinBlocks. \({\text {E}\left( T\right) } = \ln (2) n^2 \pm {O\left( n\right) }\) holds.
Consider \(x \in \{0, 1\}^n\) selected uniformly at random. For each \({i {\in } \{0, 1, \dots , n{-}1\}}\) we have that a block ends at x[i] if \(x[i] \not = x[(i+1) \bmod n]\) holds. Thus, x[i] is the end of a block with probability 1 / 2 and we see that the expected number of blocks equals n / 2. Let I denote the number of blocks. An application of Chernoff bounds yields that for any constant \(\varepsilon \) with \(0< \varepsilon < 1\) we have \({\text {Pr}\left( I \ge (1-\varepsilon ) n/2\right) } = 1-e^{-{\varOmega \left( n\right) }}\).
4.3 Fixed Budget Analysis
It has been pointed out that the notion of optimisation time does not always capture the nature of how randomised search heuristics are applied in practice. As a result, fixed budget analysis has been introduced as an alternative theoretical perspective [19, 20]. Let \(x_t\) denote the current population after t rounds of contiguous hypermutation and selection. In fixed budget analysis we want to analyse \({\text {E}\left( f(x_t)\right) }\) for all \(t \le {\text {E}\left( T\right) }\) where \({\text {E}\left( T\right) }\) is the expected optimisation time. We do this here for MinBlocks to give a more complete picture about the performance of the (1+1) CHM. Note that a comparison of the (1+1) EA and the (1+1) CHM under the fixed budget perspective has previously been performed for some example functions [21].
We begin with a statement about the expected function value of a uniform random solution, reflecting the way (1+1) \(A\) algorithms are initialised, and prove that it is roughly \((n-2)/4 \pm 1/2\).
Theorem 10
Let \(x_0 \in \{0, 1\}^n\) be selected uniformly at random and \(f := \textsc {MinBlocks} \) from Definition 6. For the initial function value \({\text {E}\left( f(x_0)\right) } = \left\lfloor n/2\right\rfloor -(n/4)+2^{-n}\) holds.
We know from the analysis of the expected optimisation time in Sect. 4.2 that the expected number of blocks in \(x_0\) equals n / 2. Let \(z(x_0)\) denote the number of 0-blocks in \(x_0\) and remember that the number of 0-blocks is half the number of blocks. This implies \({\text {E}\left( z(x_0)\right) } = n/4\).
Let \(l=\left\lfloor n/2\right\rfloor +1\) (the value defined in Definition 6). Remember that \(z(1^n) = z(1^0) = 0\) according to our definition of a 0-block (see the definition of the notation at the beginning of Sect. 4.1). For \(x_0 \notin \{ 0^n, 1^n \}\) we have \(f(x_0) = l-1-z(x)\). Furthermore, we have \(f(1^n)=l\) and \(f(0^n) = l-1 = l-1-z(0^n)\). Thus, \(f(x_0) = l-1-z(x)\) holds always except for the case \(x_0 = 1^n\).
We now give upper and lower bounds on the expected function value after t iterations of the (1+1) CHM.
Theorem 11
For the smaller of the two lower bounds we replace the actual probabilities by the smallest probability for an increase in function value which equals \(2/(n(n+1))\). We can improve on this weak lower bound slightly by using a technique called multiplicative fixed budget drift as recently introduced by Lengler and Spooner [26]. For a lower bound we need a lower bound on the expected change in function value in one generation given the current function value. If the current bit string contains i bits, we know that the expected change equals \((2i-2)(3i-2)/(n(n+1))> 3i^2/(n(n+1)) > 3i/(n(n+1))\) where the last inequality is made because Theorem 1 in [26] requires a statement about this drift that is linear. Using this we obtain \(\left\lfloor n/2\right\rfloor +1 - \left( \left\lfloor n/2\right\rfloor +1-{\text {E}\left( f(x_0)\right) }\right) \cdot (1-3/(n(n+1)))^t\) as lower bound on the expected function value after t steps.
5 Hybridising Operators
executing one operator chosen probabilistically according to a given probability distribution \({\varvec{p}}= (p_1, \dots , p_k)\) (Algorithm 4),
- (1+1) HA-chained:
executing a chain of operators, where operators are applied probabilistically according to a given probability vector \({\varvec{p}}= (p_1, \dots , p_k)\) (Algorithm 5, note that \({\varvec{p}}\) does not have to be a probability distribution),
- (1+ k ) HA:
executing all operators in parallel and picking the best among the resulting solutions (Algorithm 6), and
- \({{\varvec{k}}} \times \mathbf (1+1)~HA \):
executing all operators sequentially, resulting in a chained sequence of operations, with selection after each operator (Algorithm 7).
The (1+1) HA-chained is probably the most commonly used method in hybrid algorithms. For instance, most Genetic Algorithms fall into this category since they use a probability \(p_c\) that crossover is applied before mutation. The (1+1) HA is commonly used in memetic algorithms and hyper-heuristics when at each step a variation operator is chosen with some probability; this strategy was called "SimpleRandom" in [5]. Theoretical work on simple hyper-heuristics that fall into this framework have previously been accomplished [24]. We also introduce two other hybridisation schemes for which the results presented in this section also hold. These are simplified versions of schemes that are applied in widely used hybrid algorithms. The (1+k) HA algorithm creates offspring by using different operators. This is a common strategy in multi-meme memetic algorithms, where multiple memes refer to multiple local improvement heuristics. The idea was called "Greedy approach" in [5]. The same idea also appears in simple island models with heterogeneous islands, that is, each island consists of one individual and uses a different variation operator. A real-world example of where such a strategy is used (albeit with populations and added complexity) is the Wegener system, a popular and effective algorithm in search-based software testing, where different islands use standard bit mutations with different mutation rates [36]. The \(k \times \)(1+1) HA is interesting as it has some resemblance to the recently introduced (1+(\(\lambda \),\(\lambda \))) GA, since both use selection between the application of the operators. However, the (1+(\(\lambda \),\(\lambda \))) GA does not fall exactly within the \(k \times \)(1+1) HA scheme. In the former, if the final solution is worse than the initial one in the sequence, then the initial solution is accepted for the next generation. On the other hand, the \(k \times \)(1+1) HA always accepts the last improving solution in the sequence.
The B-cell algorithm (BCA) [23] is an example of an algorithm that uses both SBM and CHM considered in this paper. More specifically, it uses a population of search points and creates \(\lambda \) clones for each of them. It then applies standard bit mutation to a randomly selected clone for each parent search point and subsequently applies CHM to all clones. This way one offspring of each parent is subject to a sequence of two mutations, first standard bit mutation and afterwards CHM. Jansen et al. [14] proposed a variant of the BCA that only uses CHM with constant probability \(0<p<1\) (instead of \(p=1\)) and were able to show significantly improved upper bounds on the optimisation time for this algorithm on instances of the vertex cover problem. Considering the individuals that undergo both kinds of mutation (or a (1+1)-style BCA), both these variants fit within the (1+1) HA-chained model of hybridisation (Algorithm 5). More precisely, for \({\varvec{p}}= (p_1, p_2)\) with \(p_1\) the probability to execute SBM and \(p_2\) the probability to execute CHM, we have \(p_1=p_2=1\) for the original BCA and constant \(p_1=1\) and \(0< p_2 < 1\) for the modified BCA in [14]. We remark that the improved results in [14] in fact hold as long as \(p_1 = {\varOmega \left( 1\right) }\).
In the following subsection we will first consider the general hybrid algorithmic framework and then specialise the results to the combination of CHM and SBM.
