### Divide and Conquer

```COSC 3100
Divide and Conquer
Instructor: Tanvir
What is Divide and Conquer ?
• 3rd algorithm design technique, we
are going to study
• Very important: two out of ten most
influential algorithms in the
twentieth century is based directly
on “Divide and Conquer” technique
– Quicksort (we shall study today)
– Fast Fourier Transform (we shall not
study in this course, but any “signal
processing” course is based on this idea)
Divide and Conquer
• Three steps are involved:
– Divide the problem into several
subproblems, perhaps of equal size
– Subproblems are solved, typically
recursively
– The solutions to the subproblems are
combined to get a solution to the
original problem
Real work is done in 3 different places: in partitioning; at the very tail end
of the recursion, when subproblems are so small that they are solved directly;
and in gluing together of partial solutions
Divide and Conquer (contd.)
Problem
Subproblem 1
of size n/2
Solution to
subproblem 1
of size n
Don’t assume
always breaks up
into 2, could be > 2
subproblems
Solution to the
original probelm
Subproblem 2
of size n/2
Solution to
subproblem 2
Divide and Conquer (contd.)
• In “politics” divide and rule (latin:
divide et impera) is a combination of
political, military, and economic
strategy of gaining and maintaining
power by breaking up larger
concentrations of power into chunks
that individually have less power than
the one who is implementing the
strategy. (read more on wiki: “Divide
and rule”)
Divide and Conquer (contd.)
• Let us add n numbers using divide and
conquer technique
a0 + a1 + …… + an-1
a0 + …… +
−

+ …… + an-1
Is it more efficient than brute force ?
Let’s see with an example
Div. & Conq. (add n numbers)
2
0
1
2
3
4
5
6
7
8
9
2
10
3
5
7
1
6
10
1
3
0
1
2
3
4
2
10
3
5
7
10
12
3
5
3
7
5
7
12
0
1
is same as in
brute force,
needs stack 1
for recursion…
not all divide
and conquer
works!!
1
6
6
7
2
10
10
3
1
1
10
4
3
3
1
3
4
14
15
21
27
48
Could be efficient
for parallel processors though….
Div. & Conq. (contd.)
• Usually in div. & conq., a problem instance of size n
is divided into two instances of size n/2
• More generally, an instance of size n can be
divided into b instances of size n/b, with a of
them needing to be solved
• Assuming that n is a power of b (n = bm), we get
general divide-and-conquer
– T(n) = aT(n/b) + f(n)
recurrence
– Here, f(n) accounts for the time spent in dividing an
instance of size n into subproblems of size n/b and
combining their solution
– For adding n numbers, a = b = 2 and f(n) = 1
Div & Conq. (contd.)
• T(n) = aT(n/b)+f(n), a ≥ 1, b > 1
What if a = 1?
• Master Theorem:
Have we seen it?
If f(n) є Θ(nd) where d ≥ 0 then
T(n) є
Θ(nd)
if a < bd
So, A(n) є Θ(  )
Or, A(n) є Θ(n)
Θ(ndlgn)
if a = bd
Θ(  )
if a > bd
Without going through
back-subs. we got it,
but not quite…
For adding n numbers with divide and conquer technique, the number
A(n) = 2A(n/2)+1
Here, a = ?, b = ?, d = ?
a = 2, b = 2, d = 0
Which of the 3 cases holds ? a = 2 > bd = 20, case 3
Div. & Conq. (contd.)
T(n) = aT(n/b)+f(n), a ≥ 1, b > 1
If f(n) є Θ(nd) where d ≥ 0, then
T(n) = 2T(n/2)+6n-1?
T(n) є
T(n) = 3 T(n/2) + n
a=3 > bd=21
T(n) = 3 T(n/2) + n2
a=3 < bd=22
T(n) = 4 T(n/2) + n2
a=4 = bd=22
Θ(nd)
Θ(ndlgn)
if a < bd
if a = bd
Θ(  )
if a > bd
a = 3, b = 2, f(n) є Θ(n1), so d = 1
Case 3:
T(n) є Θ(   ) = Θ( . )
a = 3, b = 2, f(n) є Θ(n2), so d = 2
Case 1:
T(n) є Θ(  )
a = 4, b = 2, f(n) є Θ(n2), so d = 2
Case 2:
T(n) є Θ(  lgn )
T(n) = 0.5 T(n/2) + 1/n
Master thm doesn’t apply,a<1, d<0
Master thm doesn’t apply f(n) not polynomial
T(n) = 2 T(n/2) + n/lgn
T(n) = 64 T(n/8) – n2lgn f(n) is not positive, doesn’t apply
T(n) = 2n T(n/8) + n a is not constant, doesn’t apply
Div. & Conq.: Mergesort
• Sort an array A[0..n-1]
A[0……n-1]
A[0……  2 − 1]
divide
A[  2 ……n-1]
sort
sort
A[  2 ……n-1]
A[0……  2 − 1]
merge
A[0……n-1]
Go on dividing recursively…
Div. & Conq.: Mergesort(contd.)
ALGORITHM Mergesort(A[0..n-1])
//sorts array A[0..n-1] by recursive mergesort
//Input: A[0..n-1] to be sorted
//Output: Sorted A[0..n-1]
if n > 1
copy A[0..  2 -1] to B[0..  2 -1]
copy A[  2 ..n-1] to C[0..  2 -1]
Mergesort(B[0..  2 -1])
Mergesort(C[0..  2 -1])
Merge(B, C, A)
B:
2
3
8
A:
9
1
C:
2
3
4
1
4
5
5
7
7
8
9
Div. & Conq.: Mergesort(contd.)
ALGORITHM Merge(B[0..p-1], C[0..q-1], A[0..p+q-1])
//Merges two sorted arrays into one sorted array
//Input: Arrays B[0..p-1] and C[0..q-1] both sorted
//Output: Sorted array A[0..p+q-1] of elements of B and C
i <- 0; j <- 0; k <- 0;
while i < p and j < q do
if B[i] ≤ C[j]
A[k] <- B[i]; i <- i+1
else
A[k] <- C[j]; j <- j+1
k <- k+1
if i = p
copy C[j..q-1] to A[k..p+q-1]
else
copy B[i..p-1] to A[k..p+q-1]
Div. & Conq.: Mergesort(contd.)
Divide:
Merge:
8
8
8
3
2
2
2
3
2
8
1
1
9
5
7
9
2
3
7
9
1
4
5
5
7
4
5
7
8
4
4
4
9
4
5
1
1
3
1
1
7
9
2
4
7
2
8
9
9
3
8
3
3
5
7
5
Div. & Conq.: Mergesort(contd.)
ALGORITHM Mergesort(A[0..n-1])
//sorts array A[0..n-1] by recursive
mergesort
//Input: A[0..n-1] to be sorted
//Output: Sorted A[0..n-1]
if n > a
copy A[0..  2 -1] to B[0..  2 -1]
copy A[  2 ..n-1] to C[0..  2 -1]
Mergesort(B[0..  2 -1])
Mergesort(C[0..  2 -1])
Merge(B, C, A)
What is the time-efficiency
of Meresort?
