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We consider the problems of maintaining an approximate maximum matching and an approximate minimum vertex cover in a dynamic graph undergoing a sequence of edge insertions/deletions. Starting with the seminal work of Onak and Rubinfeld [STOC 2010], this problem has received significant attention in recent years. Very recently, extending the framework of Baswana, Gupta and Sen [FOCS 2011], Solomon [FOCS 2016] gave a randomized dynamic algorithm for this problem that has an approximation ratio of and an amortised update time of with high probability. This algorithm requires the assumption of an oblivious adversary, meaning that the future sequence of edge insertions/deletions in the graph cannot depend in any way on the algorithm's past output. A natural way to remove the assumption on oblivious adversary is to give a deterministic dynamic algorithm for the same problem in update time. In this paper, we resolve this question.
We present a new deterministic fully dynamic algorithm that maintains a -approximate minimum vertex cover and maximum fractional matching, with an amortised update time of . Previously, the best deterministic algorithm for this problem was due to Bhattacharya, Henzinger and Italiano [SODA 2015]; it had an approximation ratio of and an amortised update time of . Our results also extend to a fully dynamic -approximate algorithm with amortized update time for the hypergraph vertex cover and fractional hypergraph matching problem where every hyperedge has at most vertices.
1 Introduction
Computing a maximum cardinality matching is a fundamental problem in computer science with applications, for example, in operations research, computer science, and computational chemistry. In many of these applications the underlying graph can change. Thus, it is natural to ask how quickly a maximum matching can be maintained after a change in the graph. As nodes usually change less frequently than edges, dynamic matching algorithms usually study the problem where edges are inserted and deleted, which is called the (fully) dynamic matching problem111Node updates are usually handled through the insertion and deletion of isolated nodes, but there has been also some work on the node insertions-only or node deletions-only problem [7].. The goal of a dynamic matching algorithm is to maintain either an actual matching (called the matching version) or the value of the matching (called the value version) as efficiently as possible.
Unfortunately, the problem of maintaining even just the value of the maximum cardinality matching is hard: There is a conditional lower bound that shows that no (deterministic or randomized) algorithm can achieve at the same time an amortized update time of and a query (for the size of the matching) time of for any small [10] (see [1] for conditional lower bounds using different assumptions). The best upper bound is Sankowski's randomized algorithm [16] that solves the value problem in time per update and per query. Thus, it is natural to study the dynamic approximate maximum matching problem, and there has been a large body [13, 2, 12, 9, 5, 6, 17] of work on it and its dual, the approximate vertex cover problem, in the last few years.
Dynamic algorithms can be further classified into two types: Algorithms that require an oblivious (aka non-adaptive) adversary, i.e., an adversary that does not base future updates and queries on the answers to past queries, and algorithms that work even for an adaptive adersary. Obviously, the earlier kind of algorithms are less general than the later. Unfortunately, all randomized dynamic approximate matching and vertex cover algorithms so far either are not known to work with an adaptive adversary [13] or do not work for an adaptive adversary [2, 17]. Solomon [17] gives the best such randomized algorithm: It achieves amortized update time (with high probability) and query time for maintaining a -approximate maximum matching and a -approximate minimum vertex cover. He also extends this result to the dynamic distributed setting (à la Parter, Peleg, and Solomon [14]) with the same approximation ratio and update cost.
In this paper we present the first deterministic algorithm that maintains an -approximation to the size of the maximum matching in amortized update time and query time. We also maintain an -approximate vertex cover in the same update time. Note that this is the first deterministic dynamic algorithm with constant update time for any non-trivial dynamic graph problem. This is significant as for other dynamic problems such as the dynamic connectivity problem or the dynamic planarity testing problem there are non-constant lower bounds in the cell probe model on the time per operation [11, 15]. Thus, our result shows that no such lower bound can exist for the dynamic approximate matching problem.
There has been prior work on deterministic algorithms for dynamic approximate matching, but they all have update time: One line of work concentrated on reducing the approximation ratio as much as possible, or at least below 2: Neiman and Solomon [12] achieved an update time for maintaining a -approximate maximum matching and -approximate minimum vertex cover. This result was improved by Gupta and Peng [9] that gave an algorithm with update time for maintaining a -approximate maximum matching. Recently, Bernstein and Stein [3] gave an algorithm with amortised update time for maintaining a -approximate maximum matching. Another line of work, and this paper fits in this line, concentrated on getting a constant approximation while reducing the update time to polylogarithmic: Bhattacharya, Henzinger and Italiano [5] achieved an update time for maintaining a -approximate maximum fractional matching and a -approximate minimum vertex cover. Note that any fractional matching algorithm solves the value version of the dynamic matching problem while degrading the approximation ratio by a factor of . Thus, the algorithm in [5] maintains a -approximation of the value of the maximum matching. The fractional matching in this algorithm was later "determinically rounded" by Bhattacharya, Henzinger and Nanongkai [6] to achieve a update time for maintaining a -approximate maximum matching.
Our method also generalizes to the hypergraph vertex (set) cover and hypergraph fractional matching problem which was considered by [4]. In this problem the hyperedges of a hypergraph are inserted and deleted over time. indicates the maximum cardinality of any hyperedge. The objective is to maintain a hypergraph vertex cover, that is, a set of vertices that hit every hyperedge. Similarly a fractional matching in the hypergraph is a fractional assignment (weights) to the hyperedges so that the total weight faced by any vertex is at most . We give an -approximate algorithm with amortized update time.
1.1 Our Techniques
Our algorithm builds and simplifies the framework of hierarchical partitioning of vertices proposed by Onak and Rubinfeld [13], which was later enhanced by Bhattacharya, Henzinger and Italiano [5] to give a deterministic fully-dynamic -approximate vertex cover and maximum matching in -amortized update time. The hierarchical partition divides the vertices into -many levels and maintains a fractional matching and vertex cover. To prove that the approximation factor is good, Bhattacharya et. al.[5] also maintain approximate complementary slackness conditions. An edge insertion or deletion can disrupt these conditions (and indeed at times the feasibility of the fractional matching), and a fixing procedure maintains various invariants. To argue that the update time is bounded, [5] give a rather involved potential function argument which proves that the update time bounded by , the number of levels, and is thus . It seems unclear whether the update time can be argued to be a constant or not.
Our algorithm is morally similar to that in Bhattacharya et. al.[5], except we are a bit stricter when we fix nodes. As in [5], whenever an edge insertion or deletion or a previous update violates an invariant condition, we move nodes across the partitioning (incurring update costs), but after a node is fixed we often ensure it satisfies a stronger condition than what the invariant requires. For example, suppose a node violates the upper bound of a fractional matching, that is, the total fractional weight it faces becomes larger than , then the fixing subroutine will at the end ensure that the final weight the node faces is significantly less than . Morally, this slack allows us to make an charging argument of the following form – if this node violates the upper bound again, then a lot of "other things" must have occurred to increase its weight (for instance, maybe edge insertions have occurred). Such a charging argument, essentially, allows us to bypass the -update time to an -update time. The flip side of the slack is that our complementary slackness conditions become weak, and therefore instead of a -approximation we can only ensure an -approximation. The same technique easily generalizes to the hypergraph setting. It would be interesting to see other scenarios where approximation ratios can be slightly traded in for huge improvements in the update time.
Very recently, and independently of our work, Gupta et al. [8] obtained a -approximation algorithm for maximum fractional matching and minimum vertex cover in a hyper-graph in amortized update time.
2 Notations and Preliminaries
Since the hypergraph result implies the graph result, henceforth we consider the former problem. The input hyper-graph has nodes. Initially, the set of hyper-egdes is empty, i.e., . Subsequently, an adversary inserts or deletes hyper-edges in the hyper-graph . The node-set remains unchanged with time. Each hyper-edge contains at most nodes. We say that is the maximum frequency of a hyper-edge. If a hyper-edge has a node as one of its endpoints, then we write . For every node , we let denote the set of hyper-edges that are incident on . In this fully dynamic setting, our goal is to maintain an approximate maximum fractional matching and an approximate minimum vertex cover in . The main result of this paper is summarized in the theorem below.
Theorem 2.1.
We can maintain an approximate maximum fractional matching and an approximate minimum vertex cover in the input hyper-graph in amortized update time.
To prove Theorem 2.1, throughout the rest of this paper we fix two parameters and as follows.