5.1 The Advantage of Hybridisation
The easiest functions for SBM and CHM are OneMax and MinBlocks, respectively. Before analysing hybrid algorithms using both operators, it is natural to consider the effect of one operator on the easiest function for the other operator. It is well known that the (1+1) CHM needs \(\Theta (n^2 \log n)\) expected time on OneMax [17, 21].
Here we consider the expected optimisation time of the (1+1) EA on MinBlocks and show that it is very inefficient.
Theorem 12
The expected optimisation time of the (1+1) EA with strict or non-strict selection on MinBlocks is at least \(n^n/2\).
The function MinBlocks has the following property: for all search points except for \(0^n\) and \(1^n\), the number of 0-blocks equals the number of 1-blocks. Notice that inverting all bits in a bit string turns all 0-blocks into 1-blocks and vice versa. Hence for all \(x \notin \{0^n, 1^n\}\) we have \(\textsc {MinBlocks} (x) = \textsc {MinBlocks} (\overline{x})\).
Let \(x_0, x_1, \dots \) be the trajectory of the (1+1) EA and \(T \in {\mathbb {N}}_0\) be the first hitting time of a search point in \(\{0^n, 1^n\}\). Since \(x_0, x_1, \dots , x_T\) has the same probability as \(\overline{x_0}, \overline{x_1}, \dots , \overline{x_T}\) and \(x_T \in \{0^n, 1^n\}\), we have \({\text {Pr}\left( x_T = 0^n\right) } = 1/2\). In this case the only accepted search point is the global optimum \(1^n\), for which all bits have to be flipped in one mutation. This has probability \(n^{-n}\) and expected waiting time \(n^n\). Combined with the probability of reaching this state, the expected optimisation time is at least \(T + n^n/2 \ge n^n/2\). \(\square \)
Note that the expected optimisation time of the (1+1) EA on MinBlocks is only by a factor of at most 2 smaller than the expected optimisation time of the (1+1) EA on its hardest function, Trap, which is almost \(n^n\) [9].
Using multiple operators, the hope is that the advantages of each operator are combined. However, this is not always true: new operators can make a hybrid algorithm follow an entirely different search trajectory and lead to drastically increased optimisation times. This behaviour was demonstrated for memetic algorithms [31] as well as for standard bit mutations cycling between different mutation rates [15] and for population based EAs where the mutation rate of each individual depends on its rank in the population [28].
We show that such effects cannot occur when dealing with easiest functions. If f is an easiest function for A which is a (1+1) \(A\), then Theorem 13 stated below allows to transfer an upper bound on the expected optimisation time of A to the four hybrid algorithms.
Theorem 13
Using Theorem 13 as well as Theorem 9, we get the following corollary concerning the SBM and CHM operators.
Corollary 14
Consider the hybrid algorithms (1+1) HA\(({\varvec{p}})\) and (1+1) HA-chained\(({\varvec{p}})\) for \({\varvec{p}}= (p_1, p_2)\) with constant \(p_1, p_2 > 0\), (1+2) HA, and \(2 \times \)(1+1) HA, all based on a (1+1) algorithm \(A_1\) using SBM and another (1+1) algorithm \(A_2\) using CHM; \(A_1\) and \(A_2\) both using strict or non-strict selection. Then the expected optimisation time of all these hybrids on OneMax and MinBlocks is \(O(n \log n)\) and \(O(n^2)\), respectively.
All hybrid algorithms are hence able to combine the advantages of both operators on the two easiest functions for its two operators. This is particularly true for the modified (1+1)-style BCA [14] with constant \(0< p_1, p_2 < 1\) discussed at the beginning of Sect. 5.
5.2 Weighted Combinations of OneMax and MinBlocks
In the previous subsection it was proven that the four different hybrid schemes (1+1) HA\(({\varvec{p}})\) for \({\varvec{p}}= (p_1, p_2)\) with constant \(p_1, p_2 > 0\), (1+1) HA-chained, (1+2) HA, and \(2 \times \)(1+1) HA, using SBM and CHM as operators are all efficient for OneMax and MinBlocks. In particular, even though SBMs alone exhibit very poor performance on MinBlocks, hybrid algorithms using CHM with arbitrarily low constant probability along with SBM are efficient. In this subsection we investigate the performance of the two operators on "hybrid" functions where the fitness depends on both the number of ones and the number of blocks.
We perform 10,000 runs for each of the above pairs of settings for \(n=100\) and depict the average optimisation times in Fig. 2. Figure 2a shows a comparison of the optimisation times of different resulting algorithms (1+1) HA\((p,1-p)\) depending on w while Fig. 2b depicts the results for different functions over the parameter p of the hybrid algorithm. Note, that for the sake of recognisability we only depict a subset of the curves in both figures while on each curve we present all available data points. We additionally perform Wilcoxon signed rank tests to assess whether the observed differences in the optimisation times are statistically significant. Fixing a value for w (Fig. 2a) we perform tests for all pairs of algorithms and the 10,000 optimisation times measured for each setting. Similarly, fixing a value for p (Fig. 2b) we perform tests for all pairs of functions. We perform Holm-Bonferroni correction to account for the large number of tests we execute.
We can see from the experiments in Fig. 2 that the case \(w=0\) is very different from runs with \(w>0\). Recall that \(w=0\) means that the algorithm is confronted with \(\textsc {MinBlocks} \) and that \(w>0\) implies that we add \(w \cdot \text {OneMax} \) to the function value (while, at the same time, reducing the function value by \(w \cdot \textsc {MinBlocks} \)). The addition of \(w \cdot \text {OneMax} \) with an arbitrarily small \(w>0\) introduces a 'search gradient' in the fitness landscape: the algorithm is encouraged to increase the number of 1-bits by a small increase in fitness (at least as long as this trend is not counteracted by an increase in the number of blocks). The effect is most pronounced for \(p=0\), i. e., for the (1+1) EA where the expected optimisation time is exponential for \(\textsc {MinBlocks} \) and becomes manageable as soon as the \(\text {OneMax} \)-component introduces a 'search gradient'. In a different context, the same effect has been discussed and analysed using different example functions [16].
5.3 On the Easiest Function for Hybrid Algorithms
In the previous subsection it was shown experimentally that as soon as a small OneMax component comes into play, the function \(f_w\) becomes easy for the (1+1) HA(p,\(1-p\)) using SBM and CHM independent of the value of p. Nevertheless, in this subsection we show that \(f_w\), for any value of w, is not an easiest function for the (1+1) HA(p,\(1-p\)). In particular, we will set \(p=1/2\) and show that the easiest function for the hybrid algorithm (1+1) HA(1 / 2,1 / 2) using SBM and CHM, and strict selection, is more complex than a mere weighted combination \(f_w\) of the two easiest functions for both operators.
He et al. [10] explain how an easiest function can be computed. We construct \(\textsc {EasiestHybrid}_p\), an easiest fitness function for the hybrid algorithm (1+1) HA(p,\(1-p\)) from Corollary 14 by implementing their construction procedure and performing the necessary computations numerically. Clearly, this is computationally feasible only for small values of n. Using the unique global optimum as a starting point and level \(L_0\) we can compute the next level of search points with next best and equal fitness by computing for each search point which does not yet have a level the expected time needed to reach \(L_0\) either directly or via a mutation to a search point that already has a level. Search points with minimal time in this round make up the next level. Note that this is actually Dijkstra's algorithm for computing shortest paths (see, e. g., [3]). Also note that the numerical computation of the actual expected waiting times is easy since the exact transition probabilities for mutations leading from one bit string to another are all known and waiting times are all simply geometrically distributed.