Input size: n = 2m
Basic op: comparison
C(n) depends on input
type…
ALGORITHM Merge(B[0..p-1], C[0..q-1], A[0..p+q-1])
//Merges two sorted arrays into one sorted array
//Input: Arrays B[0..p-1] and C[0..q-1] both sorted
//Output: Sorted array A[0..p+q-1] of elements of B
//and C
i <- 0; j <- 0; k <- 0;
while i < p and j < q do
if B[i] ≤ C[j]
A[k] <- B[i]; i <- i+1
else
A[k] <- C[j]; j <- j+1
k <- k+1
if i = p
else
copy C[j..q-1] to A[k..p+q-1]
copy B[i..p-1] to A[k..p+q-1]
C(n) = 2C(n/2) + CMerge(n) for n > 1,
C(1) = 0
CB:worst
C: (n/2)+n-1
3 4 9 for n > 1
2 (n)
5 =82Cworst
In worst-case Cworst(1) = 0
CMerge(n) = n-1
A:
3 =4nlgn-n+1
5 8 9є Θ(nlgn)
C 2 (n)
worst
Could use the master thm too!
How many comparisons
is needed for this Merge?
Div. & Conq.: Mergesort(contd.)
•
•
•
•
Worst-case of Mergesort is Θ(nlgn)
Average-case is also Θ(nlgn)
It is stable but quicksort and heapsort are not
Possible improvements
– Implement bottom-up. Merge pairs of elements, merge
the sorted pairs, so on… (does not require recursionstack anymore)
– Could divide into more than two parts, particularly useful
for sorting large files that cannot be loaded into main
memory at once: this version is called “multiway
mergesort”
• Not in-place, needs linear amount of extra
memory
– Though we could make it in-place, adds a bit more
“complexity” to the algorithm
Div. & Conq.: Quicksort
• Another divide and conquer based
sorting algorithm, discovered by C.
A. R. Hoare (British) in 1960 while
trying to sort words for a machine
translation project from Russian to
English
• Instead of “Merge” in Mergesort,
Quicksort uses the idea of
seen with “Lomuto Partition”
Div. & Conqr.: Quicksort (contd.)
A[0]…A[n-1]
A[0]…A[s-1] A[s] A[s+1]…A[n-1]
all are ≤ A[s]
all are ≥ A[s]
Notice:
A[s] is in it’s
In Mergesort
final position
all work is in
combining the
partial solutions.
Now continue working with
In Quicksort all
these two parts
work is in dividing
the problem,
Combining does not require any work!
Div. & Conqr.: Quicksort (contd.)
ALGORITHM Quicksort(A[l..r])
//Sorts a subarray by quicksort
//Input: Subarray of array A[0..n-1] defined by its
//left and right indices l and r
//Output: Subarray A[l..r] sorted in nondecreasing
//order
if l < r
s <- Partition( A[l..r] ) // s is a split position
Quicksort( A[l..s-1] )
Quicksort( A[s+1]..r )
Div. & Conqr.: Quicksort (contd.)
• As a partition algorithm we could use
“Lomuto Partition”
• But we shall use the more sophisticated
• We start by selecting a “pivot”
• There are various strategies to select the
pivot, we shall use the simplest: we shall
select pivot, p =A[l], the first element of
A[l..r]
Div. & Conqr.: Quicksort (contd.)
p
p
≤p
≥p
p
i
j
If A[i] < p, we continue incrementing i, stop when A[i] ≥ p
If A[j] > p, we continue decrementing j, stop when A[j] ≤ p
p
all ≤ p
p
all ≤ p
p
j
≤p
i
≥p
all ≤ p
j i
≤p ≥p
all ≥ p
all ≥ p
j=i
=p
all ≥ p
Div. & Conqr.: Quicksort (contd.)
ALGORITHM HoarePartition(A[l..r])
//Output: the split position
p <- A[l]
i could go out of array’s
bound, we could check
i <- l; j <- r+1
or we could put a “sentinel”
at the end…
repeat
Do you see any
repeat i <- i+1 until A[i] ≥ p
possible problem
with this pseudocode ? repeat j <- j-1 until A[j] ≤ p
swap( A[i], A[j] )
until i ≥ j
swap( A[i], A[j] ) // undo last swap when i ≥ j
swap( A[l], A[j] )
return j
More sophisticated pivot selection
that we shall see briefly makes this
“sentinel” unnecessary…
Div. & Conqr.: Quicksort (contd.)
j
i
5 3 1 9 8 2 4 7
j
i
2 3 1 4 5 8 9 7
j
i
5 3 1 9 8 2 4 7
i j
2 3 1 4 5 8 9 7
j
i
5 3 1 4 8 2 9 7
i j
2 1 3 4 5 8 9 7
i j
5 3 1 4 8 2 9 7
j i
2 1 3 4 5 8 9 7
i j
5 3 1 4 2 8 9 7
1 2 3 4 5 8 9 7
j i
5 3 1 4 2 8 9 7
1 2 3 4 5 8 9 7
2 3 1 4 5 8 9 7
i=j
1 2 3 4 5 8 9 7
j i
1 2 3 4 5 8 9 7
1 2 3 4 5 8 9 7
i j
1 2 3 4 5 8 9 7
i j
1 2 3 4 5 8 7 9
j i
1 2 3 4 5 8 7 9
1 2 3 4 5 7 8 9
1 2 3 4 5 7 8 9
1 2 3 4 5 7 8 9
Div. & Conqr.: Quicksort (contd.)
0
1
2
3
4
5
6
7
0 1 2 3 4 5 6 7
5 3 1 9 8 2 4 7
l=0,r=7
0 1 2 3 4 5 6 7
s=4
2 3 1 4 5 8 9 7
l=5,r=7
l=0,r=3
s=1
l=0,r=0
5 3 1 9 8 2 4 7
s=6
l=2,r=3
s=2
l=2,r=1
l=5,r=5
l=3,r=3
l=7,r=7
0 1 2 3
5 6 7
1 2 3 4
7 8 9
0
5
1
7
2 3
7
3 4
9
3
4
Div. & Conqr.: Quicksort (contd.)
• Let us analyze Quicksort
ALGORITHM Quicksort(A[l..r])
Time-complexity of this line ?
if l < r
i j
s <- HoarePartition ( A[l..r] ) So, n+1
5 3 1 4 8 2 9 7
comparisons
Quicksort( A[l..s-1] )
i j
when cross-over
Quicksort( A[s+1]..r )
If all splits
happen in the
middle, it is
the best-case!
5 3 1 4 2 8 9 7
j i
5 3 1 4 2 8 9 7
ALGORITHM HoarePartition(A[l..r])
//Output: the split position
So, n
p <- A[l]
comparisons
i <- l; j <- r+1
when coincide What if,
repeat
repeat i <- i+1 until A[i] ≥ p
5 3 1
repeat j <- j-1 until A[j] ≤ p
swap( A[i], A[j] )
until i ≥ j
swap( A[i], A[j] ) // undo last swap when i ≥ j
swap( A[l], A[j] )
Cbest(n) = 2Cbest(n/2)+n for
return j
Cbest(1) = 0
i,j
4 5 8 9 7
n > 1
Div. & Conqr.: Quicksort (contd.)
