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Sorting Algorithms

Sorting Algorithms. Insertion Sort: Θ (n 2 ) Merge Sort: Θ ( nlog (n)) Heap Sort: Θ ( nlog (n)) We seem to be stuck at Θ ( nlog (n)) Hypothesis: Every sorting algorithm requires Ω ( nlog (n)) time. Lower Bound Definitions. Merge sort: Ω ( nlog (n)) on every input Insertion sort:

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Sorting Algorithms

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  1. Sorting Algorithms • Insertion Sort: Θ(n2) • Merge Sort:Θ(nlog(n)) • Heap Sort:Θ(nlog(n)) • We seem to be stuck at Θ(nlog(n)) • Hypothesis: Every sorting algorithm requires Ω(nlog(n)) time.

  2. Lower Bound Definitions • Merge sort: Ω(nlog(n)) on every input • Insertion sort: • 1, 2, 3, 4,…, n-1, n  O(n) time • n, n-1, n-2, …, 2, 1  O(n2) time • Hypothesis: For every sorting algorithm A and every integer n, there is some input of length n on which A requires Ω(nlog(n)) time.

  3. Proving Lower Bounds • What if there is some absurd O(n) algorithm? E.g. • Square every third element • For every prime j, A[j] = 2^A[j] • For every j, look up A[j]’th word in the August 2013 New York Times • Etc.

  4. Comparison sorting • Want algorithms that work on any input • e.g. insertion/merge/heap sort • Think about sorting: uses comparisons. • Comparison based sorting algorithm only relies on comparisons + moves • Running time = Ω(# of comparisons)

  5. New Hypothesis Every comparison-based sorting algorithm requires Ω(nlog(n)) comparisons in the worst case.

  6. Required Comparisons • O(nlog(n)) comparisons suffices • merge sort, heap sort • Trivial: need Ω(n/2) comparisons

  7. New Hypothesis Every comparison-based sorting algorithm requires Ω(nlog(n)) comparisons in the worst case. Observation: Tree of height k has at most 2k leaves.

  8. Average Case Analysis • Ω(nlog(n)) comparisons in worst case • Merge sort: always Θ(nlog(n)) • Insertion sort: sometimes Θ(n), sometimes Θ(n2) • Can we get the best of both? Sometimes Θ(nlog(n)), usually O(n)? • NO: need Ω(nlog(n)) comparisons on average among all possible inputs.

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