We will maintain a hierarchical partition of the node-set into levels , where . We let denote the level of a node . We define the level of a hyper-edge to be the maximum level among its endpoints, i.e., . The levels of nodes (and therefore hyper-edges) induce the following weights on hyper-edges: for every hyper-edge . For all nodes , let be the total weight received by from its incident hyper-edges. We will satisfy the following invariant after processing a hyper-edge insertion or deletion.
Invariant 2.2.
Every node at level has weight . Every node at level has weight .
Corollary 2.3.
Under Invariant 2.2, the nodes in levels form a vertex cover in .
Suppose that there is a hyper-edge with for all . Then we also have and . So for every node , we get: . This violates Invariant 2.2. ∎
Invariant 2.2 ensures that 's form a fractional matching satisfying approximate complementary slackness conditions with the vertex cover defined in Corollary 2.3. This gives the following theorem.
Theorem 2.4.
In our algorithm, the hyper-edge weights form a -approximate maximum fractional matching, and the nodes in levels form a -approximate minimum vertex cover.
(Sketch) Say that a fractional matching, which assigns a weight to every hyper-edge , is maximal iff for every hyper-edge there is some node such that . Let be the set of all tight nodes in this fractional matching. Clearly, the set of nodes form a vertex cover in . It is well known that the sizes of such a fractional matching and the corresponding vertex cover are within a factor of each other. The key observation is that under Invariant 2.2, the fractional matching is approximately maximal, meaning that for every hyper-edge there is some node such that . Further, the set of nodes in levels are approximately tight, since each of them has weight at least . ∎
We introduce some more notations. For any vertex , let be the total up-weight received by , that is, weight from those incident hyper-edges whose levels are strictly greater than . For all levels , we let and respectively denote the values of and if the node were to go to level and the levels of all the other nodes were to remain unchanged. More precisely, for every hyper-edge and node , we define to be the maximum level among the endpoints of that are distinct from . Then we have: and . Our algorithm maintains a notion of time such that in each time step the algorithm performs one elementary operation. We let denote the weight (resp, up-weight) faced by right before the operation at time . Similarly define , and .
Different states of a node.
Before the insertion/deletion of a hyper-edge in , all nodes satisfy Invariant 2.2. When a hyper-edge is inserted (resp. deleted), it increases (resp. decreases) the weights faced by its endpoints. Accordingly, one or more endpoints can violate Invariant 2.2 after the insertion/deletion of a hyper-edge. Our algorithm fixes these nodes by changing their levels, which may lead to new violations, and so on and so forth. To describe the algorithm, we need to define certain states of the nodes.
Definition 2.5.
A node is Down-Dirty iff and . A node is Up-Dirty iff either {} or {}. A node is Dirty if it is either Down-Dirty or Up-Dirty .
Invariant 2.2 is satisfied if and only if no node is Dirty. We need another definition of Super-Clean nodes which will be crucial.
Definition 2.6.
A node is Super-Clean iff one of the following conditions hold: (1) We have and , or (2) We have , , and .
Note that a Super-Clean node with has a stronger upper bound on the weight it faces and also an even stronger upper bound on the up-weight it faces. At a high level, one of our subroutines will lead to Super-Clean nodes, and the slack in the parameters is what precisely allows us to perform an amortized analysis in the update time.
Data Structures.
For all nodes and levels , let denote the set of hyper-edges incident on that are at level . Note that for all . We will maintain the following data structures. (1) For every level and node , we store the set of hyper-edges as a doubly linked list, and also maintain a counter that stores the number of hyper-edges in . (2) For every node , we store the weights and , its level and an indicator variable for each of the states defined above. (3) For each hyper-edge , we store the values of its level and therefore its weight . Finally, using appropriate pointers, we ensure that a hyper-edge can be inserted into or deleted from any given linked list in constant time. We now state two lemmas that will be useful in analysing the update time of our algorithm.
Lemma 2.7.
Suppose that a node is currently at level and we want to move it to some level . Then it takes time to update the relevant data structures.
If a hyper-edge is not incident on the node , then the data structures associated with are not affected as moves up from level to level . Further, among the hyper-edges , only the ones with get affected (i.e., the data structures associated with them need to be changed) as moves up from level to level . Finally, for every hyper-edge that gets affected, we need to spend time to update the data structures for its endpoints. ∎
Lemma 2.8.
Suppose that a node is currently at level and we want to move it down to some level . Then it takes time to update the relevant data structures.
If a hyper-edge is not adjacent to the node , then the data structures associated with are not affected as moves down from level to level . Further, among the hyper-edges , only the ones with get affected (i.e., the data structures associated with them need to be changed) as moves down from level to level . Finally, for every hyper-edge that gets affected, we need to spend time to update the data structures for its endpoints. ∎
3 The algorithm: Handling the insertion/deletion of a hyper-edge in the input graph
Initially, the graph is empty, every node is at level , and Invariant 2.2 holds. By induction, we will ensure that the following property is satisfied just before the insertion/deletion of a hyper-edge.
Property 3.1.
No node is Dirty.
Insertion of a hyper-edge . When a hyper-edge is inserted into the input graph, it is assigned a level and a weight . The hyper-edge gets inserted into the linked lists for all nodes . Furthermore, for every node , the weights increases by . For every endpoint , if , then the weight increases by . As a result of these operations, one or more endpoints of can now become Up-Dirty and Property 3.1 might no longer be satisfied. Hence, in order to restore Property 3.1 we call the subroutine described in Figure 1.
Deletion of a hyper-edge . When a hyper-edge is deleted from the input graph, we erase all the data structures associated with it. We remove the hyper-edge from the linked lists for all , and erase the values and . For every node , the weight decreases by . Further, for every endpoint , if , then we decrease the weight by . As a result of these operations, one or more endpoints of can now become Down-Dirty, and Property 3.1 might get violated. Hence, in order to restore Property 3.1 we call the subroutine described in Figure 1.
The algorithms is simple – as long as some Dirty node remains, it runs either FIX-UP-DIRTY or FIX-DOWN-DIRTY to take care of Up-Dirty and Down-Dirty nodes respectively. One crucial aspect is that we prioritize Up-Dirty nodes over Down-Dirty ones.
FIX-DOWN-DIRTY(): Suppose that when the subroutine is called at time . By definition, we have and . We need to increase the value of if we want to ensure that no longer remains Dirty. This means that we should decrease the level of , so that some of the hyper-edges incident on can increase their weights. Accordingly, we find the largest possible level such that , and move the node down to this level . If no such level exists, that is, if even , then we move the node down to level . Note that in this case there is no hyper-edge with for such a hyper-edge would have when is moved to level . In particular, we get .
Claim 3.2.
FIX-DOWN-DIRTY() makes the node Super-Clean .
Suppose node was at level when FIX-DOWN-DIRTY() was called at time and it ended up in level . If , then , and so becomes Super-Clean after time . Henceforth assume . Since is the maximum level where , we have . Now note that since weights of hyper-edges can increase by at most a factor when one end point drops exactly one level. This implies . Together we get that after time when is fixed to level , we have .
Now we argue about the up-weights. Note that every hyper-edge that contributes to must have . The weight of such a hyper-edge remains unchanged as moves from level to . We infer that . Therefore after time when is fixed at level , we have In sum, becomes Super-Clean after time . ∎
FIX-UP-DIRTY(): Suppose that when the subroutine is called at time . At this stage, we have either {} or {}. We need to increase the level of so as to reduce the weight faced by it. Accordingly, we find the smallest possible level where and move up to level . Such a level always exists because .
Claim 3.3.
After a call to the subroutine FIX-UP-DIRTY() at time , we have .
Suppose that the node moves up from level to level . We now consider four possible cases.
Case 1. We have .
Since is the minimum possible level where , and since , we infer that . As the node moves up from level to level , the weight it faces can drop by at most a factor of . Hence, we get: . Therefore after time when the node moves to level , we have .
Case 2. We have , and there is an edge with at time . In this case we have in the beginning of time-step , since the edge with has weight . The rest of the proof is exactly similar to Case 1.
Case 3. We have , there is no edge with at time , and .
The value of does not change as moves up from level to level . Thus, we get: , for the node is Up-Dirty at level at time . Since the node does not move further up than level , we get: .
Case 4. We have , there is no edge with at time , and .
Since the node does not stop at level , we get: . Hence, we infer that is the minimum possible level where . As the node moves up from level to level , the weight it faces can drop by at most a factor of . Hence, we get: . Therefore, we have .
It is clear that if and when FIX-DIRTY() terminates, we are in a state which satisfies Invariant 2.2. In the next section we show that after hyper-edge insertions and deletions, the total update time is indeed and so our algorithm has -amortized update time.
4 Analysis of the algorithm
Starting from an empty graph , fix any sequence of updates. The term "update" refers to the insertion or deletion of a hyper-edge in . We show that the total time taken by our algorithm to handle this sequence of updates is . We also show that our algorithm has an approximation ratio of .
