Divide and Conquer My T. UF

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1 Introduction to Algorithms Divide and UF

2 General Approach Divide the problem into a number of subproblems that are smaller instances of the same problem Conquer the subproblems by solving them recursively (recursive case). If the size of the subproblem is small enough, solve it directly (base case). Combine the solutions to the subproblems into the solution for the original problem 2

3 Analyze Running time Depend on how the problem is divided. Typically sub problems have the same size. Depend on dividing and combining time Solving the recurrence gives us the asymptotic running time. Where f(n) is dividing and combining time c, a, b, n 0 are constants 3

4 Example 4

5 Methods to solve recurrence Substitution method: guess a bound and then use mathematical induction to prove our guess correct. Recursion-tree method: use a tree to represent the recurrent formula then compute the cost at each level of the tree. Master Theorem: provides bounds for recurrences of the form 5

6 Substitution method Guess a bound and then use mathematical induction to prove our guess is correct. Example: Guess: Proof by induction: suppose then : 6

7 Substitution method Hard to make a good guess Guess the bound based on the similar recurrence Example: Similar recurrence : Guess: Pitfalls: different constants in the induction Example: guess then we have 7

8 Substitution method Transform to guess easier Example: Let Denote Then we have: 8

9 Recursion-tree method Method: Build the recursion tree, each node represents the additional cost of a single sub problem know its sub problems solutions Compute the additional cost at each level Sum up cost of all level Considered factors: The height of the tree The upper (tight) bound of the cost of each level Simplify the sum 9

10 Recursion-tree method Example Note that: What if: 10

11 Recursion-tree method Another example: 11

12 Master Theorem 12

13 Using Master Theorem Example 1: then, according to case 1: Example 2: Apply case 2: Example 3: Apply case 3: 13

14 The maximum-subarray problem Input: a sequence A of n numbers Output: a nonempty, contiguous subarray of A whose values have the largest sum called maximum subarray Example: 14

15 The maximum-subarray problem brute-force search: try all possible continuous subarray, take time Divide-and-conquer: Divide the array A[low high] into two subarrays A[low mid] (left) and A[mid +1 high] (right) All possible locations of the maximum subarray: 15

16 Find the maximum cross subarray Possible locations: Right most of the left subarray Left most of the right subarray Include elements of both left and right subarrays 16

17 Find the maximum array Do we need to compute the max-left and maxright in FIND-MAX-CROSSING-SUBARRAY procedure? 17

18 Algorithm analysis FIND-MAXIMUM-SUBARRAY takes time Recurrence for the running time T(n) of FIND- MAXIMUM-SUBARRAY Apply recurrence tree method 18

19 Matrix multiplication Input: two nxn matrices A and B Output: their product C = A.B Example: Simplest algorithm: Compute as the definition Running time: 19

20 Simple divide and conquer algorithm Partition each of A, B, and C into four n/2xn/2 matrices Then 20

21 Simple divide and conquer algorithm 8 sub problems Running time: Apply master theorem: No improvement!!! 21

22 Strassen s method Try to reduce the number of sub problems by 1 Divide A, B and C into 4 sub matrices, take time Create ten matrices as sum or difference of two matrices in previous step, take Use create 7 matrices of size n/2xn/2 Compute by adding and subtracting various combinations of the P i matrices, take Recurrence formula: Apply master theorem: < 22

23 Strassen s method Compute Ss and Ps matrices 23

24 Strassen s method Compute C matrix 24

25 Correctness of Strassen s method 25

26 Summary 3 methods to solve recurrence: Substitution, Recursion-tree, Master Theorem Divide-and-conquer strategy: Analyze the problem to find a divide scheme Analyze the running time via solving recurrence Optimize the divide-and-conquer strategy: Minimize the number of sub problems Minimized the dividing and combining time Trade-off between the number of sub problems and dividing & combining time 26

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