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Chapter 2 Divide-and-Conquer

Chapter 2 Divide-and-Conquer. 2.1 Divide-and-Conquer Approach 2.2 Binary Search 2.3 Mergesort 2.4 Quicksort (Partition Exchange Sort) 2.5 Strassen’s Matrix Multiplication Algorithm 2.6 Arithmetic with Large Integers 2.7 Fast Fourier Transform (FFT) 2.8 Determining Thresholds.

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Chapter 2 Divide-and-Conquer

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  1. Chapter 2Divide-and-Conquer 2.1 Divide-and-Conquer Approach 2.2 Binary Search 2.3 Mergesort 2.4 Quicksort (Partition Exchange Sort) 2.5 Strassen’s Matrix Multiplication Algorithm 2.6 Arithmetic with Large Integers 2.7 Fast Fourier Transform (FFT) 2.8 Determining Thresholds Divide-and-Conquer

  2. The divide-and-conquer design strategy involves the following steps： Divide an instance of a problem into one or more smaller instances. Conquer (solve) each of the smaller instances. Unless a smaller instance is sufficiently small, use recursion to do this. If necessary, combine the solutions to the smaller instances to obtain the solution to the original instance. 2.1: Divide-and-Conquer Approach Divide-and-Conquer

  3. Problem： Determine whether x is in the sorted array S of size n. The steps of Binary Search can be summarized as follows： Divide the array into two subarrays about half as large. If x is smaller than the middle term, choose the left subarray; otherwise, choose the right subarray. Conquer (solve) the subarray by determining whether x is in that subarray. Unless the subarray is sufficiently small, use recursion to do this. Obtain the solution to the array from the solution to the subarray. 2.2: Binary Search Divide-and-Conquer

  4. Suppose x = 18 and we have the following array： 10 12 13 14 18 20 25 27 30 35 40 45 47 Divide the array： because x < 25, we need to search 10 12 13 14 18 20 Conquer the subarray by determining whether x is in that subarray recursively. The answer is Yes. Obtain the solution to the array from the solution to the subarray. Yes, x is in that array. Middle term Example 2.1 Divide-and-Conquer

  5. Algorithm 2.1：Binary Search (Recursive) • Problem：Determine whether x is in the sorted array S of size n. Inputs：positive integer n, sorted array of keys S indexed from1 to n, and a key x. Output：location, the location of x in S (0 if x is not in S) • indexlocation(indexlow, indexhigh) {indexmid; if (low >high ) return 0; else{mid = (low + high )/2; if (x= =S[mid]) location = mid; else if(x<S[mid]) returnlocation(low,mid1); elsereturnlocation(mid+1,high); } } Divide-and-Conquer

  6. Compare x with 25 10 12 13 14 18 20 Compare x with 13 14 18 20 Compare x with 18 Because x= 18, xis in S. Recursive Steps of Example 2.1 10 12 13 14 18 20 25 27 30 35 40 45 47 Because x< 25, choose left subarray. Because x> 13, choose right subarray. Divide-and-Conquer

  7. Comparison in recursive call Comparison at top level Analysis of Algorithm 2.1 • Worst-Case Time Complexity By applying Master Theorem, a = 1, b = 2, d = 0, so a = bd Hence, Divide-and-Conquer

  8. 2.3 Mergesort • What is Mergesort? 1.Dividethe array into two subarrays each with n/2 items. 2.Conquer (solve) each subarray by sorting it recursively, unless the array is sufficiently small. 3.Combine the solutions to the subarrays by merging them into a single sorted array. Divide-and-Conquer

  9. Example 2.2 Suppose the array contains these numbers in sequence： 27 10 12 20 25 13 15 22 1. Divide the array： 27 10 12 20 and25 13 15 22 2. Sort each array： 10 12 20 27 and13 15 22 25 3. Merge the subarrays： 10 12 13 15 20 22 25 27 Divide-and-Conquer