Relevant counters.
We define three counters and . The first two counters account for the time taken to update the data structures while the third accounts for the time taken to find the index in both FIX-DOWN-DIRTY() and FIX-UP-DIRTY(). Initially, when the input graph is empty, all the three counters are set to zero. Subsequently, we increment these counters as follows.
Suppose node moves from level to level upon a call of FIX-UP-DIRTY(). Then for every hyper-edge with , we increment by one.
Suppose node moves from level to level upon a call of FIX-DOWN-DIRTY(). Then for every hyper-edge with , we increment the value of by one. Furthermore, we increment the value of by .
The next lemma upper bounds the total time taken by our algorithm in terms of the values of these counters. The proof of Lemma 4.1 appears in Section 4.5.
Lemma 4.1.
Our algorithm takes time to handle a sequence of updates.
We will show that and , which will imply an amortized update time of for our algorithm. Towards this end, we now prove three lemmas that relate the values of these three counters.
Lemma 4.2.
We have: .
Lemma 4.3.
We have: .
Lemma 4.4.
We have: .
The proofs of Lemmas 4.2, 4.3 and 4.4 appear in Sections 4.2, 4.3 and 4.4 respectively. All these three proofs use the concepts of epochs, jumps and phases as defined in Section 4.1. The main result of our paper (see Theorem 2.1) now follows from Theorem 2.4, Lemma 4.1 and Lemma 4.5.
4.1 Epochs, jumps and phases
Fix any node . An epoch of this node is a maximal time-interval during which the node stays at the same level. An epoch ends when either (a) the node moves up to a higher level due to a call to FIX-UP-DIRTY, or (b) the node moves down to a lower level due to a call to the subroutine FIX-DOWN-DIRTY. These events are called jumps. Accordingly, there are Up-Jumps and Down-Jumps. Next, we define a phase of a node to be a maximal sequence of consecutive epochs where the levels of the node keep on increasing. The phase of a node is denoted by . Suppose that a phase consists of consecutive epochs of at levels . Then we have: . By definition, the epoch immediately before must have level larger than implying FIX-DOWN-DIRTY() landed at level . Similarly, the epoch subsequent to is smaller than implying FIX-DOWN-DIRTY() is called again.
4.2 Proof of Lemma 4.2
Suppose that a node moves down from (say) level to level at time (say) due to a call to the subroutine FIX-DOWN-DIRTY. Let and respective denote the increase in the counters and due to this event. We will show that , which will conclude the proof of the lemma. By definition, we have:
Let denote the set of hyper-edges incident on that contribute to the increase in due to the Down-Jump of at time . Specifically, we have: . Each edge contributes a weight towards the node-weight . Thus, we get: . The last inequality holds since is Down-Dirty in the beginning of time-step . Rearranging the terms, we get: . The lemma now follows from equation (4).
4.3 Proof of Lemma 4.3
Suppose we call FIX-DOWN-DIRTY() at some time . Let just before the call, and let be the epoch with level of being . Let at time . By definition, increases by during the execution of FIX-DOWN-DIRTY(); let us call this increase . Thus, we have:
Consider the time between and let us address how can decrease in this time while 's level is fixed at . Either some hyper-edge incident on is deleted, or some hyper-edge incident on it decreases its weight. In the latter case, the level of such an hyper-edge must increase above . Let denote the number of hyper-edge deletions incident on during the time-interval . Let denote the increase in the value of during the time-interval due to the hyper-edges incident on . Specifically, at time , we have . Subsequently, during the time-interval , we increase the value of by one each time we observe that a hyper-edge increases its level to something larger than . Note that throughout the time-interval . Hence, each time we observe an unit increase in , this decreases the value of by at most . Just before time , the node made either an Up-Jump, or a Down-Jump. Hence, Claims 3.3 and 3.2 imply that . As at time , we infer that has dropped by at least during the time-interval . In order to account for this drop in , the value of must have increased by at least . Since at time , at time we get: . Hence, (5) gives us:
Each time the value of increases due to FIX-DOWN-DIRTY on some node, inequality (6) applies. If we sum all these inequalities, then the left hand side (LHS) will be exactly equal to the final value of , and the right hand side (RHS) will be at most . The factor appears in front of because each hyper-edge deletion can contribute times to the sum , once for each of its endpoints. Similarly, the factor appears in front of because whenever the level of an hyper-edge moves up due to the increase in the level of some endpoint , this contributes at most times to the sum , once for every other endpoint . Since LHS RHS, we get: . This concludes the proof of the lemma.
4.4 Proof of Lemma 4.4
Fix a node and consider a phase where goes through levels . Thus, the node enters the level at time (say) due to a call to FIX-DOWN-DIRTY. For , the node performs an Up-Jump at time (say) from the level to the level , due to a call to FIX-UP-DIRTY. This implies that . The phase ends, say, at time when the node again performs a Down-Jump from the level due to a call to FIX-DOWN-DIRTY.
Let denote the total increase in the value of the counter due to the phase . For , let denote the increase in the value of the counter due to the Up-Jump of at time . Thus, we have:
We define two more counters: , . The former counter equals the number of hyper-edge insertions/deletions incident on during the time-interval . The latter counter equals the increase in the value of due to the hyper-edges incident on during the time-interval . Alternately, these two counters can be defined as follows. At time , we set and . Subsequently, whenever at any time , a hyper-edge incident on gets inserted into or deleted from the input graph, we increment the value of by one. Further, whenever at any time , a hyper-edge incident on gets its level decreased because of a Down-Jump of some node , we increment the value of by one.
Since enters the level at time due to a call to FIX-DOWN-DIRTY, Claim 3.2 implies that:
Our main goal is to upper bound in terms of the final values of the counters and .
Claim 4.6.
For , .
By Claim 3.3 we have , that is, the total weight incident on after it has gone through FIX-UP-DIRTY at time is at most . Now, each hyper-edge which contributes to has weight, right after time , precisely . Putting together, we get . ∎
Using the above claim, we get the following upper bound on the sum of all but the last .
Claim 4.7.
We have: .
If , then we have an empty sum , and hence the claim is trivially true. For the rest of the proof, we suppose that , which implies that . Thus, we get:
To continue with the proof, summing over the inequalities from Claim 4.6, we get:
Since the node performs an Up-Jump at time from level (see equation (9)), the node must be Up-Dirty at that time. It follows that . From equation (8), we have . Thus, during the time interval the value of increases by at least . This can be either due to (a) some hyper-edge incident to being inserted, or (b) some hyper-edge gaining its weight because of some endpoint going down. The former increases and the latter increases . Furthermore, the increase in due to every such hyper-edge is at most . This gives us the following lower bound.
The claim follows from equation (11). ∎
Claim 4.8.
We have: .
From equations 1, 7 and Claims 4.7, 4.8, we get:
Using equation (12), now we can prove Lemma 4.4. For every phase of a node , as per equation (12) we can charge the increase in to the increase in corresponding to hyper-edges incident of . Summing up over all nodes and phases, the LHS gives while the RHS gives . The coefficient before comes from the fact that every hyper-edge insertion can contribute times to the RHS, once for each of its endpoints. The coefficient before comes from the fact that whenever the level of a hyper-edge decreases due to the Down-Jump of a node , this event contributes at most times to the RHS: once for every other endpoint . Thus, we get:
4.4.1 Proof of Claim 4.8
We fork into two cases.
Case 1:
Case 2:
We start by noting that the weight of a node is always less than at every time.
Claim 4.9.
We have: at every time .
The crucial observation is that fixing an Up-Dirty node never increases the weight of any node. Furthermore, a Down-Dirty node gets fixed only if no other node is Up-Dirty (see Figure 1).
In the beginning of time-step , the input graph is empty, and we clearly have . By induction, suppose that in the beginning of some time-step . Now, during time-step , the weight can increase only if one of the following events occur:
(a) A hyper-edge containing gets inserted into the graph. This increase the value of by at most one. Thus, we have .
(b) We call the subroutine FIX-DOWN-DIRTY for some node . Note that fixing a Down-Dirty node can increase the weight by at most one, and hence this can increase the weight of a neighbour of also by at most one. It again follows that .
If , then we are back in the same situation as in time-step . Otherwise, if , then is Up-Dirty in the beginning of time-step . In this case, no Down-Dirty node gets fixed (and no hyper-edge gets inserted) until we ensure that becomes smaller than one. Hence, the value of always remains smaller than . ∎
Claim 4.10.
We have: .
While making the Up-Jump at time , the node does not stop at level . The claim follows. ∎
Claim 4.11.
We have: .
Suppose that the claim does not hold. Then we get:
The first inequality holds since the weights of the hyper-edges with get scaled by at least a factor of when moves from level to , and the rest can only go down. The second inequality holds since by Claim 4.9 and the assumption . The last inequality holds since by equation (1). However, equation (15) contradicts Claim 4.10. ∎
Claim 4.11 states that . Since , equation (8) implies that . Thus during the time-interval , the value of increases by at least . This increase can occur in three ways: (1) a hyper-edge is inserted with before the Up-Jump at time (which contributes to ), (2) some hyper-edge gains weight due to a Down-Jump of some node (say) , and after the Down-Jump (which contributes to ), and (3) some hyper-edge had