T(n) = aT(n/b)+f(n), a ≥ 1, b > 1
If f(n) є nd with d ≥ 0, then
T(n) є
Θ(nd)
if a < bd
Θ(ndlgn)
if a = bd
Θ(  )
if a > bd
ALGORITHM Quicksort(A[l..r])
if l < r
s <- Partition( A[l..r] )
Quicksort( A[l..s-1] )
Quicksort( A[s+1]..r )
j
2
Cbest(n) = 2Cbest(n/2)+n for n > 1
Cbest(1) = 0
ij j j j
5 6 8 9
j
i
5 6 8 9
Using Master Thm, Cbest(n) є Θ(nlgn)
What is the worst-case ?
Cworst(n) = (n+1) + (n-1+1) + … + (2+1) = (n+1) + … + 3
= (n+1) + … + 3 + 2 + 1 – (2 + 1) = +1
- 3
1
=
+1 +2
2
- 3 є Θ(n2) !
6
5+1=6
4+1=5
j
9
3+1=4
ij
8 9
2+1=3
i
8
So, Quicksort’s fate
depends on its average-case!
Div. & Conqr.: Quicksort (contd.)
• Let us sketch the outline of averageALGORITHM Quicksort(A[l..r])
case analysis…
if l < r
s <- Partition( A[l..r] )
Cavg(n) is the average number of key-comparisons
Quicksort( A[l..s-1] )
made by the Quicksort on a randomly ordered array
Quicksort( A[s+1]..r )
of size n
After n+1 comparisons, a partition can
happen in any position s (0 ≤ s ≤ n-1)
Let us assume that
partition split can
happen in each position s
with equal probability 1/n
After the partition, left part has s elements,
Right part has n-1-s elements
s
0
n-1
Cavg(n) = Expected[ Cavg(s) + Cavg(n-1-s) + (n+1) ]
p
Average over all possibilities
s elements
n-1-s elements

−
Cavg(n) =
[ Cavg(s) + Cavg(n−1−s) + (n+1) ]
=
Cavg(0) = 0, Cavg(1) = 0 C (n) ≈ 1.39nlgn
avg
Div. & Conqr.: Quicksort (contd.)
• Recall that for Quicksort, Cbest(n) ≈ nlgn
• So, Cavg(n) ≈ 1.39nlgn is not far from Cbest(n)
• Quicksort is usually faster than Mergesort or
Heapsort on randomly ordered arrays of nontrivial
sizes
• Some possible improvements
• Randomized quicksort: selects a random element as pivot
• Median-of-three: selects median of left-most, middle, and
right-most elements as pivot
• Switching to insertion sort on very small subarrays, or not
sorting small subarrays at all and finish the algorithm with
insertion sort applied to the entire nearly sorted array
• Modify partitioning: three-way partition
These improvements can speed up by 20% to 30%
Div. & Conqr.: Quicksort (contd.)
• Weaknesses
– Not stable
– Requires a stack to store parameters of
subarrays that are yet to be sorted, the
stack can be made to be in O(lgn) but
that is still worse than O(1) space
efficiency of Heapsort
DONE with Quicksort!
Div. & Conq. : Multiplication of
Large Integers
• We want to efficiently multiply two very
large numbers, say each has more than 100
decimal digits
• How do we usually multiply 23 and 14?
• 23 = 2*101 + 3*100, 14 = 1*101 + 4*100
• 23*14 = (2*101 + 3*100) * (1*101 + 4*100)
• 23*14 = (2*1)102 + (2*4+3*1)101+(3*4)100
• How many multiplications? 4 = n2
Div. & Conq. : Multiplication of
Large Integers
• 23*14 = (2*1)102 + (2*4+3*1)101+(3*4)100
We can rewrite the middle term as:
(2*4+3*1) = (2+3)*(1+4) - 2*1 - 3*4
What has been gained?
We have reused 2*1 and 3*4 and now need
one less multiplication
If we have a pair of 2-digits numbers a and b
a = a1a0 and b = b1b0
we can write c = a*b = c2102+c1101+c0100
c2 = a1*b1
c0 = a0*b0
c1 = (a1+a0)*(b1+b0)-(c2+c0)
Div. & Conq. : Multiplication of Large
Integers
If we have a pair of 2-digits numbers a and b
a = a1a0 and b = b1b0
we can write c = a*b = c2102+c1101+c0100
c2 = a1*b1 , c0 = a0*b0
c1 = (a1+a0)*(b1+b0)-(c2+c0)
If we have two n-digits numbers,
a and b (assume n is a power of 2)
a:
a1
a0
b:
b1
b0
We can write,
a = a110n/2 + a0
b = b110n/2 + b0
a = 1234 = 1*103+2*102+3*101+4*100
= (12)102+(34)
(12)102+(34) = (1*101+2*100)102+3*101+4*100
Apply the same idea
recursively to get c2, c1, c0
until n is so small that you can
you can directly multiply
n/2 digits
c = a*b = 1 10 2 + 0 ∗ 1 10 2 + 0
= 1 ∗ 1 10 + 1 ∗ 0 + 0 ∗ 1 10
= 2 10 + 1 10 2 + 0
Why?
c2 = a1*b1
c0 = a0*b0
c1 = (a1+a0)*(b1+b0)-(c2+c0)
2
+ 0 ∗ 0
Div. & Conq. : Multiplication of Large
Integers
Notice: a1, a0, b1, b0 all are n/2
digits numbers
So, computing a*b requires
three n/2-digits multiplications
Recurrence for the number of
Multiplications is
M(n) = 3M(n/2) for n > 1
M(1) = ?
c2 = a1*b1
2
+ 0
1 subtraction
c0 = a0*b0
c1 = (a1+a0)*(b1+b0)-(c2+c0)
Assume n = 2m
M(n) = 3M(n/2) = 3[ 3M(n/22) ] = 32M(n/22)
And subtractions?
A(n) = 3A(n/2)+cn for n > 1
A(1) = 0
c = a*b = 2 10 + 1 10
Using Master Thm,
A(n) є Θ(nlg3)
M(n) = 3mM(n/2m) = ?
M(n) = 3m = 3lgn = nlg3
Why?
M(n) ≈ n1.585
Let   = x
=
=
x =
Div. & Conq. : Multiplication of Large
Integers
• People used to believe that multiplying two ndigits number has complexity Ω(n2)
• In 1960, Russian mathematician Anatoly
Karatsuba, gave this algorithm whose asymptotic
complexity is Θ(n1.585)
• A use of large number multiplication is in modern
cryptography
• It does not generally make sense to recurse all
the way down to 1 bit: for most processors 16- or
32-bit multiplication is a single operation; so by
this time, the numbers should be handed over to
built-in procedure
Next we see how to multiply
Matrices efficiently…
Div. & Conq. : Strassen’s Matrix
Multiplication
• How do we multiply two 2×2 matrices ?
