Fundamental Structures of Computer Science II

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Fundamental Structures of Computer Science II Sorting - 2 CS 202 – Fundamental Structures of Computer Science II Bilkent University Computer Engineering Department CS 202, Spring 2003 Fundamental Structures of Computer Science II Bilkent University

Fundamental Structures of Computer Science II Heapsort For a binary heap, building the heap takes O(N) time. In a binary heap, deleteMin gives the smallest element. Running time of O(logN). Basic strategy for heap sort given N elements Build a binary heap from N elements (O(N)) Perform N deleteMin operations (N*logN) Put the result of deleteMins in a new array. The new array is sorted. CS 202, Spring 2003 Fundamental Structures of Computer Science II Bilkent University

Fundamental Structures of Computer Science II Heapsort The problem is that: This strategy uses a new array This means waste of space for large N. A way for using only one array. When minimum item is deleted, put it to the end of the array. After N deleteMins: The original array will have the elements in sorted order from max to min. If you want items in increasing order (from min to max), then use a Max-Heap. CS 202, Spring 2003 Fundamental Structures of Computer Science II Bilkent University

Fundamental Structures of Computer Science II Example Assume the input that is to be sorted: 31, 41, 59, 26, 53, 58, 97 N = 7. We need first to build a binary heap (a Max-Heap) A max-Heap is heap that has the largest item in root. CS 202, Spring 2003 Fundamental Structures of Computer Science II Bilkent University

Fundamental Structures of Computer Science II Example 97 1 2 3 4 5 6 7 8 9 10 53 59 97 53 59 26 41 58 31 58 31 26 41 deleteMax Max-Heap after BuildHeap() 59 1 2 3 4 5 6 7 8 9 10 53 58 59 53 58 26 41 31 97 31 97 26 41 Heap after first DeleteMin() CS 202, Spring 2003 Fundamental Structures of Computer Science II Bilkent University

Fundamental Structures of Computer Science II Example After 5 more deleteMins, the array will look like the following. 1 2 3 4 5 6 7 8 9 10 26 31 41 53 58 59 97 This is a sorted array CS 202, Spring 2003 Fundamental Structures of Computer Science II Bilkent University

template <class Comparable> i is running from N/2 to 0 template <class Comparable> void heapsort( vector<Comparable> & a ) { /* 1*/ for( int i = a.size( ) / 2; i >= 0; i-- ) /* buildHeap */ /* 2*/ percDown( a, i, a.size( ) ); /* 3*/ for( int j = a.size( ) - 1; j > 0; j-- ) /* 4*/ swap( a[ 0 ], a[ j ] ); /* deleteMax */ /* 5*/ percDown( a, 0, j ); } j is running from N -1 to 1 Swap first and last elements CS 202, Spring 2003 Fundamental Structures of Computer Science II Bilkent University

template <class Comparable> Point from where to start percolate down Logical size of heap template <class Comparable> void percDown( vector<Comparable> & a, int i, int n ) { int child; Comparable tmp; /* 1*/ for( tmp = a[ i ]; leftChild( i ) < n; i = child ) /* 2*/ child = leftChild( i ); /* 3*/ if( child != n - 1 && a[ child ] < a[ child + 1 ] ) /* 4*/ child++; /* 5*/ if( tmp < a[ child ] ) /* 6*/ a[ i ] = a[ child ]; else /* 7*/ break; } /* 8*/ a[ i ] = tmp; CS 202, Spring 2003 Fundamental Structures of Computer Science II Bilkent University

Fundamental Structures of Computer Science II heapsort() 1 2 3 4 5 6 7 8 9 97 53 59 26 41 58 31 j=6 a.size = 7 swap a[0] with a[6] 1 2 3 4 5 6 7 8 9 31 53 59 26 41 58 97 a.size = 7 percolatedown(a, 0, 6) CS 202, Spring 2003 Fundamental Structures of Computer Science II Bilkent University