  10. Algorithms 2.2：Mergesort • Problem：Sort n keys in nondecreasing sequence. Inputs：positive integer n, array of keys S index from 1 to n. Output：the array S containing the keys in nondecreasing sequence. • voidmergesort(intn, keytype S[]) { if ( n > 1) { const inth = n/2 , m = n  h; keytype U[1..h] , V[1..m] ; copy S[1] through S[h] to U[1] through U[h]; copy S[h+1] through S[n] to V[1] through V[h]; mergesort(h, U); mergesort(m, V); merge(h, m, U, V, S); } } Divide-and-Conquer

  11. 27 10 12 20 25 13 15 22 27 10 12 20 25 13 15 22 25 13 15 22 27 10 12 20 10 27 12 20 13 25 15 22 10 12 20 27 13152225 1012131520222527 Recursive Steps for Example 2.2 27 10 12 20 25 13 15 22 Divide-and-Conquer

  12. Algorithm 2.3：Merge • voidmerge(inth, intm, const keytype U[], const keytype V[], keytypeS[]) { index i, j, k ; i = 1; j = 1; k = 1; while ( i <= h&&j <= m) { if (U[i] < V[j]) { S[k] = U[i]; i++; } else { S[k] = V[j]; j++; } k++; } if (i > h) copy V[j] through V[m] to S[k] through S[h+m]; else copy U[i] through U[h] to S[k] through S[h+m]; } Divide-and-Conquer

  13. Analysis of Algorithm 2.3 • Worst-Case Time Complexity (Merge) Merge U[h] and V[m] to S[h+m] One of the worst-case： S[]={U[1],U[2],…,U[h1],V[1],V[2],…,V[m],U[h]} The number of comparisons of U[i] and V[j]： h+m1 Hence, the worst-case time complexity is W(h,m)=h+m1 Divide-and-Conquer

  14. 2 mergersort 1 merge Analysis of Algorithm 2.2 • Worst-Case Time Complexity (Mergesort) Let W(n) be the time complexity to sort S[n]. Divide S[n] into 2subarrays with size n/2, call mergesort for each subarray, and then merge them. W(n) = 2W(n/2) + (n/2+n/21) = 2W(n/2) + n1 Using Master Theorem, a = 2, b = 2, d = 1, so a = bd , W(n)  (n log2 n) Divide-and-Conquer

  15. Algorithm 2.4：Mergesort 2 • voidmergesort2 (indexlow, indexhigh) { indexmid; if (low < high) { mid = (low + high)/2 ; mergesort2(low , mid); mergesort2(mid +1 , high); merge2(low, mid, high); } } Divide-and-Conquer

  16. Algorithm 2.5：Merge 2 • voidmerge2(indexlow,indexmid, indexhigh) { indexi, j, k ; keytype U[low..high]; i = low; j = mid + 1; k= low; while (imid&& j  high ) { if (S[i] < S[j]) { U[k] = S[i]; i ++; } else { U[k] = S[j]; j ++; } k ++; } if (i > mid) move S[j] through S[high] to U[k] through U[high]; else move S[i] through S[mid] to U[k] through U[high]; move U[low] through U[high] to S[low] through S[high]; } Divide-and-Conquer

  17. Pivot item 15 22 13 27 12 10 20 25 10 15 15 15 10 13 13 13 13 12 13 12 22 12 12 27 15 10 27 15 12 22 20 22 22 22 27 27 10 10 25 20 20 20 20 25 25 25 25 27 2.4 Quicksort (Partition Exchange Sort) Example 2.3 Suppose the array：1522 13 27 12 10 20 25 Partition the array：all items < pivot are to the left and all items> pivot to the right 2.Sort the subarrays： Divide-and-Conquer

  18. Steps of Quicksort Divide-and-Conquer

  19. Algorithm 2.6：Quicksort • Problem:Sort n keys in nondecreasing order. • Inputs：positive integer n, array of keys S index • from 1 to n. • Outputs: the array S containing the keys in • nondecreasing order. • voidquicksort (indexlow, indexhigh) • { indexpivotpoint; • if (high > low) • { partition(low, high, pivotpoint) • quicksort ( low, pivotpoint－1) ; • quicksort (pivotpoint＋1, high) • } • } Divide-and-Conquer