Fitness level landscape of the \(\textsc {EasiestHybrid}_{1/2}\) function for \(n=6\)
\(\mathbf {P_{L_7}}\)
\(\mathbf {P_{L_8}}\)
|1-blocks| = 0, \(x=\{1^n\}\)
|1-blocks| = 0, \(x=\{0^n\}\)
|1-blocks| = 1; length = {5}
\( 1/2 \cdot (\mathbf {3} P_{\mathrm {CHM}} + 2 P_{\mathrm {SBM}_1} + 4 P_{\mathrm {SBM}_3})\)
\( 1/2 \cdot (\mathbf {2} P_{\mathrm {CHM}} + 2 P_{\mathrm {SBM}_1} + 4 P_{\mathrm {SBM}_3})\)
|0-blocks| = 1; length = {5}
\( 1/2 \cdot (\mathbf {3} P_{\mathrm {CHM}} + 4 P_{\mathrm {SBM}_3} + 2 P_{\mathrm {SBM}_5})\)
\( 1/2 \cdot (\mathbf {2} P_{\mathrm {CHM}} + 4 P_{\mathrm {SBM}_3} + 2 P_{\mathrm {SBM}_5})\)
|1-blocks| = 1; length = {4}
|0-blocks| = 1; length = {4}
|1-blocks| = 1; length = {3}
\( 1/2 \cdot (\mathbf {2} P_{\mathrm {CHM}} + \mathbf {1} P_{\mathrm {SBM}_1} + 4 P_{\mathrm {SBM}_3} + 1 P_{\mathrm {SBM}_5})\)
\( 1/2 \cdot (\mathbf {4} P_{\mathrm {CHM}} + 6 P_{\mathrm {SBM}_3})\)
|1-blocks| = 2; length = {3,1}
|1-blocks| = 2; length = {2,2}
|0-blocks| = 2; length = {3,1}
|1-blocks| = 2; length = {2,1}
|0-blocks| = 2; length = {2,2}
|1-blocks| = 3; length = {1,1,1}
We consider Algorithm (1+1) HA(p,\(1-p\)) which at each step executes either an SBM or a CHM with probability \(p=1/2\) and report in Table 1 the different fitness levels \(L_0,\dots , L_{12}\) of \(\textsc {EasiestHybrid}_{1/2}\) when \(n=6\). From the table it can be noticed that both levels \(L_7\) and \(L_8\) contain bit strings with two 1-blocks and four 1-bits. However, the lengths of the two 1-blocks are different (i. e., three and one for \(L_7\) and two and two for \(L_8\)). The reason why levels \(L_7\) and \(L_8\) are distinct is highlighted in the last two columns of Table 1, which report the probabilities of reaching levels of higher fitness, respectively from levels \(L_7\) and \(L_8\). For any point of the search space belonging to level \(L_7\) there are three different CHMs leading to points in \(L_2\) and \(L_3\) while there are only two CHMs from points in \(L_8\). These extra CHMs lead to higher transition probabilities, hence lower runtimes, to reach levels \(L_2\) and \(L_3\) from \(L_7\), compared to \(L_8\). Also the probability of reaching level \(L_6\) is higher from \(L_7\), mainly because the level may be reached by flipping only one bit while at least three bits need to be flipped for bit strings in \(L_8\). Since the transition probabilities to the remaining levels of higher fitness are the same, the expected runtime from \(L_7\) is lower than that from \(L_8\), explaining why the two levels are distinct (recall that, by construction, the levels are ordered according to increasing expected runtimes).
Concerning the number of different fitness levels, these increase as the problem size n increases. Already for \(n=10\) \(\textsc {EasiestHybrid}_{1/2}\) has 78 different levels, more than the product of the number of OneMax and MinBlocks levels for \(n=10\) (i. e. 66). It is indeed the increase in number of fitness levels as n grows that makes it hard to give a precise definition of the \(\textsc {EasiestHybrid}_{1/2}\) function. We leave this as an open problem for future work.
6 Conclusions
We have extended the analysis of easiest function classes from standard bit mutations to the contiguous somatic hypermutation (CHM) operator used in artificial immune systems. Albeit the recent advances in their theoretical foundations [21, 29, 35] no such results were available concerning artificial immune system operators. With the runtime and fixed budget analyses of the (1+1) CHM on MinBlocks, the corresponding easiest function, we established a lower bound on the (1+1) CHM 's performance on any function. We also showed that MinBlocks is exponentially hard for the standard (1+1) EA, complementing the known result that the (1+1) CHM performs asymptotically worse by a factor of \({\Theta \left( n\right) }\) compared to the (1+1) EA on OneMax. Furthermore, we proved that several hybrid algorithms combining the (1+1) CHM and the (1+1) EA solve both MinBlocks and OneMax only at a constant factor slower than the pure algorithms.
Experimental work revealed that a fitness function consisting of a weighted combination of MinBlocks and OneMax is easy to optimise for both pure operators and hybrid variants even when the OneMax weight component is very small. Nevertheless, after providing the exact fitness landscape of the easiest function for the (1+1) HA(1 / 2,1 / 2), \(\textsc {EasiestHybrid}_{1/2}\) for small instance sizes, we observed that its structure is more complex than a simple weighted combination of OneMax and MinBlocks. We leave constructing and analysing easiest functions for other operators that fit the (1+1) \(A\) scheme for future work. Similarly, the question about the easiest functions for different schemes of hybridisation remains open.
- 1.
In drift analysis, a drift function is a non-negative function d(x) such that \(d(x)=0\) for an optimal point and \(d(x)\ge 0\) for a non-optimal point. The expected drift at point x is \({\text {E}\left( d(x) - d(y)\right) }\), where y is the next iteration's search point. The expected drift is denoted by \(\varDelta _f(x)\) when maximising f(x), or \(\varDelta (x)\) in short.
- 2.
Hardest functions are formally only defined for functions and algorithms with finite expected optimisation times. However, we regard functions leading to infinite expected times as being harder than those with finite expected times.
The research leading to these results has received funding from the European Union Seventh Framework Programme (FP7/2007-2013) under Grant Agreement No. 618091 (SAGE) and by the EPSRC under Grant Agreement No. EP/M004252/1. We thank the anonymous reviewers of this manuscript and the previous GECCO 2015 paper for their very useful and constructive comments.
- 1.Alanazi, F., Lehre, P.K.: Runtime analysis of selection hyper-heuristics with classical learning mechanisms. In: Proceedings of the IEEE Congress on Evolutionary Computation (CEC 2014), pp. 2515–2523. IEEE (2014)Google Scholar
- 2.Clark, E., Hone, A., Timmis, J.: A markov chain model of the B-cell algorithm. In: Proceedings of the International Conference on Artificial Immune Systems (ICARIS 2005), LNCS 3627, pp. 318–330. Springer (2005)Google Scholar
- 3.Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms. MIT Press, London (2001)zbMATHGoogle Scholar
- 4.Corus, D., He, J., Jansen, T., Oliveto, P.S., Sudholt, D., Zarges, C.: On easiest functions for somatic contiguous hypermutations and standard bit mutations. In: Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2015), pp. 1399–1406. ACM (2015)Google Scholar
- 5.Cowling, P., Kendall, G., Soubeiga, E.: A hyperheuristic approach to scheduling a sales summit. In: Burke, E., Erben, W. (eds) Proceedings of the Third International Conference on Practice and Theory of Automated Timetabling (PATAT 2000), pp. 176–190. Springer (2001)Google Scholar
- 6.de Castro, L.N., Timmis, J.: Artificial Immune Systems: A New Computational Intelligence Approach. Springer, Berlin (2002)zbMATHGoogle Scholar
- 7.Doerr, B., Doerr, C., Ebel, F.: From black-box complexity to designing new genetic algorithms. Theoret. Comput. Sci. 567, 87–104 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
- 8.Doerr, B., Johannsen, D., Winzen, C.: Drift analysis and linear functions revisited. In: Proceedings of the IEEE Congress on Evolutionary Computation (CEC 2010), pp. 1967–1974 (2010)Google Scholar
- 9.Droste, S., Jansen, T., Wegener, I.: On the analysis of the (1+1) evolutionary algorithm. Theoret. Comput. Sci. 276, 51–81 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
