SortingLowerBound - Every possible input permutation must...

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Sorting Lower Bound 1 Sorting Lower Bound Sorting Lower Bound 2 Comparison-Based Sorting (§10.4) Many sorting algorithms are comparison based. ± They sort by making comparisons between pairs of objects ± Examples: bubble-sort, selection-sort, insertion-sort, heap-sort, merge-sort, quick-sort, . .. Let us therefore derive a lower bound on the running time of any algorithm that uses comparisons to sort n elements, x 1 , x 2 , …, x n . Is x i < x j ? yes no Sorting Lower Bound 3 Counting Comparisons Let us just count comparisons then. Each possible run of the algorithm corresponds to a root-to-leaf path in a decision tree x i < x j ? x a < x b ? x m < x o ? x p < x q ? x e < x f ? x k < x l ? x c < x d ? Sorting Lower Bound 4 Decision Tree Height The height of this decision tree is a lower bound on the running time
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Unformatted text preview: Every possible input permutation must lead to a separate leaf output. ± If not, some input …4…5… would have same output ordering as …5…4…, which would be wrong. Since there are n!=1*2*…*n leaves, the height is at least log (n!) minimum height (time) log ( n !) x i < x j ? x a < x b ? x m < x o ? x p < x q ? x e < x f ? x k < x l ? x c < x d ? n ! Sorting Lower Bound 5 The Lower Bound Any comparison-based sorting algorithms takes at least log (n!) time Therefore, any such algorithm takes time at least That is, any comparison-based sorting algorithm must run in Ω (n log n) time. ). 2 / ( log ) 2 / ( 2 log ) ! ( log 2 n n n n n = ≥...
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