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Foundations of Algorithms, Fourth Edition Richard Neapolitan, Kumarss Naimipour Chapter 2 Divide-and-Conquer

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##### Foundations of Algorithms, Fourth Edition Richard Neapolitan, Kumarss Naimipour Chapter 2 Divide-and-Conquer

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**Foundations of Algorithms, Fourth Edition**Richard Neapolitan, Kumarss Naimipour Chapter 2 Divide-and-Conquer**Divide and Conquer**• In this approach a problem is divided into sub-problems and the same algorithm is applied to every subproblem( often this is done recursively) • Examples • Binary Search (review algorithm in book) • Mergesort (review algorithm in book) • Quicksort**Figure 2.1 : The steps down by a human when searching with**Binary Search. (Note: x = 18)**Complexity of Binary Search**Since this and many other divide and conquer algorithms are recursive you will recall that we can determine their complexity using recurrence relations. For Binary Search we have T(n) = T(n/2) + 1 =[T(n/4)+1]+1 = T(n/22) + 2 =[T(n/8+1]+ 2 = T(n/23)+ 3 … =T(n/2k)+k**What is T(n/2k)+k**We if we let k get larger until n=2k then we see that k = log2n. Why? Consequently the relation becomes T(n) = T(1) + log2n T(n) = log2n Since n/2k is 1 if they are equal and T(1) =1**MergeSort**Recall in this algorithm we divide the array into two equal parts and sort each half prior to merging. The recurrence relation is clearly T(n) = 2T(n/2) + n Recall that Merging is O(n) right?**T(n/2)**T(n/4) O(n) Figure 2.2: The steps done by a human when sorting with Mergesort.**T(n) = 2T(n/2) + n**T(n) = 2T(n/2) + n = 2[ 2T(n/22) + n/2] + n = 22T(n/22) + 2n = 2[2T(n/23) + n/22] +2n =23T(n/23) + 3n … =2kT(n/2k) + kn If n=2k then we have T(n) = nT(1) + (log2n)n = n+ nlog2n = O(nlog2n)**QuickSortWorks in situ!**Void quicksort(int low, int high) { int pivot; if (high > low){ partition(low, high, pivot); quicksort(low, pivot-1); quicksort(pivot+1,high); }**Figure 2.3: The steps done by a human when sorting with**Quicksort. The subarrays are enclosed in rectangles whereas the pivot points are free.**PartitionStudy this carefully**void partition (int low, int high, int&pivot) { intI,j, pivotitem; pivotitem = S[low]; // select left item (hmmm) j=low; for (i=low+1; i<=high; i++) There are many ways if (S[i] < pivotitem){ to write this function! j++; All have a complexity swap S[i] and S[j]; of O(n). } pivot= j; swap S[low] and S[pivot]; }**Complexity of Quicksort**The complexity of this algorithm depends on how good the pivot value selection is . If the value is always in the middle of array then the best case complexity is T(n) = n + 2T(n/2) Which we already have determined is T(n) = n log2 n**Worst case for Quicksort**This clearly will occur if each pivot value is less than (or greater) all the elements of the array. IE the array is split into 1 and n-1 size pieces. This gives a recurrence relation of T(n) = T(1) + T(n-1) + n-1 Time to sort left array right array partition**Worst Case analysis**T(n) = T(1) + T(n-1) + n-1 = T(n-1) + n Assume the answer is n(n-1)/2 check it out ! n(n-1)/2 = 0 + (n-1)(n-2)/2 + n-1 = (n-1)(n-2)/2 + 2(n-1)/2 =((n-1)(n-2)+ 2(n-1))/2 = (n-1)(n-2+2)/2 = n(n-1)/2 ☺**Quick Sort Analysis**Quicksort’s worst case is θ(n2) Does this mean that quick sort is just as bad as say selection sort, insertion sort and/or bubble sort. No! Its all about average case performance. The average case performance for these three is θ(n2) as well. What is the average case complexity for QS?**Average Case Analysisassume prob. pivotpoint is p**See HW 22 p 86 for above conversion**Average case continued**Multiplying by n Subtracting these equations we have**Average case QS continued**Assume we get , Applying some simple math we have Which give )**Matrix Multiplication (Strassen)**Lets look at the product of two 2 by 2’s Clearly after you do the homework #26 m1=(a11+a22)(b11+b22) m5=(a11+a12)b22 m2=(a11+a22)b11 m6=(a21+a11)(b11+b12) m3=a11(b12+b22) m7=(a12+a22)(b21+b22) m4=a22(b21+b11) will give the following!**And the answer is**Original method 8 mult, four add/sub Strassen’s method 7 mult, and 18 add/sub Hmmmm! So what’s the big deal?**Big Matrices 2n by 2n**Where C11 is the upper left hand corner of the matrix of size n/2 by n/2. The others are similarly defined. Now m1=(A11+A22)(B11+B22) Is the sum and product of matrices**Our function is then**void Strassen(int n, A,B, C)// these are nxn mats { if (n<= threshold) computer C=AxB normally Partition A and B into eight submatrices strassen(n/2, A11+A22, B11+B22, M1); strassen(n/2, A21+A22, B11, M2) etc // making 7 recursive calls } NOT EIGHT!**Complexity**T(n) = 7T(n/2) + cn2 Which is T(n) = θ(nlg7) = O(n2.81) Using the general theorem. The best know is Coppersmith and Winograd with a time complexity of O(n2.376) Why am I using big O here?**Recalling General TheoremSee page 588**Assume for n>1 and n a power of b, T(1)=d**Just a side note**Suppose we has 8 recursive calls instead of 7 in the above case. Then the recurrence relation would be T(n) = 8T(n/2) + cn2 This has a complexity of what?**When not to use divide and conquer**• An instance of size n is divided into two or more instances each almost of size n. • An instance of size n is divided into almost n instances of size n/c, where c is a constant