- 10.He, J., Chen, T., Yao, X.: On the easiest and hardest fitness functions. IEEE Trans. Evol. Comput. 19(2), 295–305 (2015)CrossRefGoogle Scholar
- 11.He, J., Yao, X.: A study of drift analysis for estimating computation time of evolutionary algorithms. Nat. Comput. 3(1), 21–35 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
- 12.Jansen, T.: Analyzing Evolutionary Algorithms: The Computer Science Perspective. Springer, Berlin (2013)CrossRefzbMATHGoogle Scholar
- 13.Jansen, T., De Jong, K.A., Wegener, I.: On the choice of the offspring population size in evolutionary algorithms. Evol. Comput. 13(4), 413–440 (2005)CrossRefGoogle Scholar
- 14.Jansen, T., Oliveto, P.S., Zarges, C.: On the analysis of the immune-inspired B-cell algorithm for the vertex cover problem. In: Proceedings of the International Conference on Artificial Immune Systems (ICARIS 2011), LNCS 6825, pp. 117–131. Springer (2011)Google Scholar
- 15.Jansen, T., Wegener, I.: On the choice of the mutation probability for the (1+1) EA. In: Proceedings of the 6th International Conference on Parallel Problem Solving from Nature (PPSN 2000), LNCS 1917, pp. 89–98. Springer (2000)Google Scholar
- 16.Jansen, T., Wiegand, R.P.: The cooperative coevolutionary (1+1) EA. Evol. Comput. 12(4), 405–434 (2004)CrossRefGoogle Scholar
- 17.Jansen, T., Zarges, C.: Analyzing different variants of immune inspired somatic contiguous hypermutations. Theoret. Comput. Sci. 412(6), 517–533 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
- 18.Jansen, T., Zarges, C.: Computing longest common subsequences with the B-cell algorithm. In: Proceedings of the International Conference on Artificial Immune Systems (ICARIS 2012), LNCS 7597, pp. 111–124. Springer (2012)Google Scholar
- 19.Jansen, T., Zarges, C.: Fixed budget computations: a different perspective on run time analysis. In: Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2012), pp. 1325–1332. ACM (2012)Google Scholar
- 20.Jansen, T., Zarges, C.: Performance analysis of randomised search heuristics operating with a fixed budget. Theoret. Comput. Sci. 545, 39–58 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
- 21.Jansen, T., Zarges, C.: Reevaluating immune-inspired hypermutations using the fixed budget perspective. IEEE Trans. Evol. Comput. 18(5), 674–688 (2014)CrossRefGoogle Scholar
- 22.Jansen, T., Zarges, C.: Analysis of randomised search heuristics for dynamic optimisation. Evol. Computation. 23(4), 513–541 (2015)CrossRefGoogle Scholar
- 23.Kelsey, J., Timmis, J.: Immune inspired somatic contiguous hypermutations for function optimisation. In: Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2003). LNCS 2723, pp. 207–218. Springer (2003)Google Scholar
- 24.Lehre, P.K., Özcan, E.: A runtime analysis of simple hyper-heuristics: to mix or not to mix operators. In: Proceedings of the Twelfth workshop on Foundations of Genetic Algorithms (FOGA 2013), pp. 97–104. ACM (2013)Google Scholar
- 25.Lehre, P.K., Witt, C.: General drift analysis with tail bounds. CoRR, abs/1307.2559 (2013)Google Scholar
- 26.Lengler, J., Spooner, N.: Fixed budget performance of the (1+1) EA on linear functions. In: Proceedings of the 2015 ACM Conference on Foundations of Genetic Algorithms (FOGA 2015), pp. 52–61 (2015)Google Scholar
- 27.Neri, F., Cotta, C., Moscato, P. (eds.): Handbook of Memetic Algorithms. Springer, Berlin (2013)Google Scholar
- 28.Oliveto, P.S., Lehre, P.K., Neumann, F.: Theoretical analysis of rank-based mutation—combining exploration and exploitation. In: Proceedings of the IEEE Congress on Evolutionary Computation (CEC 2009), pp. 1455–1462 (2009)Google Scholar
- 29.Oliveto, P.S., Sudholt, D.: On the runtime analysis of stochastic ageing mechanisms. In: Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2014), pp. 113–120. ACM (2014)Google Scholar
- 30.Ross, P.: Hyper-heuristics. In: Burke, E.K., Kendall, G. (eds.) Search Methodologies, pp. 611–638. Springer, Berlin (2014)CrossRefGoogle Scholar
- 31.Sudholt, D.: The impact of parametrization in memetic evolutionary algorithms. Theoret. Comput. Sci. 410(26), 2511–2528 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
- 32.Sudholt, D.: Hybridizing evolutionary algorithms with variable-depth search to overcome local optima. Algorithmica 59(3), 343–368 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
- 33.Sudholt, D.: A new method for lower bounds on the running time of evolutionary algorithms. IEEE Trans. Evol. Comput. 17(3), 418–435 (2013)CrossRefGoogle Scholar
- 34.Sudholt, D., Zarges, C.: Analysis of an iterated local search algorithm for vertex coloring. In: Proceedings of the 21st International Symposium on Algorithms and Computation (ISAAC 2010), LNCS 6506, pp. 340–352. Springer (2010)Google Scholar
- 35.Timmis, J., Hone, A., Stibor, T., Clark, E.: Theoretical advances in artificial immune systems. Theoret. Comput. Sci. 403(1), 11–32 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
- 36.Wegener, J., Baresel, A., Sthamer, H.: Evolutionary test environment for automatic structural testing. Inf. Softw. Technol. 43(14) 841–854 (2001)Google Scholar
- 37.Witt, C.: Tight bounds on the optimization time of a randomized search heuristic on linear functions. Comb. Probab. Comput. 22(2), 294–318 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
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text:
I stayed up way too late last night watching "Cliffhanger" with Hubster.
I always do this to him, I surf around the channels until I find something that will hold my ADD brain in place for a few minutes, just long enough to get Hubster sucked in as well, and then I announce I'm going to bed. This drives him nuts.
"How can you not want to know what happens," he always asks?
Last night, he persuaded me to stay and try to watch the film through to the end. He did this by basically sitting on me and Whoopsie.
I have to say, that, overall, watching Sylvester Stallone as a rock climbing guide could have been a hell of a lot worse. I would love to have been able to nail a toe hold like he did with those steel enforced boots, but it's a movie and I'll let that slide. What saved it for me this time was the dubbing.
Normally, I can not stand it when English or American movies are dubbed over in French. Not because I don't like hearing French or following the translation, it's that the dubbing sometimes does not match what I know to be the true voice of the actor, and hence, really screws up my whole opinion of a movie. For example, Harrison Ford dubed over in "Six Days, Seven Nights" sounds like a loopy frat boy who sidelined with the Chipmunks in an all boys choir rather then the honey voiced, weather beaten drunk he played in English.
Watching "Dirty Harry" dubbed over nearly kills me too. Clint Eastwood does not sound like that! No, not at all! And Julia Roberts in "Erin Brockovich," same thing. Even cartoons can send me over the edge with this change in tonality. "Noddy" (aka "Oui-Oui") just sounds way too chipper in French!
That being said, Sly was saved last night. The French voice used for him actually made him sound intelligent, sensitive, coherent, and oh my gosh, human rather than a bumbling hulk of a man trying to speak through cotton balls shoved in his mouth. It almost seemed like Sly could act!
In fact, the dubbing was so good, I actually got into the movie and stayed squished under Hubster until the very end. Seeing John Lithgow playing a psycho path is always nice right before bed and I thank the French voice of Mr. Lithgow for really helping me appreciate the nuances of well spoken French cuss words.
In that light, dubbing does have it's good points. Heck, since I dropped out of my French classes last year, I gotta keep up with my vocabulary somehow, don't I?

