Divide and Conquer

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Divide and Conquer 8item_t a[n]maxmin ( i, j, max, min ){if ( i == j ){max = min = a[i]return}if ( i == j - 1 ){if ( a[i] < a[j] ){max = a[j]min = a[i]}else{max = a[i]min = a[j]}return}// Global array// Only one element// Only two elements// Dividemid = floor ( ( i + j ) / 2 )maxmin ( i, mid, max, min )maxmin ( mid+1, j, max1, min1 )// Conquer}if ( max < max1 ) max = max1if ( min > min1 ) min = min1return• Analyzing divide and conquer maxmin described by the recurrence⎧⎨ T n + T n + 2 n > 22 2T n = 1 n = 2⎩0 n = 1Let n = 2 k , for some k > 0T n = 2T n2 + 2= 2(2T n4 + 2) + 2= 2 2 T n2 2 + 22 + 2= 2 2 (2T n2 3 + 2) + 22 + 2= 2 3 T n2 3 + 23 + 2 2 + 2..= 2 k−1 T n∑k−1+ 2 i2 k−1i=1
Divide and Conquer 9k−1∑= 2 k−1 +i=12 i= 2 k−1 + 2(k−1)+1 − 12 − 1= 2 k−1 + 2 k − 1= n 2 + n − 1= 3n 2 − 1Merge sort• Recursive algorithm• Algorithm can be described by the following recurrence{ 2T n + cn n > 1, c is a constantT n =2a n = 1, a is a constantQuick sort• The array to be sorted is partitioned at a pivot element such that the elements at indices less than that of the pivot are lessthan the pivot while the elements with indices greater than the pivot are larger than the pivot• Quicksort eliminates the need for a subsequent merge as required by merge sort• Performance analysis– Worst case when elements are already in sorted order• Randomized quick sort– Select pivot as median of three– Select a random element as the pivot (Las Vegas algorithm)Selection• Selecting kth smallest element using partition from quicksortStrassen’s matrix multiplication• Given two n × n matrices A and B; their product C is given byC ij =– It requires n multiplications for each element in C– Total time to perform above computation is Θ(n 3 )• Divide and conquer strategy– Assume that n = 2 kn∑A ik B kjk=1
Page 1 and 2: Divide and ConquerGeneral methodDiv
Page 3 and 4: Divide and Conquer 3- Pitfall∗ T
Page 5 and 6: Divide and Conquer 5· Case 2: n lo
Page 7: Divide and Conquer 7return ( r );el
Page 11 and 12: Divide and Conquer 11• Structure
Page 13 and 14: Divide and Conquer 13p3 = p, for wh

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