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Spend $50 in beef and get 5lbs of yellow potatoes! We have some combo packs available along with our ground beef special of 5lbs. for $30. Don't forget we carry lots of different types of cuts! Steaks, roasts, patties, stir fry and more...Read more
Blueberry Update 8/20/19
This week we will only be open on Friday 8/23/19 from 9-12pm and Saturday 8/24/19 from 8am-4pm. We are giving the berries a little rest and time to ripen more! Also we will no longer be taking orders for flats. It is u-pick only at this time...Read more
Blueberry Update 8/16/19
Today Friday 8/16/19 our u-pick blueberry patch will be CLOSED! Tomorrow Saturday 8/17/19 we will be open from 8-12pm Thank you, Mama'sRead more

subset: commoncrawl


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Home :Lancaster Bowling League - 2020
Lancaster Bowling League - 2020
Lancaster Bowling League - 2020
16/03/2020 - Please see below an excerpt from a BCGBA Message, sent to county secretaries, Management and Officers, Leagues and members.
Following the Government Announcement this afternoon, it is with great sadness that we now have to suspend all bowling for the forseeable future.
All BCGBA meetings should now be postponed as un-necessary travel should be avoided. Contact in Pubs, Clubs and restaurants to be avoided.
Taking this all into account all leagues should now suspend matches until such time as we are given an all clear to resume competition.
All BCGBA and BCGLCC Competitions are now suspended until further notice this should include all qualifier events.
Online Bowls
League News
Cup Draw
League Tables
Leading Merits
Club Zone
League Fixtures - 2020
Select a date below
to view fixtures
07/04/20- Week 1
14/04/20- Week 2
21/04/20- Week 3
28/04/20- Week 4
05/05/20- Week 5
12/05/20- Week 6
19/05/20- Week 7
26/05/20- Week 8
02/06/20- Week 9
16/06/20- Week 10
23/06/20- Week 11
07/07/20- Week 12
14/07/20- Week 13
21/07/20- Week 14
28/07/20- Week 15
04/08/20- Week 16
11/08/20- Week 17
18/08/20- Week 18
League Tables
Week 15 - Commencing Tuesday 28th July 2020
Division 1
Tuesday 28/07/20-(19:00)
Bare A
Warton A
Carnforth A
Bowling Green A
Lune Road A
Skerton B
Trimpell A
Bowling Green B
Division 2
Tuesday 28/07/20-(19:00)
Bolton-Le-Sands A
Lansil B
Lune Road B
Luneside B
Lansil A
Skerton A
Station B
Slyne A
Overton A
Division 3
Tuesday 28/07/20-(19:00)
Bay Mixed
Trimpell B
Bolton-Le-Sands B
Luneside A
Cumberland 'A'
Skerton C
Station A
Carnforth B
University B
University A
Division 4
Tuesday 28/07/20-(19:00)
Bare C
Bare Snooker
Overton B
Slyne B
Bare B
Warton B
Regent Loyals
Thursday 30/07/20-(19:00)
Lune Road C
Cumberland 'B'
Online Bowls
© 1998 - 2020 RmK


text:
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|Figure 1: Woodbury, TN At least High School Education
|Figure 2: Woodbury, TN Bachelors Degree or Better Education
|Figure 3: Advertisement
|Figure 4: Woodbury, TN School Dropout Rate
|Figure 5: Woodbury, TN Education Attainment Breakdown
|Figure 6: Higher Education Attainment (100%=All People with Bachelor or better)
|Figure 7: Woodbury, TN Bachelors Degrees Field of Study
|Figure 8: Woodbury, TN Bachelors Degree Obtained
|Figure 9: Woodbury, TN Education Attainment by Level Comparison (Age 25+)
|Figure 10: Woodbury, TN Education Attainment Detailed Comparison (Age 25+)
|Figure 11: Woodbury, TN Detailed Education Attainment Breakout by Age Group (Age 18+)
|Figure 12: Woodbury, TN Detailed Education Attainment Breakout by Race (Age 25+)
|Figure 13: Woodbury, TN Detailed Male and Female breakdown of Educational Attainment
|Figure 14: Woodbury, TN School Enrollment by Aggregate Categories
|Figure 15: Woodbury, TN Overall Public vs. Private School Enrollment
|Figure 16: Woodbury, TN Public vs. Private K-8 School Enrollment
|Figure 17: Woodbury, TN Public vs. Private High School Enrollment
|Figure 18: Woodbury, TN Public vs. Private College Enrollment
|Figure 19: Woodbury, TN Public vs. Private Graduate or Professional School Enrollment
|Figure 20: Woodbury, TN Total Enrolled in Schools
|Figure 21: Advertisement
|Figure 22: Woodbury, TN Public vs. Private Preschool
|Figure 23: List of Schools in the Woodbury, TN Area (2013)
|Figure 24: Woodbury, TN School Enrollment (2013)
|Figure 25: Student to Teacher Ratios (2013) - Low Scores Are Better
|Figure 26: Woodbury, TN School Racial Mix (2013)
|Cities marked with an asterisk ("*") should resemble a city or town but do not have their own government (i.e. Mayor, City Council, etc.) These places should be recognizable by the local community but their boundaries have no legal status. Technically these include both Census Designated Places (CDP) and Census County Divisions (CCD) which are defined by the Census Bureau along with local authorities. (For more information, see:
Census Designated Place or "CDP")
and Census County Division "CCD".)
For comparison purposes, the US and state value are provided as well as in some areas a "Combined Statistical Area" or CSA. These are large groupings of adjacent metropolitan areas identified by the Census Bureau based on social and economic ties. (See: Combined Statistical Area)
|Data sources - Mouse over icon in upper right corner of each chart for information.
|Woodbury Tennessee, Murfreesboro, Tennessee and Surrounding Area Comments and Discussion