subset: commoncrawl


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5. India Mart: It is same as Yellow pages here you can also list your business at Free and in very less time, India Mart also recieves around million of searches in one day, and it is in competition with other listing sites.
6. Sulekha: Sulekha is also the online business listing site with great UI, It receives basically thousanbds of searches in one day. The registeration process is quite easy and is also of FREE of cost.
7. Asklaila : Asklaila comes into the competition in the last 3 months only, Asklaila recieves thousands of Searches per day, here they give great great lead generation process. The registeration process is of FREE of cost and is quite easy to use.
8. Indiacom : Indiacom is alsoone of the online business listing site where it helps you in generating business through online. It receives basically thousands of searches in one day. The registeration process is quite easy and is also of FREE of cost.
9. Yalwa: Yalwa is also the online business listing site with great features, It also receives thousands of searches in one day, here the registeration process is quite easy and is also of FREE of cost.
10. Indianyellowpages: Indianyellowpages is also the online business listing site with good PR and Alexa rank here you can list your business online and with that you can generate business also but that is a paid process.
Below is the Top 30 Business listing sites:
1. https://www.yellowbot.com/
2. http://www.salespider.com/
3. http://biz.prlog.org/
4. http://searchrunners.com/FreeListing
5. http://biz.burrp.com/
6. http://www.getit.in/
7. http://yellowpages.webindia123.com
8. http://www.asklaila.com/
9. http://www.lacartes.com
10. http://www.hotfrog.in/AddYourBusiness.aspx
11. http://resources.afaqs.com/addupdate.html?add=1
12. http://www.indiacom.com/yellowpage/addyourlist.asp
13. http://www.yalwa.in/
14. http://www.whereincity.com/
15. http://www.startlocal.in/cgi-bin/directory/add.cgi
16. http://www.eindiabusiness.com/login/sign-up.html
17. https://admin.cylex.in/firma_default.aspx?step=0&d=cylex.in
18. http://businesswithindia.in/free_listing.php
19. http://directory.ac/business-listing.html
20. http://www.eximdata.com/add_company.php
21. http://wowcity.com/home.php?location=Delhi,%20Delhi
22. http://www.bollywoodshaadis.com/signup?cpurl=add-business
23. http://www.vcsdata.com/bizlisting.html
24. http://www.hudku.com/
25. http://www.dealerdice.com/vendors/register
26. http://www.gidonline.com/Home
27. http://exportersindia.net/
28. http://www.businesslistdirectory.com/get-listed-free
29. http://www.grotal.com/UserAccount/UserLogin.aspx?Log=AddBsns
30. http://www.jantareview.com/Noida/add-a-new-business
31. http://zatse.in/provider/addprovider
32. http://www.infoline.in/freelisting.aspx
37. http://www.mysheriff.co.in/users/
38. http://addyp.com/
39. http://www.onepageprofile.com/reg.htm
40. http://www.gurgaononcall.com/AddFreeList.aspx
41. http://www.onepageprofile.com/reg.htm
42. http://www.gurgaononcall.com/AddFreeList.aspx
43. http://searchrunners.com/FreeListing
44. http://www.greaternoidadirectory.com/wp-login.php?redirect_to=%2Fsubmit%2F
45. http://merchants.clearindia.com/
46. http://yellowpages.goodlinksindia.com/add-business.php
47. http://www.indisearch.com/register
48. http://www.yellowpages99.com/add_your_business.html
49. http://www.ncrpages.in/reg-buss.html
50. http://www.attamarketonline.com/registration.php


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It is a great benefit to have the very real sales force that is my fellow chapter members – a sales force that knows me, my product, my values and desires, and actively looks for business on my behalf.
BNI has a professional format which works, irrespective of whether you are an experienced networker or not. BNI will teach you the fundamentals of networking to help you build a better business, including how to win more business and how to build your business.
If you want bigger, better clients, then BNI is exactly what you've been looking for. BNI will help you think on your feet and boost your business confidence. It will improve your public speaking skills and help you define and articulate your business's real competitive advantage.
Over time, you will develop an incredible network of professional contacts, many of whom will become your good friends. You will gain great practical advice and experience from the members within your BNI group and apply it to make your business more effective and more successful.
- Anthony Brindley
The best BNI that I have been to!Paul Green
There's a good mix of people, which are always welcoming.
From a business point, the amount of business that moves around the table is really good.Shital Odedra
Welcome to BNI Diamond
Business networking in Milton Keynes.