Fundamental Structures of Computer Science II percDown() 1 2 3 4 5 6 7 8 9 31 53 59 26 41 58 97 i=0 tmp = 31, n=6 1 2 3 4 5 6 7 8 9 59 53 26 41 58 97 1 2 3 4 5 6 7 8 9 59 53 26 41 58 97 i=2 tmp = 31, n=6 1 2 3 4 5 6 7 8 9 59 53 58 26 41 31 97 Go out of for loop! CS 202, Spring 2003 Fundamental Structures of Computer Science II Bilkent University

Fundamental Structures of Computer Science II heapsort() 1 2 3 4 5 6 7 8 9 59 53 58 26 41 31 97 j=5 Swap(a[0], a[5]) 1 2 3 4 5 6 7 8 9 31 53 58 26 41 59 97 j=5 percolateDown(a, 0, 5) CS 202, Spring 2003 Fundamental Structures of Computer Science II Bilkent University

Fundamental Structures of Computer Science II percDown() 1 2 3 4 5 6 7 8 9 31 53 58 26 41 59 97 i=0 1 2 3 4 5 6 7 8 9 58 53 31 26 41 59 97 CS 202, Spring 2003 Fundamental Structures of Computer Science II Bilkent University

Fundamental Structures of Computer Science II MergeSort CS 202, Spring 2003 Fundamental Structures of Computer Science II Bilkent University

Fundamental Structures of Computer Science II MergeSort Runs in O(NlogN) worst case running time. A recursive algorithm Based on merging to sorted lists List 1: 2,3,5,9, 10 List 2: 1, 4, 6,7, 11 Merged List: 1,2,3, 4,5,6,7,10,11 It is enough to do one pass over the list 1 and list 2 CS 202, Spring 2003 Fundamental Structures of Computer Science II Bilkent University

Fundamental Structures of Computer Science II Merging Algorithm Merge Array A Array B Array C 1 13 24 26 2 15 27 38 Actr Bctr Cctr We maintain 3 counters: Actr, Bctr, Cctr CS 202, Spring 2003 Fundamental Structures of Computer Science II Bilkent University

Fundamental Structures of Computer Science II Pseudo-code for merge While both Actr and Bctr did not reach to end of arrays { if A[Actr] < B[Bctr] then C[Cctr] = A[Actr] Actr++ else C[Cctr] = B[Bctr] Bctr++ } If Actr did not reach end of array A copy rest of A to C If B ctr did not reach end of array B copy rest of B to C CS 202, Spring 2003 Fundamental Structures of Computer Science II Bilkent University

Fundamental Structures of Computer Science II Array A Array B Array C 1 13 24 26 2 15 27 38 Actr Bctr Cctr Array A Array B Array C 1 13 24 26 2 15 27 38 1 Actr Bctr Cctr 1 13 24 26 2 15 27 38 1 2 Actr Bctr Cctr CS 202, Spring 2003 Fundamental Structures of Computer Science II Bilkent University

Fundamental Structures of Computer Science II 1 13 24 26 2 15 27 38 1 2 13 Actr Bctr Cctr 1 13 24 26 2 15 27 38 1 2 13 15 Actr Bctr Cctr 1 13 24 26 2 15 27 38 1 2 13 15 24 Actr Bctr Cctr CS 202, Spring 2003 Fundamental Structures of Computer Science II Bilkent University

Fundamental Structures of Computer Science II Exhausted 1 13 24 26 2 15 27 38 1 2 13 15 24 26 Actr Bctr Cctr Copy rest of B to C 1 13 24 26 2 15 27 38 1 2 13 15 24 26 27 38 Actr Bctr Cctr Result of merge CS 202, Spring 2003 Fundamental Structures of Computer Science II Bilkent University

MergeSort Algorithm Description Mergesort (A[1:N]) If N = 1 then return Else Mergesort (A[1:N/2]) Mergesort (A[N/2+1:N] Merge (A[1:N/2], A[N/2+1:N]) CS 202, Spring 2003 Fundamental Structures of Computer Science II Bilkent University

Fundamental Structures of Computer Science II example Mergesort the following array 10 5 20 7 3 30 1 9 CS 202, Spring 2003 Fundamental Structures of Computer Science II Bilkent University