text:
Transformers, Robot Heroes, Generation I Series Wave 1.5
Values Guide Help
What is the DASH Values Guide?
The DASH Values Guide is an encyclopedia of Action Figures organized by category with photos, reviews, values and more!
Search our database to find information about the items you have or want.
Add items to your collection & wanted list. Receive updates when an item is for sale.
Showing 1 - 2 of 2
10 Have It
1 Want It
Item Values
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$8 - $10
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$8 - $10
Un-Packaged Value
$5 - $5
The "value" of items is based on a number of factors. Please review how we have compiled our value ranges for figures and use our values as a rough guideline.
We'd LOVE to hear from you!


text:
The pandemic is far from over and kids need help.
Take a bite of nostalgia and stick to your diet.
Feed the beasts!
Your kids will love these homemade snacks!
That you'll enjoy, too.
And what criteria to look for.
Do you have a picky eater on your hands?
Switch up your kids meals with these ideas!
Children deserve to eat nutritious foods!
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Don't miss a thing—stay on top of your goals with the Eat This! E-mail Newsletter.


text:
A partnership with Ticketmaster Sport is a sound investment for your business.
Not only have we been doing this for over 30 years, but we've the specialist sports knowledge and know-how to help you navigate the challenges you face in your business on a daily basis. Allowing you to focus your time on what really matters – giving your fans the very best experience they deserve.
Find out what you expect when you partner with Ticketmaster Sport.
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Our state-of-the-art sports technology is specially built and customised for sport and proven to meet the demands of even the most highest profile sports events. Following a recent major re-platforming project, our technology is now even more scalable, supportable and customisable than ever before. Our feature-rich technology provides you with the latest in digital ticketing, a new access control system, season ticket upgrades online, a tool to manage your allocations with ease amongst many other unique features. We're also developing Our engineering team are constantly looking at ways to further develop our products, with regular upgrades to the technology and at absolutely no extra cost to you.
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We're renowned for our service levels and support, so when you join the Ticketmaster Sport family, you're guaranteed 360 support, when you need it. From the moment you decide to partner with us, you'll be looked after by our onboarding team, experts trained in project management, to get you up and running as quickly and efficiently as possible. Our training team are on hand to provide you with comprehensive training packages and you'll also be appointed a Client Account Manager to look after all your needs on a day-to-day basis, including event day support. Our Support desk is on hand 24/7, including weekends, for those times when you need that extra support. What's more, you can expect our service and support to be included within the partnership and at no extra cost to you.
Industry Experience
The world of sport presents a unique set of challenges, which is why we only work with the best talent in the business. Ticketmaster Sport has been at the forefront of sport for over 30 years and our experience is second to none. We've worked across events of every size, scale and degree of complexity giving us the ability to provide our client partners with valuable insight, tailored support and advice that can be trusted. We're proud of our team, that includes industry specialists in every field; from ticketing, technology, operations and project management to sports marketing specialists, data science, design and customer service. Our team have first-hand specialist knowledge and industry experience so you can rest assured you're in a safe pair of hands.
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Our dedicated sports marketing expertise works collaboratively with you, to understand your goals and to help you get the most out of your partnership with us. We employ a strategic approach, using our state of the art marketing tools, the latest in data science and advanced marketing techniques to provide you with comprehensive marketing campaigns to reach beyond your objectives and to ultimately help you drive ticket sales and additional revenue. Our leading reach both in the UK and Internationally means we can put you in front of a giant global audience and, what's more, our pioneering affiliate programme helps to further extend this reach by giving your events exposure to those all important new fans.
Get in touch with a member of the team today, to find out how we can help grow your business.


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Vaimo is a global full service omni-channel agency helping brands, retailers and manufacturers all over the world to drive success in digital commerce. Delivering solutions around B2B and B2C Digital Commerce, PIM, Marketing & CRM, Integrations, Order Management, Mobile & Apps, Omnichannel, Business Intelligence and Security, Vaimo delivers a wide spectrum of services related to commerce and is committed to forging long-term relationships with its clients.
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Integrations for every ecommerce tech stack.
Nosto integrates with a selection of best in class technology providers that bring additional value to our clients. Enhance your Nosto journey further; select a partner of ours below!


text:
From SWRPG Blake Sector
Revision as of 22:53, 7 January 2019 by AngelWeathers1 (Talk | contribs)
I would like to introduce myself to you, I am Jacob Duncanson. Wisconsin has always been my living place. To play domino is something her husband doesn't actually like but she does. The work she's been occupying for several years is an business office supervisor and she will not alter it at any time quickly. You can always discover his site listed here:


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Amanda Laura Bynes (born April 3, 1986 in Thousand Oaks, California) is an American actress. She started out a child star in All That and The Amanda Show and was widely seen as a comic talent. She appeared in quite a few high-profile works in the late '90s and 'aughts, but announced a hiatus from acting in 2010 due to personal problems. She announced in March 2020 that she is pregnant with her first child.
- All That (1996-2000): Various roles
- Figure It Out (1997-1999): Panelist
- The Amanda Show (1999-2002): Host/various roles
- Rugrats (2001-2002): Taffy (voice role)
- Big Fat Liar (2002): Kaylee
- What I Like About You (2002-2006): Holly Tyler
- What a Girl Wants (2003): Daphne Reynolds
- Robots (2005): Piper Pinwheeler
- Love Wrecked (2005): Jenny Taylor
- She's the Man (2006): Viola Hastings
- Hairspray (2007): Penny Pingleton
- Sydney White (2007): Sydney White
- Easy A (2010): Marianne Bryant