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Mortar round-auto fin deploy
auto deploying fins on a 50mm mortar round. a rubber band activates the fins after the round exits the barrel.
Last edited by Spuddin on Wed Aug 08, 2007 6:15 pm, edited 1 time in total.
SWEET!! I've always wanted to do something like this! Great paint job, and nice video, too.
field-legal paintball semi
that looks like a really good idea. maybe a lil time consuming for ammo though, but if it works, then its worth it.
After waching the video more closely, does the rocket start to tumble at the peak of the flight?
Would it work better with bigger fins? They look like they are only a half inch or so. If that was increased to an inch or more would they work better, or would that be to much weight in back?
your right, it did tumble because not enough weight in the nose. i had this same problem with another shell this size, i added bondo in the nose and it flew straight. i gotta add some weight to this one for sure.
Make some jigs for the parts and it wouldn't take as long to make them. Knock out 20 or so of each thing at a time and you'd be in business.
I'm weird, I know it, you don't need to tell me.
Woah, I just realized it's August already! Frickin' summer, I never know what day it is or what the date is. Oh well.
Back on topic: Nice job, but aren't those fins a bit skinny to help as much as they could?
Looks good, but like you said needs more weight on the front.And another set of fins wouldn't be bad.
"Some people wonder all their lives if they've made a difference. The Marines don't have that problem"-Ronald Reagan
"KA BLAAM! Elimination! Lack of education."-Big Mama, Fox and the Hound, Disney
i think you might benefit from some fatter fins. And it would deploy faster if the rubber band was fastened to the end of the fin. good job though
searching for a modern day savior from another place,inclined toward charity,everyone's begging for an answer,without regard to validity,the searching never ends,it goes on and on for eternity
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In July I made Cynthia's first dress, a second long sleeve and a petticoat.
Both of the dresses were made of the same fabric that I had stashed for a long time. Last night while cleaning out my sewing room for memorial weekend, I thought to try them back on Cynthia. And alas they fit!


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"If you walk the footsteps of a stranger,
you'll learn things you never knew you never knew"
Disney's Pocahontas
Two aspects of my life, as a painter and humanitarian, will merge this July as I share my paintings and raise funds for my charity at a solo exhibition and fundraiser at the Gallery Underground in Arlington, VA. I choose the name Colors of the World, a take on the Disney song from the movie Pocahontas, Colors of the Wind, because, in my opinion, the song and my art resonate with the concept of empathy....... "We are all connected to each other, in a circle, in a hoop that never ends" When traveling, I am fascinated by the unique colors and sense of energy that permeate a region. However, I am not drawn to paint the monuments or tourist attractions, but rather the snapshots of beauty that can be found in moments of every day life - moments that stir universal emotions. Every painting tells a story, a place I visited, a person I met, a life that touched mine. These people have brightened my life, expanded my world and I try to give back by helping their communities though the charity Painting Brighter Futures. Proceeds from each sale will go to Painting Brighter Futures projects to support groups on the ground helping children in Ghana , Kenya and here in the US. You can learn more on the website www.PaintingBrighterfutures.com.
Hope to see you there.
Reception July 10 from 5-8pm.
Hope to see you there.
Reception July 10 from 5-8pm.
June 29 – July 31, 2015
Gallery Underground, Arlington VA
Opening Reception July 10, 6-8pm


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At The Table Subscription
At the Table is Vanishing Inc.'s twice-monthly online lecture service. We invite some of the top lecturers in the industry to perform and lecture, and you watch live from the comfort of your home. Once it is over you are given a downloadable file and you can enjoy it again and again.
Subscribe here to receive access to this month's two lectures. Your subscription will then automatically renew each month (you can cancel at any time). If you sign up during September, you won't be charged this month, but if you don't cancel the subscription by the first of next month you'll be charged $9.95 for next month's lectures.
First up this month, a rising star in magic, Josh Janousky has earned a reputation for sharing material that is both extremely clever and highly workable. Josh's magic can be found in the working repertoires of many industry legends and has been tested in real world situations thousands of times! At just 27 years old, he has taken the world of magic by storm and shows no signs of slowing down!
Despite his very young age, Hyoungmin Kim has proven himself at various magic conventions worldwide. His light-hearted performance style is always entertaining and incredibly refreshing to watch! In this lecture, you will experience effects that he has perfected from his years of competing and performing, along with new techniques he has developed to share for the first time!
Join us as we welcome Josh Janousky and Hyoungmin Kim for their amazing lectures, At The Table!
So far, my 'At the Table' subscription has been more than value for money! If there was a 6 star rating, it would deserve that. The variety of material covered has been particularly satisfying. The Daniel Garcia lecture has been my highlight. I look forward to what's in store over the coming lectures.Quentin Green
I mean really. Where else can you spend over two hours up close with some of the top professionals in the business, AND learn some first rate effects for basically $12.50 per show? It feels like a personal conversation with the performer and I have enjoyed getting to know the person as well as learning some of the illusions they perform. It has changed my impression (in a good way) of several performers. There are thing to take away from every program.I am a subscriber and I hope you will become one too. Very worthwhile.
Unbeatable value for money.One caveat: If you don't watch it live - be prepared to wait 24 hours or more for the download.Other than that, it's great!
This is probably the best investment I've made in the last year. The quality does vary but that is probably just my taste. of all of the lectures I have watched (well over 20), there have only been a smattering of sub-par ones. Even if you only watch one of the lectures a month, it's well worth it. Plus, they'll stay in your account area so you can watch them at any time anyway.
We ship to you directly from within the UK. While most other UK magic shops hide what happens behind the scenes, we want you to know exactly how things work. Here's the timeline:
Typically takes 2 - 3 days
We send UK orders in batch to the UK from our USA warehouse every day. This way you avoid expensive international shipping fees and taxes.
We keep our most popular products in stock in the UK so often this step is skipped.
We send packages out via MyHermes the same day that we receive them. This is a popular, secure, trackable courier service.
Typically takes 2 days
As soon as the package is dispatched you'll receive a link to track it on this page. While they don't offer any guarantees, MyHermes tend to deliver your order within two days.
At Vanishing Inc. Magic shop we're proud to produce great magic books, tricks, downloads and DVDs.
Every item is carefully selected under the watchful eye of cofounders, and professional magicians Joshua Jay and Andi Gladwin. We don't stock every magic trick released, but instead hand pick the best!
We accept:
Prices in USD. Switch to UK site
© Copyright Vanishing Inc. Magic. shop 2019. Privacy policy


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13 SharesShareTweetSharePinThat's exactly where TopMate 20000mah Huge Capacity Power Bank, Three Output Ports(Two 2.1A One 1A), Dual LED Lights is going. It's a mean, badass machine that wants to make sure you're always juiced. So now that you know what kind of monsters I like, let's see how well it does in real life. Sure, it's got 3-ports, however, can …Read More »
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text: Guitarist Victor Samalot is best known for his work with the local prog rock outfit Rare Blend but he also plays jazz/rock/pop as a solo act. This particular song has some twang to it, as well, and comes off as it's the song to a old-time Western movie. The intricate guitar work allows Samalot to show off his six-string skills and the song's keyboard flourishes help give it that soundtrack vibe.


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Inventory # AOL0914-E12386-001
Joseph Rosenthal
OSA PDCC RCA 1921 - 2018 Canadian
Mother & Child
oil on board
signed and on verso signed and titled
30 x 24 in 76.2 x 61cm
Private Collection, Toronto
All prices are in Canadian Dollars.
Although great care has been taken to ensure the accuracy of the information posted, errors and omissions may occur. All bids are subject to our Terms and Conditions of Business.


