5.2 divide and conquer

Comp 122, Spring 2004
Divide and Conquer
(Merge Sort)
Divide and Conquer
(Merge Sort)

Comp 122- 2
Divide and Conquer
 Recursive in structure
 Divide the problem into sub-problems that are
similar to the original but smaller in size
 Conquer the sub-problems by solving them
recursively. If they are small enough, just solve
them in a straightforward manner.
 Combine the solutions to create a solution to
the original problem

Comp 122- 3
An Example: Merge Sort
Sorting Problem: Sort a sequence of n elements into
non-decreasing order.
 Divide: Divide the n-element sequence to be
sorted into two subsequences of n/2 elements each
 Conquer: Sort the two subsequences recursively
using merge sort.
 Combine: Merge the two sorted subsequences to
produce the sorted answer.

Comp 122- 5
Merge Sort – Example
18 26 32 6 43 15 9 1
18 26 32 6 43 15 9 1
18 26 32 6 43 15 9 1
2618 632 1543 19
18 26 32 6 43 15 9 1
18 26 32 6 43 15 9 1
18 26 326 15 43 1 9
6 18 26 32 1 9 15 43
1 6 9 15 18 26 32 43
18 26
18 26
18 26
32
32
6
6
32 6
18 26 32 6
43
43
15
15
43 15
9
9
1
1
9 1
43 15 9 1
18 26 32 6 43 15 9 1
18 26 632
626 3218
1543 19
1 915 43
16 9 1518 26 32 43
Original Sequence Sorted Sequence

Comp 122- 6
Merge-Sort (A, p, r)
INPUT: a sequence of n numbers stored in array A
OUTPUT: an ordered sequence of n numbers
MergeSort (A, p, r) // sort A[p..r] by divide & conquer
1 if p < r
2 then q ← (p+r)/2
3 MergeSort (A, p, q)
4 MergeSort (A, q+1, r)
5 Merge (A, p, q, r) // merges A[p..q] with A[q+1..r]
Initial Call: MergeSort(A, 1, n)

Comp 122- 7
Procedure Merge
Merge(A, p, q, r)
1 n1 ← q – p + 1
2 n2 ← r – q
3 for i ← 1 to n1
4 do L[i] ← A[p + i – 1]
5 for j ← 1 to n2
6 do R[j] ← A[q + j]
7 L[n1+1] ← ∞
8 R[n2+1] ← ∞
9 i ← 1
10 j ← 1
11 for k ←p to r
12 do if L[i] ≤ R[j]
13 then A[k] ← L[i]
14 i ← i + 1
15 else A[k] ← R[j]
16 j ← j + 1
Sentinels, to avoid having to
check if either subarray is
fully copied at each step.
Input: Array containing
sorted subarrays A[p..q]
and A[q+1..r].
Output: Merged sorted
subarray in A[p..r].

Comp 122- 8
j
Merge – Example
6 8 26 32 1 9 42 43… …A
k
6 8 26 32 1 9 42 43
k k k k k k k
i i i i
∞ ∞
i j j j j
6 8 26 32 1 9 42 43
1 6 8 9 26 32 42 43
k
L R

Comp 122- 9
Correctness of Merge
Merge(A, p, q, r)
1 n1 ← q – p + 1
2 n2 ← r – q
3 for i ← 1 to n1
4 do L[i] ← A[p + i – 1]
5 for j ← 1 to n2
6 do R[j] ← A[q + j]
7 L[n1+1] ← ∞
8 R[n2+1] ← ∞
9 i ← 1
10 j ← 1
11 for k ←p to r
14 i ← i + 1
16 j ← j + 1
Loop Invariant for the for loop
At the start of each iteration of the
for loop:
Subarray A[p..k – 1]
contains the k – p smallest elements
of L and R in sorted order.
L[i] and R[j] are the smallest elements of
L and R that have not been copied back into
A.
Initialization:
Before the first iteration:
•A[p..k – 1] is empty.
•i = j = 1.
•L[1] and R[1] are the smallest
elements of L and R not copied to A.

Comp 122- 10
Correctness of Merge
Merge(A, p, q, r)
1 n1 ← q – p + 1
2 n2 ← r – q
3 for i ← 1 to n1
4 do L[i] ← A[p + i – 1]
5 for j ← 1 to n2
6 do R[j] ← A[q + j]
7 L[n1+1] ← ∞
8 R[n2+1] ← ∞
9 i ← 1
10 j ← 1
11 for k ←p to r
14 i ← i + 1
16 j ← j + 1
Maintenance:
Case 1: L[i] ≤ R[j]
•By LI, A contains p – k smallest elements
of L and R in sorted order.
•By LI, L[i] and R[j] are the smallest
elements of L and R not yet copied into A.
•Line 13 results in A containing p – k + 1
smallest elements (again in sorted order).
Incrementing i and k reestablishes the LI
for the next iteration.
Similarly for L[i] > R[j].
Termination:
•On termination, k = r + 1.
•By LI, A contains r – p + 1 smallest
elements of L and R in sorted order.
•L and R together contain r – p + 3 elements.
All but the two sentinels have been copied
back into A.

