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Mergesort

Department of Computer and Information Science, School of Science, IUPUI. Mergesort. Dale Roberts, Lecturer Computer Science, IUPUI E-mail: droberts@cs.iupui.edu. Computational Complexity. Framework to study efficiency of algorithms. Example = sorting.

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Mergesort

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  1. Department of Computer and Information Science,School of Science, IUPUI Mergesort Dale Roberts, Lecturer Computer Science, IUPUI E-mail: droberts@cs.iupui.edu

  2. Computational Complexity • Framework to study efficiency of algorithms. Example = sorting. • MACHINE MODEL = count fundamental operations. • count number of comparisons • UPPER BOUND = algorithm to solve the problem (worst-case). • N log2 N from mergesort • LOWER BOUND = proof that no algorithm can do better. • N log2 N - N log2 e • OPTIMAL ALGORITHM: lower bound ~ upper bound. • mergesort

  3. a1 < a2 YES NO a2 < a3 a1 < a3 YES NO YES NO a1 < a3 a2 < a3 YES NO YES NO Decision Tree printa1, a2, a3 printa2, a1, a3 printa1, a3, a2 printa3, a1, a2 printa2, a3, a1 printa3, a2, a1

  4. Comparison Based Sorting Lower Bound • Theorem. Any comparison based sorting algorithm must use (N log2N) comparisons. • Proof. Worst case dictated by tree height h. • N! different orderings. • One (or more) leaves corresponding to each ordering. • Binary tree with N! leaves must have height • Food for thought. What if we don't use comparisons? • Stay tuned for radix sort. Stirling's formula

  5. Mergesort Analysis • How long does mergesort take? • Bottleneck = merging (and copying). • merging two files of size N/2 requires N comparisons • T(N) = comparisons to mergesort N elements. • to make analysis cleaner, assume N is a power of 2 • Claim. T(N) = N log2 N. • Note: same number of comparisons for ANY file. • even already sorted • We'll prove several different ways to illustrate standard techniques.

  6. void merge(Item a[], int left, int mid, int right) <999>{ int i, j, k; for (<999>i = mid+1; <6043>i > left; <5044>i--) <5044>aux[i-1] = a[i-1]; for (<999>j = mid; <5931>j < right; <4932>j++) <4932>aux[right+mid-j] = a[j+1]; for (<999>k = left; <10975>k <= right; <9976>k++) if (<9976>ITEMless(aux[i], aux[j])) <4543>a[k] = aux[i++]; else <5433>a[k] = aux[j--]; <999>} void mergesort(Item a[], int left, int right) <1999>{ int mid = <1999>(right + left) / 2; if (<1999>right <= left) return<1000>; <999>mergesort(a, aux, left, mid); <999>mergesort(a, aux, mid+1, right); <999>merge(a, aux, left, mid, right); <1999>} Mergesort prof.out Profiling Mergesort Empirically Striking feature:All numbers SMALL! # comparisonsTheory ~ N log2 N = 9,966Actual = 9,976

  7. Sorting Analysis Summary • Running time estimates: • Home pc executes 108 comparisons/second. • Supercomputer executes 1012 comparisons/second. • Lesson 1: good algorithms are better than supercomputers. • Lesson 2: great algorithms are better than good ones. Insertion Sort (N2) Mergesort (N log N) computer thousand million billion thousand million billion home instant 2.8 hours 317 years instant 1 sec 18 min super instant 1 second 1.6 weeks instant instant instant Quicksort (N log N) thousand million billion instant 0.3 sec 6 min instant instant instant

  8. Acknowledgements • Sorting methods are discussed in our Sedgewick text. Slides and demos are from our text’s website at princeton.edu. • Special thanks to Kevin Wayne in helping to prepare this material.

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