1
2
3
5
3
4
1
4
=
5
13
13
31
How many multiplications
V. Strassen in 1969 found out, he can
do the above multiplication in the following
way:
m1+m4-m5+m7 m3+m5
c
c
00
01
a00 a01
b00 b01
=
=
m2+m4
m1+m3-m2+m6
c10 c11
a10 a11
b10 b11
m1 = (a00+a11)*(b00+b11)
m2 = (a10+a11)*b00
m4 = a11*(b10-b00)
m3 = a00*(b01-b11)
m5 = (a00+a01)*b11
m6 = (a10-a00)*(b00+b01) m7 = (a01-a11)*(b10+b11)
7 mults
Div. & Conq. : Strassen’s Matrix
Multiplication
• Let us see how we can apply Strassen’s idea for
multiplying two n×n matrices
Let A and B be two n×n matrices where n is a power of 2
A00 A01
B00 B01
A10 A11
B10
B11
=
Each block is (n/2)×(n/2)
You can treat blocks
as if they were numbers
to get the C = A*B
C00 C01
C10
C11
E.g., In Strassen’s
method,
M1 = (A00+A11)*(B00+B11)
M2 = (A10+A11)*B00
etc.
Div. & Conq. : Strassen’s Matrix
Multiplication
ALGORITHM Strassen(A, B, n)
//Input: A and B are n×n matrices
//where n is a power of two
//Output: C = A*B
if n = 1
return C = A*B
else
A00 A01
Partition A =
A10 A11
and B =
Recurrence for
# of multiplications is
M(n) = 7M(n/2) for n > 1
M(1) = ?
B00 B01
B10
B11
where the blocks Aij and Bij are (n/2)-by-(n/2)
M1 <- Strassen(A00+A11, B00+B11, n/2)
M2 <- Strassen(A10+A11, B00, n/2)
M3 <- Strassen(A00, B01-B11, n/2)
M4 <- Strassen(A11, B10-B00, n/2)
M5 <- Strassen(A00+A01, B11, n/2)
M6 <- Strassen(A10-A00, B00+B01, n/2)
M7 <- Strassen(A01-A11, B10+B11, n/2)
C00 <- M1+M4-M5+M7
C01 <- M3+M5
C10 <- M2+M4
C11 <- M1+M3-M2+M6
return C =
C00 C01
C10
C11
For n = 2m,
M(n) = 7M(n/2) = 72M(n/22)
M(n) = 7m M(n/2m)
= 7m = 7lgn
= nlg7 ≈ n2.807
A(n) = 7A(n/2)+18(n/2)2 for n > 1
A(1) = 0
Using Master thm,
A(n) є Θ(nlg7) better
than brute-force’s Θ(n3)
DONE WITH STRASSEN!
Div. & Conq.: Closest pair
• Find the two closest points in a set of n
points
Traffic
Control:
detect two
vehicles
most likely
to collide
ALGORITHM BruteForceClosestPair(P)
//Input: A list P of n (n≥2) points p1(x1,y1),
//p2(x2,y2), …, pn(xn,yn)
//Output: distance between closest pair
d <- ∞
for i <- 1 to n-1 do
for j <- i+1 to n do
d <- min( d, sqrt(  −
2
+  −
2
))
return d
Idea: consider each pair of points and keep track of the
pair having the minimum distance
There are
n
(−1)
pairs, so time-efficiency is in Θ(n2)
=
2
2
Div. & Conq.: Closest pair (contd.)
• We shall apply “divide and conquer”
technique to find a better solution
Let P be the
set of points
sorted by
x-coordinates
Any idea how to divide and then
conquer?
x=m
dl
let Q be the
same points sorted
by y-coordinates
Left portion
dr
Right portion
Solve right and left
portions recursively
and then combine
the partial solutions
How should we
combine?
d = min {dl, dr} ?
Does not work,
because one point
can be in left portion
and the other could be
in right portion having
distance < d between
them…
Div. & Conq.: Closest pair (contd.)
x=m
d = min{dl, dr}
We wish to find a pair
having distance < d
It is enough to consider
the points inside the symmetric
vertical strip of width 2d around
the separating line! Why?
dl
dr
Left portion
Right portion
d
d
Because the distance
between any other pair
of points is at least d
We shall scan through S, updating
But S can
contain all
points, right? dmin = d using brute-force.
Let S be the list of points
inside the strip of width 2d
obtained from Q, meaning ?
Let p(x, y) be a point in S.
for a point p’(x’, y’) to have a
chance to be closer to p than dmin,
p’ must “follow” p in S and the
difference bewteen their
y-coordinates must be less than dmin
Div. & Conq.: Closest pair (contd.)
x=m
dl
dr
Left portion
Let p(x, y) is a point in S.
For a point p’(x’, y’) to have a
chance to be closer to p than dmin,
p’ must “follow” p in S and
the difference bewteen their
y-coordinates must be less
than dmin Why?
It seems, this rectangle
Right portion
d
Geometrically,
p’ must be in
the following
rectangle
My claim is “this”
is the most you can
put in the rectangle…
x=m
d
d
dmin
d
Now comes the crucial observation,
everything hinges on this one…
How many points can there
be in the dmin-by-2d rectangle?
can contain many
points, may be all…
p
x=m
d
d
d
One of these 8
being p, we need
to check 7 pairs
to find if any pair
has distance < dmin
Div. & Conq.: Closest pair (contd.)
ALGORITHM EfficientClosestPair(P, Q)
//Solves closest-pair problem by divide and conquer
//Input: An array P of n ≥ 2 points sorted by x-coordinates and another array Q of same
//points sorted by y-coordinates
The algorithm spends linear time
in dividing and merging, so assuming
//Output: Distance between the closest pair
n = 2m, we have the following
if n ≤ 3
recurrence for runnning-time,
return minimal distance by brute-force
else
T(n) = 2T(n/2)+f(n) where f(n) є Θ(n)
copy first  2 points of P to array Pl
copy the same  2 points from Q to array Ql
Applying Master Theorem,
copy the remaining  2 points of P to array Pr
T(n) є Θ(nlgn)
copy the same  2 points from Q to array Qr
dl <- EfficientClosestPair( Pl, Ql )
dr <- EfficientClosestPair( Pr, Qr )
d <- min{ dl, dr }
Dividing line
Brute froce
m <- P[  2 -1].x
Inside the
Points in 2*d
copy all points of Q for which |x-m| < d into array S[0..num-1]
2*d width strip
width strip
dminsq <- d2
for i <- 0 to num-2 do
k <- i+1
while k ≤ num-1 and ( S[k].y – S[i].y )2 < dminsq
dminsq <- min( (S[k].x-S[i].x)2+(S[k].y-S[i].y)2 , dminsq)
k <- k+1
return sqrt( dminsq ) // could easily keep track of the pair of points
```
### Content Structure
The text passage is a detailed lecture or course material on the "Divide and Conquer" algorithm design technique. It covers various algorithms and problems that use this technique, including Quicksort, Mergesort, multiplication of large integers, Strassen's matrix multiplication, and the closest pair problem. The passage includes explanations, pseudocode, and time complexity analyses for these algorithms.

### Analysis
multiple_questions_with_answers

### explanation
1. **multiple questions**: the passage contains several mathematical questions and problems, such as:
   - how to add n numbers using the divide and conquer technique.
   - time complexity analysis of various divide and conquer algorithms (e.g., mergesort, quicksort).
   - how to multiply large integers efficiently.
   - how to multiply matrices using strassen's method.
   - how to find the closest pair of points in a set.

2. **answers provided**: for each of these questions, the passage provides detailed explanations, pseudocode, and time complexity analyses. for example:
   - it explains the divide and conquer approach for adding n numbers and compares it with the brute force method.
   - it provides the time complexity analysis for mergesort and qu
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
### Divide and Conquer

```COSC 3100
Divide and Conquer
Instructor: Tanvir
What is Divide and Conquer ?