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Introduce Black History to the primary curriculum
Responsible department: Department for Education
To make Black History part of the school curriculum for Primary education. It helps to enrich the development of the spiritual, moral, social and cultural needs of all ethnic groups.
We believe Black History should be made a mandatory element of the national curriculum.
I support this petition but have a question…
Who will be responsible for teaching black history in the schools?
It is important that they hire teachers who are knowledgeable and well versed in the subject matter. It is also important that the curriculum isn't just solely focused on the transatlantic slave trade, black America's history, black entertainers and black sports personalities.
CLICK HERE to sign the petition.
Categories: UK News


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model: allison browning
22 january 2007
beauty eden peach portrait temptation woman
I love the framing and colour of this work….is there more to the series?? Would be cool to see more!
this really evokes some emotion and aptly titled, great composition and colours.well done :- )
@jessica: sorry for the extremely late reply. there are more images from this shoot, but most of them are on my flickr, not on here: http://flickr.com/photos/bellybuttongirl/sets/7...
@tony: thank you.
Loading more work by Bronwen Hyde...
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July 30, 2010
Foggy, With a (110%) Chance of Nostalgia
end finally came this June. Even after all these years of anticipation, it seemed unreal to see the school stripped of its sign. [Edit: I've since learned the school district may not have been responsible for this removal. Vandals!].
Speaking of the (missing) sign, you can see it in this old photograph (below) I came across while sorting through a box of treasures from my dear late friend and mentor, Jean Hill, a talented artist and life-long resident of Wooden Valley. He taught art at the school for many years.
Perhaps it was these BB guns which were responsible for the dimples in the ornament at the top of the flag pole (below).
Posted by The Monkey Flower Group at 8:29 AM
2 comments:
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You are here: Fairness.com > Resources > More Workers Face Pay Cuts, Not Fur...
More Workers Face Pay Cuts, Not Furloughs
- Date: August 3, 2010
- QUOTE: Local and state governments, as well as some companies, are squeezing their employees to work the same amount for less money in cost-saving measures that are often described as a last-ditch effort to avoid layoffs.
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This article includes some ideas for the young teacher about what advice to offer students in order to get them started on composing an exposition. Writing a pep talk remains in many means identical.
The suggestions below are not in any kind of purchase of importance. You will definitely have to choose which ones you may make use of along with your class. You might need to have to simplify the advised suggestions.
Each concept could require you to show a lesson on it with you really displaying effective ways to tackle that, e.g. the concept of 'beyond, found and future'.
Recommended Techniques/Approaches
1. Brainstorm the label from the essay or even speech. Only make a note of any idea that enters into your thoughts no matter how absurd or even unnecessary you believe this might be.
2. Use the following basic team of tips to aid offer ideas.o Exactly how, when, where, why and/or whato When - previously, currently, soono When - previous, found, futureo Family, connections, friends, friends, strangerso Local, local, state/provincial, nationwide, internationalo Home, institution, outside university
3. Use Google.com to investigate your subject matter, when possible.
4. Create a simple plan/plot from the records you have acquired - a synopsis.
5. Plan and write a draft from your intro.
6. Plan as well as write a draft from your closure.
7. Ensure there is a link in between the start and the ending.
8. Put one idea/part of the account right into each paragraph.
9. Write all the very first paragraphes from each paragraph to make sure the 'story' resides in the correct series.
10. Broaden each very first sentence in to a paragraph to fill in the 'account' as you go.
11. When writing your initial draft, leave a blank line after each composed line to aid the modifying procedure. The teacher should illustrate exactly what is indicated listed here for much younger students.
12. Utilize the language as well as terminology of the topic discipline from the topic.
THIRTEEN. Vary the technique you start each sentence to develop interest.
14. The 1st draft ought to be recorded straightforward foreign language as well as words to obtain your 'information' all over. In the editing procedure, include words that provide added significance, focus and also emotional state.
15. It is necessary to read your final draught aloud to your own self to make sure that this helps make sense and obtains the 'information' throughout perfectly.
Additional Suggestions For Pep Talk Writing
16. As a speech, it is actually necessary to note that most sound speakers deliver their speech at one hundred words a minute. That will definitely calculate the duration from the speech according to the time you have accessible.
17. In a speech, it is crucial not to feature way too many concepts. That is much better to possess, mention 3 ideas for your concepts and also assist those tips firmly.
18. Present your simple motif along with a tough overview and also enhance your motif in your end.
19. Time your speech to observe if it matches the moment enabled. In a competition pep talk, it is actually far better to be a little quick on the opportunity than to become very long.
TWENTY. In a competition speech, there will definitely be actually an alert alarm therefore know where in your pep talk you need to go to then. Then readjust the rate from your pep talk or even leave out component of that to complete on time.
Once you have actually reviewed all these pointers, you will definitely know that not all will agree with for more youthful trainees. You will should undergo the tips to find those that match the age, experience as well as capability from the students you teach.
essays online With many of the tips, you will definitely should illustrate to your course just how to utilize all of them. Along with older trainees, you could possibly decide on an essay or speech subject and also partner with the course on an approach to develop a draft plan. The pupils can carry on through to the last composing stage.
If you show your trainees these strategies, then they may never truly claim to you, "I do not know how you can start."
As a pep talk, this is actually crucial to keep in mind that many speakers provide their pep talk at one hundred words a moment. In a speech, this is actually vital certainly not to include too numerous ideas. Time your pep talk to see if it matches right into the opportunity allowed. In a competitors speech, there are going to be actually a precaution bell therefore understand where in your speech you must be actually at that time. Along with older pupils, you can decide on an essay or even pep talk subject matter as well as job with the lesson on a tactic to develop a draft strategy.
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Cheaper GeForce 9500 GT not due until Q4By Justin Mann
Nvidia's GeForce 9500 GT units based upon a 55nm process won't arrive until 4Q 2008. Other cards are already being manufactured with a 55nm process, but Nvidia is not yet ready to bring the 9500 GT to market with the new process. It's supposed to be a replacement for the low end of the 8000 series cards, particularly the 8600 GT, and will be an "entry-level" offering.
The reduction in process comes with a reduction in price, with a retail price estimated to be as low as $65. AMD's CPU half may still be struggling, but the recent slew of GPUs coming out from them has given Nvidia quite a bit of shock, and may be in part responsible for the low cost of Nvidia's latest offerings.


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Cancellations, Refunds, and Credits
After School Program – A four week notice is required for after school program changes (withdrawals, reduction in weekly attendance schedule). All requests for cancellations or changes must be made in writing or by email.
Summer Camps – Request for registration cancellations for summer camps must be made in writing or by email. Mindframe Education will refund program fees less a $95 administrative fee per session for cancellations received before June 1st. There are no refunds after June 1. A family may elect to receive a credit instead of a refund before or after June 1. This credit will be in the full amount of program fees paid, and may be used within the next two years for Mindframe Education for any immediate family member.
Late Pickup Fees
Mindframe Education understands that an unexpected event may occur which results in late pickups. Please call us if you are running late. In the event of a late pickup, past the 15 minute grace period, your account will be charged $10 .
Mindframe Education does not discriminate on the basis of gender, race/ethnicity, sexual orientation, physical ability, or cultural or religious background.
Behavior Code
1. Keep hands, feet, and objects to yourself. 2. No running in the classroom or the school building. 3. Eat only in designated areas at designated times. 4. No cursing or swearing. 5. Follow teachers' instructions. Students who fail to meet these expectations after several verbal reminders will be removed from class and their parents notified. In the unlikely event that a student's behavior represents a safety issue, Mindframe Education will remove the student from the program and there will be no refund of program fees.
Authorization to Obtain Emergency Medical Care
By enrolling in our program, you authorize Mindframe Education to obtain emergency medical care if any emergency occurs when the parent/guardian(s) cannot be located immediately.
Medication Policy
Prescription and non-prescription medications are not permitted with the exception of epipens and inhalers.
Act of God
In the event that classes are delayed or end early due to an act of God or civil authority or other reason beyond the control of Mindframe Education, no adjustment of tuition will be made.
Student Attendance
Mindframe Education Summer and After School Programs operate as exempt from Child Day Program Licensure under §63.2-1715.2 of the Code of Virginia. We require a parent or legal guardian to sign-in and sign-out their children when dropping off and picking up from our facility. Students with a valid driver's license may drive themselves to the facility, and sign themselves in / out. For any outdoor / off-site activities, a Mindframe camp supervisor and/or instructor will accompany and supervise the students at all times. While classes are in session, students are not allowed to leave the facility unless accompanied by a Mindframe Supervisor or instructor. Section §63.2-1715.2 of the Code of Virginia provides for flexibility for meeting the needs of "children with disabilities". Please contact us if you believe your child has a disability which requires you to sign him or her in and out of the program.
Photography and Video Release
By enrolling in our program, you allow Mindframe Education to take photographs and video of your child, which may be used in our catalog, website, and other promotional materials. If you do not want us to photograph or take video of your child, please send a written request via email to firstname.lastname@example.com before attendance begins with the name of your child.