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Dedicated Online Games for a Legendary US Guitarist – Jimi Hendrix
If you ask music lovers to name the most iconic guitar player that they have heard in the history of pop music, you will hear the name of none other than the Jimi Hendrix. The legendary musician who bid goodbye to this world at a very young age is written in golden words in the history of popular music. This legend has now been immortalized with an amazing online slots game named after him. He still has a cult following across the world and such was his mastery over the instrument that many people who tried to copy him after his demise failed miserably in doing so. There are very few people who have brought such revolutionary changes to the world of pop music like Jimi Hendrix.
The Jimi Hendrix named online games: A real tribute to the legend
Jimi Hendrix was not just a guitar player; he was one guy who managed to revolutionise the way guitar was played till then. His use of fuzz and wah-wah techniques drove music lovers mad, and people queued up for long hours to hear him play. One could say that it was his techniques that led to the creation of the heavy metal and hard rock. Try out the amazing online slots game at UK's top online casino and get a feel for the 70's background theme as you play and get to watch the super cool graphics of the great legend.
Experience purple haze Jimi Hendrix style
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Jimi Hendrix's Cross-Town traffic
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EducationUSA@USIEF Chennai invites you to an information session on Writing a winning essay/SOP on Thursday 08, 2019 at 3 p.m. at the American Center, American Consulate Building, 220, Anna Salai, Chennai 600 006.
Want to know the tips for writing an essay/ SOP? Join us for this session with Shraddha Singaurey, Recruitment Adviser at Wayne State University and we guide you in this process
Please note that all fields should be completed in the form below. On submission, you will receive a confirmation message on the screen. Students and parents are welcome. Each person must register independently.
Since our office is located inside the American Consulate, please adhere to the following entry regulations to enter the building:
1. Indian citizens are required to bring an original, valid Government issued photo ID like driving license, passport, voter ID, Aadhar card or PAN card. Photocopied, laminated ID cards will not be accepted. The security guard will retain the ID and return it to you when you leave the building. American citizens are required to bring their passport.
2. Visitors below 17 years of age should bring their valid, original school or college photo ID card and must be accompanied by an adult throughout the visit. The accompanying adult must carry a separate ID card.
3. Please do not bring laptops, flash drive/USB stick or any electronic items with you to the Consulate; these will not be allowed into the Consulate and there is currently no provision to store these at the security/entrance.
4. The entrance to EducationUSA @ USIEF office is through the U.S. Consulate - Cathedral Road gate.
5. The security guard will collect your mobile phones for safekeeping and return it when you leave.
6. There is no two-wheeler or car parking facility inside the U.S. Consulate building. Please also bear in mind that vehicles cannot be parked at the entrance or near the Consulate.
7. Please arrive 15 minutes in advance of the event to allow time for security processes.
If you have any questions, please call us on (044) 2857 4134 or email us on email@example.com
Thank you.
EducationUSA Centre at USIEF, Chennai
Click here to register!
Thank you Edutrust for saving me in Insurance


text:
Make your comment by making a reply in this section.
Action Against Drug is a global initiative started By Raymond Mafukidze.Support the fight against drugs by making a donation through Western Union Money Transfer.Name:Raymond MafukidzeCity:HarareCountry:ZimbabweSend the collection details to this email address Email:firstname.lastname@example.com can also make a donation with bitcoin,address given on top part of this site
It is time to take action against drugs.Action against drug dealing and drug abuse.Drugs are destroying people and lives.All you need to do is to sign up on this website and become a member.Your membership is your vote.If you have any meaningful contributions you can go to the Members Comments area and comment by making a reply.People who are addicted to drugs will now be helped through effective ways.Sign up right now! I and share this thread.If you want help from drug abuse problems ckick here http://thedrugabusefightingkit.xobor.com
Drug dealers love to be told,"no you can't sell drugs".The more you challenge them head on the more they find other ways of selling drugs.If a government should crack down on drug dealers like never before,they will just operate through some rotten apple judges,military,police and even hospital officials.They know that they can do anything anywhere as long as the price is right.People are bought by drug dealers.Why are drug lords only caught in South America and Hong Kong mostly? Are there no drug lords in Europe,America,Australian,Canada etc? It is very complex.One of the sure and best ways is to admit that they are a lawless group and then to work on their conscious because after all is said and done everyone has a conscious.Continue here DO DRUG DEALERS HAVE A CONSCIOUS?
It is a fact that the major drug dealers are the very wealthy people in society.Some of these people have well known legit companies that they are well known for.The purpose of this post and page is not to expose people and companies and names.What good will that do? The main purpose is to ask these people," DO YOU HAVE A CONSCIOUS?" " DO DRUG DEALERS HAVE A CONSCIOUS?" if this questioned is asked long enough a good number of these "big" drug dealers will become conscious of the fact that like most people they too have a conscious after all.Yes,there is big money in selling drugs but the major drug dealers are already rich through drugs or were already rich prior to selling drugs.We now have two major problems;1) DRUG ABUSE.This shortens the hopes and lives of many people.Someone told me in April 2019 that his friend Marcel was heavily into drug abuse.Marcel died just two weeks later.Why did that person in America choose to tell me this story like i could help him or his friend? There are more questions than anwers.It is time to take action against drug abuse.2) DRUG INFLOWS INTO SCHOOLS,NEIGHBOURHOODS AND NATIONS.The pimps and the street drug dealers are selling drugs because of the big drug dealers that bring in the supplies.Do these big drug dealers have a conscious at all?My heart is with people like Marcel and maybe after all like the faith his friend put in me,i can help them after all.It will not be enough to only to help the victims of drug abuse but to also address the issue of rampant drug availabilty.If you are into taking drugs and you know you want to stop but you find it hard to stop,please go through this drug abuse fighting method http://thedrugabusefightingkit.xobor.comSocieties are being torn to pieces,lives are lost but do drug dealers have a conscious?


text:
Purchasing Process
- Duration in hours: 2 x 4h
- Module level: Foundation to Skill
- [SC01-MOD02] - Purchasing Process from 25 ao�t 2020 at 08:00 to 25 ao�t 2020 at 12:00 Register
Overview: From the identification and expression of a need to the evaluation and follow-up of Vendor's performance, this module is a detailed study of the Purchasing Process within the loop of the continuous improvement. Golden rules give a frame of Good Practices.
Audience: This course is primarily designed for Buyers but it is also recommended to requester from Technical Department who will learn how to integrate Lead Time in their schedule, will know how to differentiate a SOR from a SOW, and better express their need in order to optimize the Purchasing Process.
Deliverables: Upon completion of this module Trainees will have a full understanding and control of the Purchasing Process whatever they are in Procurement functions or prescribers in Technical Departments. This module intends to improve the Purchasing Process and increase its efficiency by better interacting Requesters and Buyers.
- Golden Rules & PDCA cycle: 7 Golden Rules in Purchasing – PDCA Cycle applied to Purchasing Process.
Schedule and define the needs: Internal & External Lead Time – Definition of the "Right" need: SQSCI Criteria – SOR/SOW – Technical/Functional Specifications.
Select a Supplier: Qualification of Vendor – Category Management – OEM – Bidder's List.
Purchase: Consultation – Evaluation – Cost Structures – Clarification – Negotiation – Award – Purchasing Documents.
Follow-up and receive: Expediting & Reception of the Purchase Order.
Evaluate Vendor's performance: Vendor's Evaluation Process – Vendor's Progress Plan.


text:
10 Georges St-Pierre Fights We Never Got
Georges St-Pierre has gone down in history as one of the best, if not the
Georges St-Pierre has gone down in history as one of the best, if not the
Georges St-Pierre recently spoke with MMA TRUFAN about his life and career for a new
UFC today announced that former UFC welterweight and middleweight champion Georges St-Pierre will be inducted
For years, MMA fans wanted to see Georges St-Pierre move up to middleweight to battle
Injuries are commonplace when it comes to sports and physical activities. Unfortunately, the human body
Former UFC welterweight champion Tyron Woodley is now training at TriStar Gym in Montreal with
Back at UFC 217, Michael Bisping was looking to defend his middleweight title against Georges
Kevin Lee has been searching for a new place to train full-time and he has
Tyron Woodley was making his case of being the greatest welterweight ever. He had racked
Back in late February, MMA legend, Georges St-Pierre announced his retirement from the sport after