• 3rd algorithm design technique, we
are going to study
• Very important: two out of ten most
influential algorithms in the
twentieth century is based directly
on “Divide and Conquer” technique
– Quicksort (we shall study today)
– Fast Fourier Transform (we shall not
study in this course, but any “signal
processing” course is based on this idea)
Divide and Conquer
• Three steps are involved:
– Divide the problem into several
subproblems, perhaps of equal size
– Subproblems are solved, typically
recursively
– The solutions to the subproblems are
combined to get a solution to the
original problem
Real work is done in 3 different places: in partitioning; at the very tail end
of the recursion, when subproblems are so small that they are solved directly;
and in gluing together of partial solutions
Divide and Conquer (contd.)
Problem
Subproblem 1
of size n/2
Solution to
subproblem 1
of size n
Don’t assume
always breaks up
into 2, could be > 2
subproblems
Solution to the
original probelm
Subproblem 2
of size n/2
Solution to
subproblem 2
Divide and Conquer (contd.)
• In “politics” divide and rule (latin:
divide et impera) is a combination of
political, military, and economic
strategy of gaining and maintaining
power by breaking up larger
concentrations of power into chunks
that individually have less power than
the one who is implementing the
strategy. (read more on wiki: “Divide
and rule”)
Divide and Conquer (contd.)
• Let us add n numbers using divide and
conquer technique
a0 + a1 + …… + an-1
a0 + …… +
−

+ …… + an-1
Is it more efficient than brute force ?
Let’s see with an example
Div. & Conq. (add n numbers)
2
0
1
2
3
4
5
6
7
8
9
2
10
3
5
7
1
6
10
1
3
0
1
2
3
4
2
10
3
5
7
10
12
3
5
3
7
5
7
12
0
1
is same as in
brute force,
needs stack 1
for recursion…
not all divide
and conquer
works!!
1
6
6
7
2
10
10
3
1
1
10
4
3
3
1
3
4
14
15
21
27
48
Could be efficient
for parallel processors though….
Div. & Conq. (contd.)
• Usually in div. & conq., a problem instance of size n
is divided into two instances of size n/2
• More generally, an instance of size n can be
divided into b instances of size n/b, with a of
them needing to be solved
• Assuming that n is a power of b (n = bm), we get
general divide-and-conquer
– T(n) = aT(n/b) + f(n)
recurrence
– Here, f(n) accounts for the time spent in dividing an
instance of size n into subproblems of size n/b and
combining their solution
– For adding n numbers, a = b = 2 and f(n) = 1
Div & Conq. (contd.)
• T(n) = aT(n/b)+f(n), a ≥ 1, b > 1
What if a = 1?
• Master Theorem:
Have we seen it?
If f(n) є Θ(nd) where d ≥ 0 then
T(n) є
Θ(nd)
if a < bd
So, A(n) є Θ(  )
Or, A(n) є Θ(n)
Θ(ndlgn)
if a = bd
Θ(  )
if a > bd
Without going through
back-subs. we got it,
but not quite…
For adding n numbers with divide and conquer technique, the number
A(n) = 2A(n/2)+1
Here, a = ?, b = ?, d = ?
a = 2, b = 2, d = 0
Which of the 3 cases holds ? a = 2 > bd = 20, case 3
Div. & Conq. (contd.)
T(n) = aT(n/b)+f(n), a ≥ 1, b > 1
If f(n) є Θ(nd) where d ≥ 0, then
T(n) = 2T(n/2)+6n-1?
T(n) є
T(n) = 3 T(n/2) + n
a=3 > bd=21
T(n) = 3 T(n/2) + n2
a=3 < bd=22
T(n) = 4 T(n/2) + n2
a=4 = bd=22
Θ(nd)
Θ(ndlgn)
if a < bd
if a = bd
Θ(  )
if a > bd
a = 3, b = 2, f(n) є Θ(n1), so d = 1
Case 3:
T(n) є Θ(   ) = Θ( . )
a = 3, b = 2, f(n) є Θ(n2), so d = 2
Case 1:
T(n) є Θ(  )
a = 4, b = 2, f(n) є Θ(n2), so d = 2
Case 2:
T(n) є Θ(  lgn )
T(n) = 0.5 T(n/2) + 1/n
Master thm doesn’t apply,a<1, d<0
Master thm doesn’t apply f(n) not polynomial
T(n) = 2 T(n/2) + n/lgn
T(n) = 64 T(n/8) – n2lgn f(n) is not positive, doesn’t apply
T(n) = 2n T(n/8) + n a is not constant, doesn’t apply
Div. & Conq.: Mergesort
• Sort an array A[0..n-1]
A[0……n-1]
A[0……  2 − 1]
divide
A[  2 ……n-1]
sort
sort
A[  2 ……n-1]
A[0……  2 − 1]
merge
A[0……n-1]
Go on dividing recursively…
Div. & Conq.: Mergesort(contd.)
ALGORITHM Mergesort(A[0..n-1])
//sorts array A[0..n-1] by recursive mergesort
//Input: A[0..n-1] to be sorted
//Output: Sorted A[0..n-1]
if n > 1
copy A[0..  2 -1] to B[0..  2 -1]
copy A[  2 ..n-1] to C[0..  2 -1]
Mergesort(B[0..  2 -1])
Mergesort(C[0..  2 -1])
Merge(B, C, A)
B:
2
3
8
A:
9
1
C:
2
3
4
1
4
5
5
7
7
8
9
Div. & Conq.: Mergesort(contd.)
ALGORITHM Merge(B[0..p-1], C[0..q-1], A[0..p+q-1])
//Merges two sorted arrays into one sorted array
//Input: Arrays B[0..p-1] and C[0..q-1] both sorted
//Output: Sorted array A[0..p+q-1] of elements of B and C
i <- 0; j <- 0; k <- 0;
while i < p and j < q do
if B[i] ≤ C[j]
A[k] <- B[i]; i <- i+1
else
A[k] <- C[j]; j <- j+1
k <- k+1
if i = p
copy C[j..q-1] to A[k..p+q-1]
else
copy B[i..p-1] to A[k..p+q-1]
Div. & Conq.: Mergesort(contd.)
Divide:
Merge:
8
8
8
3
2
2
2
3
2
8
1
1
9
5
7
9
2
3
7
9
1
4
5
5
7
4
5
7
8
4
4
4
9
4
5
1
1
3
1
1
7
9
2
4
7
2
8
9
9
3
8
3
3
5
7
5
Div. & Conq.: Mergesort(contd.)
ALGORITHM Mergesort(A[0..n-1])
//sorts array A[0..n-1] by recursive
mergesort
//Input: A[0..n-1] to be sorted
//Output: Sorted A[0..n-1]
if n > a
copy A[0..  2 -1] to B[0..  2 -1]
copy A[  2 ..n-1] to C[0..  2 -1]
Mergesort(B[0..  2 -1])
Mergesort(C[0..  2 -1])
Merge(B, C, A)
What is the time-efficiency
of Meresort?