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Have you ever seen Kudzu growing on the side of a hill? I grew up in the Mountains of Eastern Kentucky and you know Kudzu, a vine of asian origin, is amazing it grows, grows and covers mountains and whatever is in its way. I haven't thought about kudzu in forever but yesterday it came to mind.
You see, this time last year I had a great idea. I was going to meet with the CBA and develop a program to help attorneys in trouble. It was a simple idea and my goals were very basic. Get the information out there, educate others about the issues and keep lawyers from self destructing. It was a no brainer and an unspoken truth – lawyers are unhappy people. Many people acknowledge this fact, but no one had ever taken the next step to do anything about the issue.
So, what started out as a simple idea just keeps growing, growing and now has momentum all its own. You guessed it, just like Kudzu. Now, the issue is what should be done and when. Do we just provide information and articles, do we set up a support group or groups for lawyers now or wait and what about law students? Dealing with so many competing things and so many personalities is like herding cats as a colleague and friend told me recently. Perhaps, the idea was good, but requires some rethinking or at least an organizational structure. After all, having the idea is the first step in creating something, what comes after that is work.
My main issue is inclusion, for others it is to find the solution. I want to give people a way to reach their own resolution and for the to be feel safe sharing what has happened in their life. Others don't seem to be interested in this and just want to develop a "fix" to the problem. You know lets all live a balanced life. How do you the two types of solutions/actions. Is it enough that a person's your heart and ideals are in the right place even if their fixer mentality means that they wish t0 dictate solutions to people. And what about the fact that the solution based approach ignores the human suffering that is behind the issue. Saying you don't want to hear someone's problems is not acceptable. There must be room at the table for those who have suffered from depression, stress induced mental and physical illness. Isn't this the purpose of the project?
Yesterday, I found myself defending the rights of others to tell their story and the need to have those stories be a part of the solution. The fixers among us fail to see the value in this and frankly, don't feel it is worth their time to hear. I can't seem to shake my frustration and anger at the idea that members of a group devoted to Health and Well-being can't see the value in first hand accounts about the downside of practicing law. So, what to do? Create a place for those who wish to share – a support group? Is it our job as a committee to create programing and that's it? Who are we to say what the solution is? Isn't that a personal thing for each person to find on their own.
I am not sure what the answer is, but I can tell you there must always be room for those who want to share their struggle and who want to be a part of the process of finding a solution. Neither the Committee as a whole or any one of its members has the solution to these issues/problems. There is no quick fix. The lack of a quick fix is what this is about.
There are no hard and fast rules on how to have a balanced life. If their were, then life would come with a check list and we all know that is not how life works. As much as we like to think that we can control our life, we can't. The point is to lead people in the right direction and support them along the way. Who better to help with this process than those who have been through it who have struggled with illness and come out on the other side.


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HAPPY BIRTHDAY SISTER #3!!! I hope you have a 'flippin' good birthday! I am SUPER excited to see you in just one short week. I hop your finals go well. I hope you have a wonderful day and can't wait to celebrate your birthday once you are home!
Love and Peace


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Thursday, July 16th, 2015. BMG: The love of money is the root of all evil, according to the Bible, but it can also be a source of good.
To control and channel our youngest population's attitudes toward money, the Ministry of Education and international firms PWC and Peace-works are hosting a series of camps on financial literacy for 800 students in Belize City.
Students attending the camp in Belize City report that they learned how to budget, save wisely, spend carefully and manage their money.
Andrea Fishman of PWC says financial literacy is really the foundation for growth for a lot of these kids, giving them the opportunity to learn more about the foundations of finance, savings.
And helping them learn more about what those opportunities are and how saving now can help them for their future. The camp is child-friendly and geared toward development.
Photo courtesy of Channel 7 News.
© 2015, BreakingBelizeNews.com. This article is the copyrighted property of Breaking Belize News. Written permission must be obtained before reprint in online or print media. REPRINTING CONTENT WITHOUT PERMISSION AND/OR PAYMENT IS THEFT AND PUNISHABLE BY LAW.


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The holidays are just around the corner. Rather than worry about finding last-minute gifts, this year you might consider giving something sensual and flexible: a gift certificate for a professional massage.
According to the U.S. Department of Labor, the message therapy industry is projected to grow by 20 percent from 2006 to 2016. Perhaps we are all a little more stressed these days? More likely is that more people are discovering the benefits of a professional massage. Wheaton residents, take note: The following are five reasons why a gift certificate to Elements Therapeutic Massage/Wheaton makes the ideal holiday gift for just about anyone.
1. Stress Relief
We all get a little stressed out around the holidays, what with all the traveling, visitors, gift shopping and deadlines at work. Massage therapy is a great way to alleviate some of the seasonal stress. Do you know a loved one who could benefit from a nice post-holiday massage? The week after Christmas and before New Year's is a great time to visit a Elements in Wheaton – especially for those whose New Year's resolution is to take better care of themselves.
2. Flexibility
Another benefit of giving a gift certificate for a massage is that the Chicago land area is a great area for massage therapy! With so many skilled massage therapists to choose from, it's easy to find one that will fit into your loved one's busy schedule. There are also many different massage styles to choose from, depending on what the recipient needs most.
3. Overall Health
One aim of massage therapy is to promote the overall health of the patient. Rather than focus on just one area of the body, a thorough massage from an Elements massage therapist will leave your entire body feeling better. The fact is, the body is a network of interrelated organs, bones, nerves and muscles; by relieving tension in one area, the body as a whole benefits.
4. Injury Recovery
Is there an athlete in your family who has suffered a sports-related injury? Massage therapy can help with convalescence and provide a more enjoyable recovery. There is a specific style of massage, known as a sports massage, that's commonly administered to athletes recovering from an injury. Have the recipient check with his or her doctor first to make sure the injury is ready for a massage.
5. Back Pain
Last, but not least: back pain. We all experience it at some point or another. This is one of the more universal reasons people get massages. Unwarranted back pains can be a serious stress inducer. In some cases, a good massage is all you need to reduce the pain and loosen up some of those knots.