  20. 15 22 13 27 12 10 20 25 15 10 15 15 13 13 13 13 12 12 12 22 15 27 10 27 12 22 22 22 27 10 10 27 20 20 20 20 25 25 25 25 Algorithm 2.7：Partition voidpartition (indexlow, indexhigh, index& pivotpoint) { indexi, j; keytypepivotitem; pivotitem = S[low]; j = low; for (i = low + 1; i  high; i ++) if (S[i] < pivotitem) { j ++; exchange S[i] and S[j]; } pivotpoint = j; exchange S[low] and S[pivotpoint]; } Divide-and-Conquer

  21. Steps of Partition Divide-and-Conquer

  22. Analysis of Algorithm 2.7 (Partition) • Every-Case Time Complexity (Partition) Basic operation：the comparison of S[i] with pivotitem. Input size: n = high low+1, the number of items in the subarray. Every item except the first is compared. Hence, T(n) = n－1. Divide-and-Conquer

  23. Time to sort right subarray Time to sort left subarray Time to partition Analysis of Algorithm 2.6 • Worst-Case Time Complexity (Quicksort) W(n) = W(0) ＋ W(n1) ＋n1 = W(n1) ＋n1 = n(n1)/2  W(n)  (n2) Divide-and-Conquer

  24. Average time to sort subarrays when pivotpoint is p Time to partition Analysis of Algorithm 2.6 • Average-Case Time Complexity Divide-and-Conquer

  25. 15 22 13 27 12 10 20 25 15 15 10 10 13 10 13 12 13 15 12 27 12 27 27 22 22 22 20 20 20 25 25 25 Another Partition Method voidpartition2 (indexlow, indexhigh, index& pivotpoint) { indexi, j; keytypepivotitem; pivotitem = S[low]; i = low; j = high+1; while (i< j) {while (i < high && S[i+1] < pivotitem) i++; while (j > low &&S[j1] > pivotitem) j; i++;j; if (i < j) exchange S[i] and S[j]; } pivotpoint = j; exchange S[low] and S[pivotpoint]; } Divide-and-Conquer

  26. 2.5 Strassen’s Matrix Multiplication Algorithm Example 2.4：product of 2x2 matrices, A and B Strassen let Divide-and-Conquer

  27. Strassen’s Matrix Multiplication：n×n matrices where Cij, Aij and Bij are all(n/2) ×(n/2)matrices. Example 2.5：Find the product of A and B. Sol： Divide-and-Conquer

  28. Algorithm 2.8:Strassen • Problem:Find the product of two n×n matrices • where n is a power of 2. • Inputs：aninteger n that is power of 2, and two • matrices A and B. • Outputs: the product C of A and B. • voidstrassen (intn, nn_matrixA, B, nn_matrix&C) • { if (nthreshold) • compute C = AB using the standard algorithm; • else • { partitionA into four submatrices A11, A12, A21, A22 ; • partitionB into four submatrices B11, B12, B21, B22 ; • compute C = AB using the Strassen’s method; • } • } Divide-and-Conquer

  29. Analysis of Algorithm 2.8 • Every-Case Time Complexity(no. of multiplications) By Master Theorem： Divide-and-Conquer

  30. Analysis of Algorithm 2.8 • Every-Case Time Complexity(no. of ＋/－) BY Master Theorem： Divide-and-Conquer

  31. Comparison Standard Algorithm Strassen’s Algorithm Multiplications Additions/ Subtraction Divide-and-Conquer

  32. 2.6 Arithmetic with Large Integers Representation of Large Integers Array of Integers：each element store a digital Ex：543127 S[]={0, 5, 4, 3, 1, 2, 7} -543127 S[]={1, 5, 4, 3, 1, 2, 7} Addition, Subtraction, and Division All can be done in Linear Time. Divide-and-Conquer

  33. Multiplication of Large Integers Split the integer of n digits into two integers, one with n/2digits, one with n/2 digits Ex：567832=567×103 + 832 9423723=9423×103 + 723 Example 2.6：567832×9423723=? Sol： Divide-and-Conquer