Input size: n = 2m
Basic op: comparison
C(n) depends on input
type…
ALGORITHM Merge(B[0..p-1], C[0..q-1], A[0..p+q-1])
//Merges two sorted arrays into one sorted array
//Input: Arrays B[0..p-1] and C[0..q-1] both sorted
//Output: Sorted array A[0..p+q-1] of elements of B
//and C
i <- 0; j <- 0; k <- 0;
while i < p and j < q do
if B[i] ≤ C[j]
A[k] <- B[i]; i <- i+1
else
A[k] <- C[j]; j <- j+1
k <- k+1
if i = p
else
copy C[j..q-1] to A[k..p+q-1]
copy B[i..p-1] to A[k..p+q-1]
C(n) = 2C(n/2) + CMerge(n) for n > 1,
C(1) = 0
CB:worst
C: (n/2)+n-1
3 4 9 for n > 1
2 (n)
5 =82Cworst
In worst-case Cworst(1) = 0
CMerge(n) = n-1
A:
3 =4nlgn-n+1
5 8 9є Θ(nlgn)
C 2 (n)
worst
Could use the master thm too!
How many comparisons
is needed for this Merge?
Div. & Conq.: Mergesort(contd.)
•
•
•
•
Worst-case of Mergesort is Θ(nlgn)
Average-case is also Θ(nlgn)
It is stable but quicksort and heapsort are not
Possible improvements
– Implement bottom-up. Merge pairs of elements, merge
the sorted pairs, so on… (does not require recursionstack anymore)
– Could divide into more than two parts, particularly useful
for sorting large files that cannot be loaded into main
memory at once: this version is called “multiway
mergesort”
• Not in-place, needs linear amount of extra
memory
– Though we could make it in-place, adds a bit more
“complexity” to the algorithm
Div. & Conq.: Quicksort
• Another divide and conquer based
sorting algorithm, discovered by C.
A. R. Hoare (British) in 1960 while
trying to sort words for a machine
translation project from Russian to
English
• Instead of “Merge” in Mergesort,
Quicksort uses the idea of
seen with “Lomuto Partition”
Div. & Conqr.: Quicksort (contd.)
A[0]…A[n-1]
A[0]…A[s-1] A[s] A[s+1]…A[n-1]
all are ≤ A[s]
all are ≥ A[s]
Notice:
A[s] is in it’s
In Mergesort
final position
all work is in
combining the
partial solutions.
Now continue working with
In Quicksort all
these two parts
work is in dividing
the problem,
Combining does not require any work!
Div. & Conqr.: Quicksort (contd.)
ALGORITHM Quicksort(A[l..r])
//Sorts a subarray by quicksort
//Input: Subarray of array A[0..n-1] defined by its
//left and right indices l and r
//Output: Subarray A[l..r] sorted in nondecreasing
//order
if l < r
s <- Partition( A[l..r] ) // s is a split position
Quicksort( A[l..s-1] )
Quicksort( A[s+1]..r )
Div. & Conqr.: Quicksort (contd.)
• As a partition algorithm we could use
“Lomuto Partition”
• But we shall use the more sophisticated
• We start by selecting a “pivot”
• There are various strategies to select the
pivot, we shall use the simplest: we shall
select pivot, p =A[l], the first element of
A[l..r]
Div. & Conqr.: Quicksort (contd.)
p
p
≤p
≥p
p
i
j
If A[i] < p, we continue incrementing i, stop when A[i] ≥ p
If A[j] > p, we continue decrementing j, stop when A[j] ≤ p
p
all ≤ p
p
all ≤ p
p
j
≤p
i
≥p
all ≤ p
j i
≤p ≥p
all ≥ p
all ≥ p
j=i
=p
all ≥ p
Div. & Conqr.: Quicksort (contd.)
ALGORITHM HoarePartition(A[l..r])
//Output: the split position
p <- A[l]
i could go out of array’s
bound, we could check
i <- l; j <- r+1
or we could put a “sentinel”
at the end…
repeat
Do you see any
repeat i <- i+1 until A[i] ≥ p
possible problem
with this pseudocode ? repeat j <- j-1 until A[j] ≤ p
swap( A[i], A[j] )
until i ≥ j
swap( A[i], A[j] ) // undo last swap when i ≥ j
swap( A[l], A[j] )
return j
More sophisticated pivot selection
that we shall see briefly makes this
“sentinel” unnecessary…
Div. & Conqr.: Quicksort (contd.)
j
i
5 3 1 9 8 2 4 7
j
i
2 3 1 4 5 8 9 7
j
i
5 3 1 9 8 2 4 7
i j
2 3 1 4 5 8 9 7
j
i
5 3 1 4 8 2 9 7
i j
2 1 3 4 5 8 9 7
i j
5 3 1 4 8 2 9 7
j i
2 1 3 4 5 8 9 7
i j
5 3 1 4 2 8 9 7
1 2 3 4 5 8 9 7
j i
5 3 1 4 2 8 9 7
1 2 3 4 5 8 9 7
2 3 1 4 5 8 9 7
i=j
1 2 3 4 5 8 9 7
j i
1 2 3 4 5 8 9 7
1 2 3 4 5 8 9 7
i j
1 2 3 4 5 8 9 7
i j
1 2 3 4 5 8 7 9
j i
1 2 3 4 5 8 7 9
1 2 3 4 5 7 8 9
1 2 3 4 5 7 8 9
1 2 3 4 5 7 8 9
Div. & Conqr.: Quicksort (contd.)
0
1
2
3
4
5
6
7
0 1 2 3 4 5 6 7
5 3 1 9 8 2 4 7
l=0,r=7
0 1 2 3 4 5 6 7
s=4
2 3 1 4 5 8 9 7
l=5,r=7
l=0,r=3
s=1
l=0,r=0
5 3 1 9 8 2 4 7
s=6
l=2,r=3
s=2
l=2,r=1
l=5,r=5
l=3,r=3
l=7,r=7
0 1 2 3
5 6 7
1 2 3 4
7 8 9
0
5
1
7
2 3
7
3 4
9
3
4
Div. & Conqr.: Quicksort (contd.)
• Let us analyze Quicksort
ALGORITHM Quicksort(A[l..r])
Time-complexity of this line ?
if l < r
i j
s <- HoarePartition ( A[l..r] ) So, n+1
5 3 1 4 8 2 9 7
comparisons
Quicksort( A[l..s-1] )
i j
when cross-over
Quicksort( A[s+1]..r )
If all splits
happen in the
middle, it is
the best-case!
5 3 1 4 2 8 9 7
j i
5 3 1 4 2 8 9 7
ALGORITHM HoarePartition(A[l..r])
//Output: the split position
So, n
p <- A[l]
comparisons
i <- l; j <- r+1
when coincide What if,
repeat
repeat i <- i+1 until A[i] ≥ p
5 3 1
repeat j <- j-1 until A[j] ≤ p
swap( A[i], A[j] )
until i ≥ j
swap( A[i], A[j] ) // undo last swap when i ≥ j
swap( A[l], A[j] )
Cbest(n) = 2Cbest(n/2)+n for
return j
Cbest(1) = 0
i,j
4 5 8 9 7
n > 1
Div. & Conqr.: Quicksort (contd.)