Fundamental Structures of Computer Science II Level 1 10 5 20 7 3 30 1 9 10 5 20 7 3 30 1 9 Divide(1) Level 2 Level 2 10 5 20 7 3 30 1 9 Mergesort(2) Mergesort(2) Merge(1) CS 202, Spring 2003 Fundamental Structures of Computer Science II Bilkent University

Fundamental Structures of Computer Science II Level 2 10 5 20 7 3 30 1 9 Level 2 10 5 20 7 Divide(2) Level 3 Level 3 10 5 20 7 Mergesort(3) Mergesort(3) Merge(2) CS 202, Spring 2003 Fundamental Structures of Computer Science II Bilkent University

Fundamental Structures of Computer Science II Level 2 10 5 20 7 3 30 1 9 Level 2 3 30 1 9 Divide (2) Level 3 Level 3 3 30 1 9 Mergesort(3) Mergesort(3) Merge(2) CS 202, Spring 2003 Fundamental Structures of Computer Science II Bilkent University

Fundamental Structures of Computer Science II Level 3 before 10 5 20 7 3 30 1 9 Level 3 after 5 10 20 7 3 30 1 9 10 5 Divide (3) Level 3 Level 3 10 5 Mergesort(4) Mergesort(4) 10 10 Merge(3) 5 10 CS 202, Spring 2003 Fundamental Structures of Computer Science II Bilkent University

Fundamental Structures of Computer Science II Level 3 before 10 5 20 7 3 30 1 9 Level 3 after 10 5 7 20 3 30 1 9 20 7 Divide (3) Level 3 Level 3 20 7 Mergesort(4) Mergesort(4) 20 7 Merge(3) 7 20 CS 202, Spring 2003 Fundamental Structures of Computer Science II Bilkent University

Fundamental Structures of Computer Science II Level 3 before 10 5 20 7 3 30 1 9 Level 3 after 10 5 20 7 3 30 1 9 3 30 Divide (3) Level 3 Level 3 3 3 Mergesort(4) Mergesort(4) 30 30 Merge(3) 3 30 CS 202, Spring 2003 Fundamental Structures of Computer Science II Bilkent University

Fundamental Structures of Computer Science II Level 3 before 10 5 20 7 3 30 1 9 Level 3 after 10 5 20 7 3 30 1 9 1 9 Divide (3) Level 3 Level 3 1 9 Mergesort(4) Mergesort(4) 1 9 Merge(3) 1 9 CS 202, Spring 2003 Fundamental Structures of Computer Science II Bilkent University

Fundamental Structures of Computer Science II Back to Level 2 before 10 5 20 7 3 30 1 9 Level 2 after 5 7 10 20 3 30 1 9 10 5 20 7 Divide (2) Level 3 Level 3 10 5 20 7 Mergesort(3) Mergesort(3) 5 10 7 20 Merge(2) 5 7 10 20 CS 202, Spring 2003 Fundamental Structures of Computer Science II Bilkent University

Fundamental Structures of Computer Science II Back to Level 2 before 10 5 20 7 3 30 1 9 Level 2 after 5 7 10 20 1 3 9 30 3 30 1 9 Divide (2) Level 3 Level 3 3 30 1 9 Mergesort(3) Mergesort(3) 3 30 1 9 Merge(2) 1 3 9 30 CS 202, Spring 2003 Fundamental Structures of Computer Science II Bilkent University

Fundamental Structures of Computer Science II Back to Level 1 before 10 5 20 7 3 30 1 9 after 1 3 5 7 9 10 20 30 10 5 20 7 3 30 1 9 Divide (1) Level 2 Level 2 10 5 20 7 3 30 1 9 Mergesort(2) Mergesort(2) 5 7 10 20 1 3 9 30 Merge(1) Result 1 3 5 7 9 10 20 30 CS 202, Spring 2003 Fundamental Structures of Computer Science II Bilkent University

Fundamental Structures of Computer Science II MergeSort The sorted array is 1,3,5,7,9,10,20,30 MergeSort is an example of Divide and Conquer Approach in Algorithm Design. CS 202, Spring 2003 Fundamental Structures of Computer Science II Bilkent University