Comp 122- 11
Analysis of Merge Sort
 Running time T(n) of Merge Sort:
 Divide: computing the middle takes Θ(1)
 Conquer: solving 2 subproblems takes 2T(n/2)
 Combine: merging n elements takes Θ(n)
 Total:
T(n) = Θ(1) if n = 1
T(n) = 2T(n/2) + Θ(n) if n > 1
⇒ T(n) = Θ(n lg n) (CLRS, Chapter 4)

Comp 122, Spring 2004
Recurrences – IRecurrences – I

Comp 122- 13
Recurrence Relations
 Equation or an inequality that characterizes a
function by its values on smaller inputs.
 Solution Methods (Chapter 4)
 Substitution Method.
 Recursion-tree Method.
 Master Method.
 Recurrence relations arise when we analyze the
running time of iterative or recursive algorithms.
 Ex: Divide and Conquer.
T(n) = Θ(1) if n ≤ c
T(n) = a T(n/b) + D(n) + C(n) otherwise

Comp 122- 14
Substitution Method
 Guess the form of the solution, then
use mathematical induction to show it correct.
 Substitute guessed answer for the function when the
inductive hypothesis is applied to smaller values –
hence, the name.
 Works well when the solution is easy to guess.
 No general way to guess the correct solution.

Comp 122- 15
Example – Exact Function
Recurrence: T(n) = 1 if n = 1
T(n) = 2T(n/2) + n if n > 1
Guess: T(n) = n lg n + n.
Induction:
•Basis: n = 1 ⇒ n lgn + n = 1 = T(n).
•Hypothesis: T(k) = k lg k + k for all k < n.
•Inductive Step: T(n) = 2 T(n/2) + n
= 2 ((n/2)lg(n/2) + (n/2)) + n
= n (lg(n/2)) + 2n
= n lg n – n + 2n
= n lg n + n

Comp 122- 16
Recursion-tree Method
 Making a good guess is sometimes difficult with
the substitution method.
 Use recursion trees to devise good guesses.
 Recursion Trees
 Show successive expansions of recurrences using
trees.
 Keep track of the time spent on the subproblems of a
divide and conquer algorithm.
 Help organize the algebraic bookkeeping necessary
to solve a recurrence.

Comp 122- 17
Recursion Tree – Example
 Running time of Merge Sort:
T(n) = Θ(1) if n = 1
T(n) = 2T(n/2) + Θ(n) if n > 1
 Rewrite the recurrence as
T(n) = c if n = 1
T(n) = 2T(n/2) + cn if n > 1
c > 0: Running time for the base case and
time per array element for the divide and
combine steps.

Comp 122- 18
Recursion Tree for Merge Sort
For the original problem,
we have a cost of cn,
plus two subproblems
each of size (n/2) and
running time T(n/2).
cn
T(n/2) T(n/2)
Each of the size n/2 problems
has a cost of cn/2 plus two
subproblems, each costing
T(n/4).
cn
cn/2 cn/2
T(n/4) T(n/4) T(n/4) T(n/4)
Cost of divide and
merge.
Cost of sorting
subproblems.

Comp 122- 19
Continue expanding until the problem size reduces to 1.
cn
cn/2 cn/2
cn/4 cn/4 cn/4 cn/4
c c c cc c
lg n
cn
cn
cn
cn
Total : cnlgn+cn

Comp 122- 20
Continue expanding until the problem size reduces to 1.
cn
cn/2 cn/2
cn/4 cn/4 cn/4 cn/4
c c c cc c
•Each level has total cost cn.
•Each time we go down one level,
the number of subproblems doubles,
but the cost per subproblem halves
⇒ cost per level remains the same.
•There are lg n + 1 levels, height is
lg n. (Assuming n is a power of 2.)
•Can be proved by induction.
•Total cost = sum of costs at each
level = (lg n + 1)cn = cnlgn + cn =
Θ(n lgn).

Comp 122- 21
Other Examples
 Use the recursion-tree method to determine a
guess for the recurrences
 T(n) = 3T(n/4) + Θ(n2
).
 T(n) = T(n/3) + T(2n/3) + O(n).

Comp 122- 22
Recursion Trees – Caution Note
 Recursion trees only generate guesses.
 Verify guesses using substitution method.
 A small amount of “sloppiness” can be
tolerated. Why?
 If careful when drawing out a recursion tree and
summing the costs, can be used as direct proof.

Comp 122- 23
The Master Method
 Based on the Master theorem.
 “Cookbook” approach for solving recurrences
of the form
T(n) = aT(n/b) + f(n)
• a ≥ 1, b > 1 are constants.
• f(n) is asymptotically positive.
• n/b may not be an integer, but we ignore floors and
ceilings. Why?
 Requires memorization of three cases.

Comp 122- 24
The Master Theorem
Theorem 4.1
Let a ≥ 1 and b > 1 be constants, let f(n) be a function, and
Let T(n) be defined on nonnegative integers by the recurrence
T(n) = aT(n/b) + f(n), where we can replace n/b by n/b or n/b.
T(n) can be bounded asymptotically in three cases:
1. If f(n) = O(nlogba–ε
) for some constant ε > 0, then T(n) = Θ(nlogba
).
2. If f(n) = Θ(nlogba
), then T(n) = Θ(nlogba
lg n).
3. If f(n) = Ω(nlogba+ε
) for some constant ε > 0,
and if, for some constant c < 1 and all sufficiently large n,
we have a·f(n/b) ≤ c f(n), then T(n) = Θ(f(n)).
Theorem 4.1
Let a ≥ 1 and b > 1 be constants, let f(n) be a function, and
Let T(n) be defined on nonnegative integers by the recurrence
T(n) = aT(n/b) + f(n), where we can replace n/b by n/b or n/b.
T(n) can be bounded asymptotically in three cases:
1. If f(n) = O(nlogba–ε
) for some constant ε > 0, then T(n) = Θ(nlogba
).
2. If f(n) = Θ(nlogba
), then T(n) = Θ(nlogba
lg n).
3. If f(n) = Ω(nlogba+ε
) for some constant ε > 0,
and if, for some constant c < 1 and all sufficiently large n,
we have a·f(n/b) ≤ c f(n), then T(n) = Θ(f(n)).
We’ll return to recurrences as we need them…

5.2 divide and conquer

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