T(n) = aT(n/b)+f(n), a ≥ 1, b > 1
If f(n) є nd with d ≥ 0, then
T(n) є
Θ(nd)
if a < bd
Θ(ndlgn)
if a = bd
Θ(  )
if a > bd
ALGORITHM Quicksort(A[l..r])
if l < r
s <- Partition( A[l..r] )
Quicksort( A[l..s-1] )
Quicksort( A[s+1]..r )
j
2
Cbest(n) = 2Cbest(n/2)+n for n > 1
Cbest(1) = 0
ij j j j
5 6 8 9
j
i
5 6 8 9
Using Master Thm, Cbest(n) є Θ(nlgn)
What is the worst-case ?
Cworst(n) = (n+1) + (n-1+1) + … + (2+1) = (n+1) + … + 3
= (n+1) + … + 3 + 2 + 1 – (2 + 1) = +1
- 3
1
=
+1 +2
2
- 3 є Θ(n2) !
6
5+1=6
4+1=5
j
9
3+1=4
ij
8 9
2+1=3
i
8
So, Quicksort’s fate
depends on its average-case!
Div. & Conqr.: Quicksort (contd.)
• Let us sketch the outline of averageALGORITHM Quicksort(A[l..r])
case analysis…
if l < r
s <- Partition( A[l..r] )
Cavg(n) is the average number of key-comparisons
Quicksort( A[l..s-1] )
made by the Quicksort on a randomly ordered array
Quicksort( A[s+1]..r )
of size n
After n+1 comparisons, a partition can
happen in any position s (0 ≤ s ≤ n-1)
Let us assume that
partition split can
happen in each position s
with equal probability 1/n
After the partition, left part has s elements,
Right part has n-1-s elements
s
0
n-1
Cavg(n) = Expected[ Cavg(s) + Cavg(n-1-s) + (n+1) ]
p
Average over all possibilities
s elements
n-1-s elements

−
Cavg(n) =
[ Cavg(s) + Cavg(n−1−s) + (n+1) ]
=
Cavg(0) = 0, Cavg(1) = 0 C (n) ≈ 1.39nlgn
avg
Div. & Conqr.: Quicksort (contd.)
• Recall that for Quicksort, Cbest(n) ≈ nlgn
• So, Cavg(n) ≈ 1.39nlgn is not far from Cbest(n)
• Quicksort is usually faster than Mergesort or
Heapsort on randomly ordered arrays of nontrivial
sizes
• Some possible improvements
• Randomized quicksort: selects a random element as pivot
• Median-of-three: selects median of left-most, middle, and
right-most elements as pivot
• Switching to insertion sort on very small subarrays, or not
sorting small subarrays at all and finish the algorithm with
insertion sort applied to the entire nearly sorted array
• Modify partitioning: three-way partition
These improvements can speed up by 20% to 30%
Div. & Conqr.: Quicksort (contd.)
• Weaknesses
– Not stable
– Requires a stack to store parameters of
subarrays that are yet to be sorted, the
stack can be made to be in O(lgn) but
that is still worse than O(1) space
efficiency of Heapsort
DONE with Quicksort!
Div. & Conq. : Multiplication of
Large Integers
• We want to efficiently multiply two very
large numbers, say each has more than 100
decimal digits
• How do we usually multiply 23 and 14?
• 23 = 2*101 + 3*100, 14 = 1*101 + 4*100
• 23*14 = (2*101 + 3*100) * (1*101 + 4*100)
• 23*14 = (2*1)102 + (2*4+3*1)101+(3*4)100
• How many multiplications? 4 = n2
Div. & Conq. : Multiplication of
Large Integers
• 23*14 = (2*1)102 + (2*4+3*1)101+(3*4)100
We can rewrite the middle term as:
(2*4+3*1) = (2+3)*(1+4) - 2*1 - 3*4
What has been gained?
We have reused 2*1 and 3*4 and now need
one less multiplication
If we have a pair of 2-digits numbers a and b
a = a1a0 and b = b1b0
we can write c = a*b = c2102+c1101+c0100
c2 = a1*b1
c0 = a0*b0
c1 = (a1+a0)*(b1+b0)-(c2+c0)
Div. & Conq. : Multiplication of Large
Integers
If we have a pair of 2-digits numbers a and b
a = a1a0 and b = b1b0
we can write c = a*b = c2102+c1101+c0100
c2 = a1*b1 , c0 = a0*b0
c1 = (a1+a0)*(b1+b0)-(c2+c0)
If we have two n-digits numbers,
a and b (assume n is a power of 2)
a:
a1
a0
b:
b1
b0
We can write,
a = a110n/2 + a0
b = b110n/2 + b0
a = 1234 = 1*103+2*102+3*101+4*100
= (12)102+(34)
(12)102+(34) = (1*101+2*100)102+3*101+4*100
Apply the same idea
recursively to get c2, c1, c0
until n is so small that you can
you can directly multiply
n/2 digits
c = a*b = 1 10 2 + 0 ∗ 1 10 2 + 0
= 1 ∗ 1 10 + 1 ∗ 0 + 0 ∗ 1 10
= 2 10 + 1 10 2 + 0
Why?
c2 = a1*b1
c0 = a0*b0
c1 = (a1+a0)*(b1+b0)-(c2+c0)
2
+ 0 ∗ 0
Div. & Conq. : Multiplication of Large
Integers
Notice: a1, a0, b1, b0 all are n/2
digits numbers
So, computing a*b requires
three n/2-digits multiplications
Recurrence for the number of
Multiplications is
M(n) = 3M(n/2) for n > 1
M(1) = ?
c2 = a1*b1
2
+ 0
1 subtraction
c0 = a0*b0
c1 = (a1+a0)*(b1+b0)-(c2+c0)
Assume n = 2m
M(n) = 3M(n/2) = 3[ 3M(n/22) ] = 32M(n/22)
And subtractions?
A(n) = 3A(n/2)+cn for n > 1
A(1) = 0
c = a*b = 2 10 + 1 10
Using Master Thm,
A(n) є Θ(nlg3)
M(n) = 3mM(n/2m) = ?
M(n) = 3m = 3lgn = nlg3
Why?
M(n) ≈ n1.585
Let   = x
=
=
x =
Div. & Conq. : Multiplication of Large
Integers
• People used to believe that multiplying two ndigits number has complexity Ω(n2)
• In 1960, Russian mathematician Anatoly
Karatsuba, gave this algorithm whose asymptotic
complexity is Θ(n1.585)
• A use of large number multiplication is in modern
cryptography
• It does not generally make sense to recurse all
the way down to 1 bit: for most processors 16- or
32-bit multiplication is a single operation; so by
this time, the numbers should be handed over to
built-in procedure
Next we see how to multiply
Matrices efficiently…
Div. & Conq. : Strassen’s Matrix
Multiplication
• How do we multiply two 2×2 matrices ?
1
2
3
5
3
4
1
4
=
5
13
13
31
How many multiplications
V. Strassen in 1969 found out, he can
do the above multiplication in the following
way:
m1+m4-m5+m7 m3+m5
c
c
00
01
a00 a01
b00 b01
=
=
m2+m4
m1+m3-m2+m6
c10 c11
a10 a11
b10 b11
m1 = (a00+a11)*(b00+b11)
m2 = (a10+a11)*b00
m4 = a11*(b10-b00)
m3 = a00*(b01-b11)
m5 = (a00+a01)*b11
m6 = (a10-a00)*(b00+b01) m7 = (a01-a11)*(b10+b11)
7 mults
Div. & Conq. : Strassen’s Matrix
Multiplication
• Let us see how we can apply Strassen’s idea for
multiplying two n×n matrices
Let A and B be two n×n matrices where n is a power of 2
A00 A01
B00 B01
A10 A11
B10
B11
=
Each block is (n/2)×(n/2)
You can treat blocks
as if they were numbers
to get the C = A*B
C00 C01
C10
C11
E.g., In Strassen’s
method,
M1 = (A00+A11)*(B00+B11)
M2 = (A10+A11)*B00
etc.