text:
A pensioner has reportedly been paid R4.7 million by the Unemployment Insurance Fund (UIF) as Covid-19 relief…
The Sowetan reported that the 65-year-old pensioner from Hammanskraal, who is a retired bus driver, was mistakenly paid the funds which were meant to go to 1,000 of his former co-workers at Unitrans.
The publication reported that UIF could not explain how the R4.7 million was paid to the pensioner on 25 April instead of it being paid to the company.
The bus company the retiree used to work for had enquired about its application for UIF Covid-19 relief when it found out that the payment had been made to the 65-year-old.
At the time, the pensioner had already spent about R100,000 of the R4.7 million.
UIF spokesperson, Makhosonke Buthelezi reportedly would not say how the money was recovered and would not explain why a case was not opened.
Buthelezi told the publication that the Covid-19 relief fund will be paid to the bus company once an investigation has been concluded.
The Asset forfeiture unit recently froze accounts linked to R5 million in Covid-19 UIF Temporary Employees Relief Scheme (Ters).
The money was allegedly paid into a wrong account before several transfers were made to other accounts in order to launder the money.
(Compiled by Makhosandile Zulu)
For more news your way, download The Citizen's app for iOS and Android.
Citizen reporter – Citizen
The post Pensioner erroneously paid R4.7m from Covid-19 relief fund appeared first on Citizen.
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text:
TOKYO (CBS.MW) -- Tokyo stocks rallied Thursday after Wall Street snapped a four-session slide and the yen came off a fresh three-year high against the dollar.
The Nikkei Average recharged up 251 points, or 2.6 percent, to end at 9,865.70, while the broader Topix climbed 1.9 percent to 971.
Most Asian markets were mixed despite relief on Wall Street, where the Dow gained 0.7 percent and the Nasdaq 1 percent on the back of an upbeat housing report and strong gains for General Electric. Read market snapshot.
South Korea fell again amid a political funding scandal implicating the country's biggest conglomerates. Taipei, Hong Kong, Bangkok, Singapore were lower while Sydney and Shanghai gained.
In Japan, the biggest gainers were in the banking and financial sectors. UFJ Holdings
gained 6.2 percent to 412,000 yen and Mizuho
Some positive domestic developments included a bigger-than-expected gain in the trade surplus for Japan last month and Tokyo Stock Exchange data showing that foreign investors remain net buyers of Japanese stocks.
Kanebo, Isuzu soar
Among other active stocks, Kanebo
jumped 16.3 percent to 114 yen in heavy trade after the Nihon Keizai Shimbun reported that the money-losing cosmetics firm will spin off its cosmetics division into a new company in March and sell a stake to rival Kao
Most-active stock Isuzu Motors
Technology stocks were firm after U.S. computer giant Hewlett-Packard reported a more than doubling of earnings in the fourth quarter as it beat analysts' forecasts. Read full story.
PC rival NEC
rose 3.8 percent to 815, while Fujitsu
The auto sector also regained some zip. Toyota
The dollar traded at 108.95 by late Tokyo, up sharply from a fresh 3-year low of 107.52 in London Wednesday but down from 109.30 yen in late Wednesday trade in New York, where the Bank of Japan reportedly intervened to prop up the greenback. Read currency report.
Asia propels Japan surplus
Japan's trade surplus rose 20.4 percent in October from year-ago levels to 1.07 trillion yen ($9.8 billion), marking gains for the fourth straight month, the finance ministry said. Economists had expected a narrower surplus of around 941 billion yen, according to a Bloomberg poll. Read full story.
The data shows that exports to Asia (and to China especially) are driving Japan's growth as the country seeks to narrow its huge surplus with the United States ahead of next year's presidential election. Recently, economists have been saying that China will replace the U.S. as Japan's top export market within a few years.
Meanwhile, the Bank of Japan started its two-day policy meeting today. While no rate changes are expected, investors will want to know whether the central bank will confirm recent reports about maintaining its ultra-easy monetary policy next fiscal year as well as introduce any market-friendly policies given recent declines in stocks and the stronger yen.
Japanese bond prices fell despite good results from a 20-year bond auction as investors went into stocks. The yield of the key 10-year JGB rose 0.040 point to 1.330 percent.
Nonresident investors remained buyers of Japanese stocks last week with 178.3 billion in net purchases, the TSE said. It was the third straight week foreigners - the biggest driver of the rally this year -- have been net buyers.
Seoul sinks
Among losers in Asia, Seoul's Kospi fell 0.3 percent to 769, Singapore's Straits Times Index shed 0.4 percent to 1,671, Taipei's Weighted Average sank 0.5 percent to 5,834, Hong Kong's Hang Seng Index eased 0.2 percent to 11,845 and Bangkok fell 0.5 percent to 615. On the flip side, Australia's All Ordinaries Index rose 0.1 percent to 3,192 and the Shanghai Composite rose 2.8 percent to 1,376.
In Seoul, stocks sank again but losses were modest compared to Wednesday's 4-percent sell-off that was precipitated by a raid at the offices of an LG group unit as part of a political funding investigation. LG Electronics LECLF rebounded 3.8 percent to 54,900 won.
The big losers were in the banking and credit-card companies amid fears that the latter may be facing a liquidity crunch. South Korea's top credit-card issuer LG Card plunged 10.7 percent to 8,500 won. Samsung Electronics
In Hong Kong, top bank HSBC
In Taipei, United Microelectronics
In Singapore, Chartered Semiconductor