  34. Algorithm 2.9 ：Larger Integer Multiplication • Problem:Multiply two large integers, u and v. • Inputs：Largeintegers u and v. • Outputs: the product of u and v. • large-integerprod(large-integeru, large-integerv) • { large-integerx, y, w, z; • int n, m; • n = max( number of digits in u, number of digits in v) • if (u = 0 || v = 0) return 0; • else if ( n threshold) returnu×v obtained in the usual way; • else{m=n/2; • x=u divide10m ; y=urem 10m ; • w=v divide10m ; z=vrem 10m; • return prod(x, w)102m (prod(x, z)+prod(w, y))10m •  prod(y, z); } • } Divide-and-Conquer

  35. Analysis of Algorithm 2.9 Worst-Case Time Complexity： By Master Theorem： Gain no profit from the criterion of Time Complexity. Then why use it? Another efficient way can be considered. Divide-and-Conquer

  36. Algorithm 2.10 • Problem:Multiply two large integers, u and v. • Inputs：Largeintegers u and v. • Outputs: the product of u and v. • large-integerprod2 (large-integeru, large-integerv) • { large-integerx, y, w, z , r, p, q; • int n, m; • n = max( number of digits in u, number of digits in v) • if (u = 0 || v = 0) return 0; • else if ( n threshold) returnu×v obtained in the usual way; • else{ m=n/2; x=u divide10m ; y=urem 10m; • w=v divide10m ; z=vrem 10m; • r=prod2(x+y, w+z); p=prod2(x, w); q=prod2(y, z); • return p102m (rp  q)10m q; • } • } Divide-and-Conquer

  37. Analysis of Algorithm 2.10 Worst-Case Time Complexity： BY Master Theorem： Divide-and-Conquer

  38. 2.7 Fast Fourier Transform (FFT) Discrete Time Fourier Series (DTFS)： Complexity of Direct Computation： For onex[n]orX[k]: (N) For allx[n]orX[k]: (N2) 0 = 2/N : fundamental frequency Divide-and-Conquer

  39. Let N = 2m DTFS Decomposition Generally used for the case N = 2p Divide-and-Conquer

  40. 8-point DTFS Divide-and-Conquer

  41. 8-point DTFS (Cont’d) Divide-and-Conquer

  42. = 8-point DTFS FFT Algorithm for N=23 Divide-and-Conquer

  43. = 4-point DTFS = 4-point DTFS = 8-point DTFS FFT Algorithm for N=23 Divide-and-Conquer

  44. = 2-point DTFS = 4-point DTFS = 8-point DTFS FFT Algorithm for N=23 Divide-and-Conquer

  45. = 2-point DTFS = 4-point DTFS = 8-point DTFS FFT Algorithm for N=23 Divide-and-Conquer

  46. Complexity of FFT Algorithm Complexity of Addition： By Master Theorem: Complexity of Multiplication： Similarly, Hence, the total complexity of FFT is(N lgN) . Divide-and-Conquer

  47. Amplitude: Phase: Complex Multiplication & Addition Complex Addition Rule: 1. R = R1+R2 2. I = I1+I2 Complex Multiplication Rule: 1.2. Data Structure for Complex: X[k] = R[k]+jI[k] X[k]: DTFS, R[k]: Real, I[k]: Imaginary Divide-and-Conquer

  48. 2.8 Determining Thresholds Q: When we are only sorting 8 keys, is Exchange Sort faster than the recursive Mergesort? A：For a small n, recursive Mergesort requires a fair amount of overhead. That is, we would like to find an optimal threshold value of n. When the number of keys is larger than this threshold value n, Mergesort is faster then Exchange Sort. Any other algorithms can be worthy only under the condition that n is larger than their respective threshold values. Divide-and-Conquer

  49. Example 2.7 The optimal threshold for Algorithm 2.5：Mergesort 2 Threshold value t satisfies Divide-and-Conquer

  50. Example 2.8 Find the optimal threshold for a divide-and-conquer algorithm with if an other iterative algorithm requires only n2s. Sol： Divide-and-Conquer

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