Div. & Conq. : Strassen’s Matrix
Multiplication
ALGORITHM Strassen(A, B, n)
//Input: A and B are n×n matrices
//where n is a power of two
//Output: C = A*B
if n = 1
return C = A*B
else
A00 A01
Partition A =
A10 A11
and B =
Recurrence for
# of multiplications is
M(n) = 7M(n/2) for n > 1
M(1) = ?
B00 B01
B10
B11
where the blocks Aij and Bij are (n/2)-by-(n/2)
M1 <- Strassen(A00+A11, B00+B11, n/2)
M2 <- Strassen(A10+A11, B00, n/2)
M3 <- Strassen(A00, B01-B11, n/2)
M4 <- Strassen(A11, B10-B00, n/2)
M5 <- Strassen(A00+A01, B11, n/2)
M6 <- Strassen(A10-A00, B00+B01, n/2)
M7 <- Strassen(A01-A11, B10+B11, n/2)
C00 <- M1+M4-M5+M7
C01 <- M3+M5
C10 <- M2+M4
C11 <- M1+M3-M2+M6
return C =
C00 C01
C10
C11
For n = 2m,
M(n) = 7M(n/2) = 72M(n/22)
M(n) = 7m M(n/2m)
= 7m = 7lgn
= nlg7 ≈ n2.807
A(n) = 7A(n/2)+18(n/2)2 for n > 1
A(1) = 0
Using Master thm,
A(n) є Θ(nlg7) better
than brute-force’s Θ(n3)
DONE WITH STRASSEN!
Div. & Conq.: Closest pair
• Find the two closest points in a set of n
points
Traffic
Control:
detect two
vehicles
most likely
to collide
ALGORITHM BruteForceClosestPair(P)
//Input: A list P of n (n≥2) points p1(x1,y1),
//p2(x2,y2), …, pn(xn,yn)
//Output: distance between closest pair
d <- ∞
for i <- 1 to n-1 do
for j <- i+1 to n do
d <- min( d, sqrt(  −
2
+  −
2
))
return d
Idea: consider each pair of points and keep track of the
pair having the minimum distance
There are
n
(−1)
pairs, so time-efficiency is in Θ(n2)
=
2
2
Div. & Conq.: Closest pair (contd.)
• We shall apply “divide and conquer”
technique to find a better solution
Let P be the
set of points
sorted by
x-coordinates
Any idea how to divide and then
conquer?
x=m
dl
let Q be the
same points sorted
by y-coordinates
Left portion
dr
Right portion
Solve right and left
portions recursively
and then combine
the partial solutions
How should we
combine?
d = min {dl, dr} ?
Does not work,
because one point
can be in left portion
and the other could be
in right portion having
distance < d between
them…
Div. & Conq.: Closest pair (contd.)
x=m
d = min{dl, dr}
We wish to find a pair
having distance < d
It is enough to consider
the points inside the symmetric
vertical strip of width 2d around
the separating line! Why?
dl
dr
Left portion
Right portion
d
d
Because the distance
between any other pair
of points is at least d
We shall scan through S, updating
But S can
contain all
points, right? dmin = d using brute-force.
Let S be the list of points
inside the strip of width 2d
obtained from Q, meaning ?
Let p(x, y) be a point in S.
for a point p’(x’, y’) to have a
chance to be closer to p than dmin,
p’ must “follow” p in S and the
difference bewteen their
y-coordinates must be less than dmin
Div. & Conq.: Closest pair (contd.)
x=m
dl
dr
Left portion
Let p(x, y) is a point in S.
For a point p’(x’, y’) to have a
chance to be closer to p than dmin,
p’ must “follow” p in S and
the difference bewteen their
y-coordinates must be less
than dmin Why?
It seems, this rectangle
Right portion
d
Geometrically,
p’ must be in
the following
rectangle
My claim is “this”
is the most you can
put in the rectangle…
x=m
d
d
dmin
d
Now comes the crucial observation,
everything hinges on this one…
How many points can there
be in the dmin-by-2d rectangle?
can contain many
points, may be all…
p
x=m
d
d
d
One of these 8
being p, we need
to check 7 pairs
to find if any pair
has distance < dmin
Div. & Conq.: Closest pair (contd.)
ALGORITHM EfficientClosestPair(P, Q)
//Solves closest-pair problem by divide and conquer
//Input: An array P of n ≥ 2 points sorted by x-coordinates and another array Q of same
//points sorted by y-coordinates
The algorithm spends linear time
in dividing and merging, so assuming
//Output: Distance between the closest pair
n = 2m, we have the following
if n ≤ 3
recurrence for runnning-time,
return minimal distance by brute-force
else
T(n) = 2T(n/2)+f(n) where f(n) є Θ(n)
copy first  2 points of P to array Pl
copy the same  2 points from Q to array Ql
Applying Master Theorem,
copy the remaining  2 points of P to array Pr
T(n) є Θ(nlgn)
copy the same  2 points from Q to array Qr
dl <- EfficientClosestPair( Pl, Ql )
dr <- EfficientClosestPair( Pr, Qr )
d <- min{ dl, dr }
Dividing line
Brute froce
m <- P[  2 -1].x
Inside the
Points in 2*d
copy all points of Q for which |x-m| < d into array S[0..num-1]
2*d width strip
width strip
dminsq <- d2
for i <- 0 to num-2 do
k <- i+1
while k ≤ num-1 and ( S[k].y – S[i].y )2 < dminsq
dminsq <- min( (S[k].x-S[i].x)2+(S[k].y-S[i].y)2 , dminsq)
k <- k+1
return sqrt( dminsq ) // could easily keep track of the pair of points
```
user
### Content Structure
The text passage is a detailed lecture or course material on the "Divide and Conquer" algorithm design technique. It covers various algorithms and problems that use this technique, including Quicksort, Mergesort, multiplication of large integers, Strassen's matrix multiplication, and the closest pair problem. The passage includes explanations, pseudocode, and time complexity analyses for these algorithms.

### Analysis
TYPE: MULTIPLE_QUESTIONS_WITH_ANSWERS

### Explanation
1. **Multiple Questions**: The passage contains several mathematical questions and problems, such as:
   - How to add n numbers using the divide and conquer technique.
   - Time complexity analysis of various divide and conquer algorithms (e.g., Mergesort, Quicksort).
   - How to multiply large integers efficiently.
   - How to multiply matrices using Strassen's method.
   - How to find the closest pair of points in a set.

2. **Answers Provided**: For each of these questions, the passage provides detailed explanations, pseudocode, and time complexity analyses. For example:
   - It explains the divide and conquer approach for adding n numbers and compares it with the brute force method.
   - It provides the time complexity analysis for Mergesort and Qu