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Everything posted by grandpa
grandpa replied to nonuniqueipaddress's topic in Workshops, Institutes, and Campsyeah, none of the fellows from uk ever went to the toc and, two teams from my high school that had people go to the toc that went to uk really didn't and, uk's debate team sucks even though they made it to semis at the toc in 2003 finally, harvard kt are the worst team ever even though they won the copland and are teaching at uk
Year end review: what worked, what didn't
grandpa posted a topic in United NationsI'm interested to hear what people thought were the main reasons they won/lost aff rounds this year. Particularly, I'm interested in people discussing specific aspects of cases they ran for a long time this year, and what strategies they found most/least effective. Just curious
Passive Millimeter Aff
grandpa replied to downward747's topic in Civil LibertiesOlmstead isn't the standard for this kind of search in the status quo. Check out Kyllo v. U.S. 533 u.s. 121 (2001). In summary (paraphrased and accurate to the best of my knowledge), the supreme court held that thermal imaging devices do violate the fourth amendment (is an illegal search)when law enforcement uses technology that isn't genrally in useabout to search that person's home without a warrant. "..where, as here, the government uses a device that is not in general public use, to explore detials of a home that would previously have been unknowable without physical intrusion, the surveillance is a search and is presumptively unreasonable w/o a warrant". (Scalia, from the majority in Kyllo).
out of round impact: "make them rewrite the case"Hey all- Ran into a critical argument that I haven't heard before several times this while judging. The critique's thesis was the U.N. is a front for western colonialism, with genocide/biopower implications. The link was basic "u.n. action = colonialism, you use the U.N. for PKO so you link." Anyways, during c-x the aff asked the neg why the neg had any kind of pre-fiat out of round implications. The neg replied, "well, our argument will persuade you to rewrite your case so it's not colonialist, and that's how our movement will start." The aff asked, "well we don't have any intention of rewriting our case based on this critique, so why do you have prefiat implications?" neg answers "well, if you don't rewrite the case, you're entrenching colonialist thinking and you lose the round based on the fact you reject our movement in the real world". Another team made this same argument about "we force them to be less _____ist and rewrite their case, starting our movement" (from a different school) and I dropped them for the same framework/theory reasons. I totally didn't buy this. The aff argued convincingly that they could NEVER rewrite their case to be 'less colonialistic' since the critique's link if off of generic u.n. action, and that there was no chance of the aff rewriting their case after the round since they just wouldn't do it, and therefore there was no prefiat/out of round implications. The aff also extended the usual policymaking/roleplaying/realism good framework defense. I vote aff, since the neg goes big for the critique in 2nr (and the block dropped a link turned disad COLD that the aff extended in the 1ar). Give an oral critique, and the neg team goes ballistic, basically arguing 'they didn't read enough evidence, we did a better job with the critique on the flow and the disads don't matter because of our out of round implications'. Am I missing something theoretically here? I think I made the right decision, but the team I judged travels nationally and went to decent camps. If I don't know what's going on, educate me. I'm not a critique expert, but I think this argument is bogus.
grandpa replied to TIMMIS AND CHRIS CHENG's topic in Great Lakesnew alliance=solid I smell top seed and 2nd and 5th speaker awards, oil mechanix ownz u
Office Space
grandpa replied to ZiggyStardust's topic in Great Lakeswhere is my stapler???
The Empire has no Impacts...
grandpa replied to Ithican Angel's topic in CritiquesHardt and Neghri, lil buddy. Your friendly neighborhood librarian will hook you up-it's either in the library or you can get it on loan from another library or from whatever college is close by.
A Serious Question for WGLF
grandpa replied to Da Fightin' Materialist's topic in CritiquesBush reelection = 911 * 484149849489846514351564984864653416531464164863546354 I don't know if that's worse than the sun explosion or dinosaurs though
I understand where you're coming from, but I think that rolling with shannahan's radical critique of debate only works for a few teams. If you're on the national circuit and you have the right panel and you were mentored by shannahan at ddi, it may work out. however, I don't think most people in this thread have the theoretical background or the judging pool to pull it off. although I understand your arguments, one thing that can make this position disadvantageous is time tradeoff. the argument is very abstract, takes a lot of time to explain, and it takes the negative a lot more time to answer the affs positions (if they read spanos indicts, talk about the context of the evidence, make realism good arguments, perm arguments, fiat theory arguments). Although teams like the Haze, WGLF, and Fullerton can pull it off, I think there are easier strategies that would work better for the vast majority of debaters.
Spending scenarios
grandpa replied to teenydude8's topic in Disadvantages and CounterplansI've heard a NASA tradeoff scenario a frightening amount of times-people love that space impacts debate
Here's what Spanos said: Dear Matt, I have been aware of the way high school (and college) debaters have mangled my work for quite some time now. I don't know how my books got into that circuit. My intentions in writing them was/is not to appeal to debate. The very idea of institutionalized debate--where it doesn't matter at all what side one takes--is anathema to my way of thinking. In fact, you could say that my books in a fundamental way are intended to demolish the phony "pluralistic" thinking that kind of debating foster. Thinking (and the language we use to make it manfifest), in other words, should make a difference in a world, especially in the United States, which calls itself civilized and free but in reality is barbarically corrupt and unfree, which always invokes the language of justice to conceal the terrible injustices it perpetrates at home and everywhere in the world, that celebrates the individual while reducing him her to what Foucault calls useful and docile body and Heidegger, "disposable reserve. And this is precisely what the kind of "thinking" debating (as practiced in debate tournaments) IS NOT. But the fact is that, like or not, my work has been appropriated to this debate scene. And there's not much I can do about it. Who could are people like you, who seem to be aware of the fact that there's something rotten in the debating state Denmark, that it actually violates the very essence of the kind of thinking I am struggling to articulate. So, I understand your renunciation of high shcool debating. But if you really want to make your opposition felt, you should return to the debate circuit and make it your purpose to challenge the reductive distortions your debater colleagues impose on my kind of thinking. _________________________ Although Shannahan is a very smart and innovative thinker, it seems Spanos himself believes his works are distorted as currently ran in debate rounds.
to the o.p.- I think all the info you need is at http://www.ndtceda.com/schoofs
grandpa replied to ixlovexmex02's topic in Help Me...I agree with Darkness and Ankur
Establishing a UNPKO in Federal Indian Country
grandpa replied to kritikingforlife's topic in United NationsExactly. This thing is the biggest turkey ever.
Establishing a UNPKO in Federal Indian Country
grandpa replied to kritikingforlife's topic in United NationsWGLF just made me spit my coffee all over my computer screen from laughing
grandpa replied to Cobra_Debate's topic in United NationsDon't run this critically-ESPECIALLY if you are a novice.
grandpa replied to killabee331's topic in Novice CenterListen to Ankur. Don't waste your time with this junk. Run another disad, a T violation, more case ev-anything but ASPEC. Why complain? If they say 'fed gvt' and 'normal means', then you jsut have more disads that link, right?
grandpa replied to hot_kabbage's topic in United NationsThere is no plan text because this case has no solvency advocate, and it is a stupid whine that was created so debaters can feel like they are cool liberal radicals.
grandpa replied to seattle's topic in Evidence TradingI've got a small file-I'm willing to trade-pm me
old resolutionsI think: the usfg should substantially increase social services to homeless individuals in the u.s. the usfg should substantially reduce worldwide pollution through its trade and/or aid policies
grandpa replied to newsasiknowit's topic in Help Me...pm me and i can get you some cites
So all the af has to do to win the round is to play a hip hop song that criticizes the system and you win under your theory? Gee, I guess there's no point in even debating anything, since the aff (under your theory) doesn't have to prove any stock issues, magically doesn't link to disads, cannot be critiqued because "helping the resistance" or "make the choice not to participate in violent representations" is the only aff burden? Also, just because you have some card that has some author (gibson-graham) saying your theory of communication is correct doesn't automatically kill the logic behind the negative arguments. Also, the position turns itself-you say participating in violent representations is bad-doesn't hip hop, I dunno, talk about violence?
please help i have a 1AC due tommarow
grandpa replied to meadfill's topic in Help Me...meadfill- If you need some RRF inherency cites, pm me and I'll get you some. 'Paw
Narrative AFFs
grandpa replied to jmiller's topic in Evidence TradingAgreed
kind of out there topicality
grandpa replied to crazyaboutbroccoli's topic in Help Me...nah, it's not pointless-at least you didn't run it in the round-then you would feel silly AND probably lose-better to ask before u do 'paw


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Published 12 days ago
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Nerd Confession
The bad boy good looks was enough, until i grew up and wondered how it would feel to have a man treat me well and expand the horizons of my mind. Someone i viewed to be just a friend listened to my troubles for months about the bad boys in my life. Little did i know, he would be the one who showed me there was another kind of man out there that would open up my mind to things i never thought of before.
Published 19 days ago
19 days ago
Black Lives Matter
Joining together sharing our view points regarding the tragic murders of innocent black citizens across the united states teaches awareness as well as our own experiences living in america. Our voices create a positive impact as we unity together as one human race. Lets draw awareness in the motive of our actions to become a better people as a whole. If you are a non black citizen, ask your self these questions:
The Swamp
Published about a month ago
about a month ago
Top underrated dramatic,TV Shows
Unfortunately television shows involving unique groups of friends with extraordinary experiences become unnoticed. Shows from people searching for happiness and life saving experiences should not become spoilers. I am here to tell you about my top three underrated television shows that deserve more credit from the television network. My top three unforgettable shows are: "Your the Worst", "The Last Airbender", and "Jane the Virgin".
Published about a month ago
about a month ago
My Furry Babies
Every time i role play with my furry babies it create adventurous times in the yard. This photo is myself in character as Pippi Long stockings, along with my ferrets, and my dog getting into a world of fun. Euros is my four year old male ferret who has black eyes posing side ways for the camera. Dj is my four year old female ferret with red eyes, who is very active. During play time she usually tries jumping off of my shoulders. Trying to hold onto DJ and make sure she doesn't get lost is quite the challenge. Playing Pippi long stockings and trying to keep up with Dj had me tumbling to my side though i was having a joyful time.
Published about a month ago
about a month ago
Comfort Warriors
How to overcome fear
Published about a month ago
about a month ago
Roaring 20's Party
Attending the Jitterbugs 1920's party seems like it was so long ago, though it felt like an ageless time in history. This is a flash back of a photo of me home after getting dressed to go to a party. Prior to the party, i researched the clothing i should have worn for the party. Once i picked out my outfit took a snap shot of myself in my nineteen twenties apparel. Then i went to the jazziest parties i have ever been to.
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