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Permuting Lower Bound

Permuting Lower Bound. Permuting N elements according to a given permutation takes I/ Os in “indivisibility” model Indivisibility model: Move of elements only allowed operation Note: We can allow copies (and destruction of elements)

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Permuting Lower Bound

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  1. I/O-efficient Algorithms and Data Structures Permuting Lower Bound Permuting N elements according to a given permutation takes I/Os in “indivisibility” model • Indivisibility model: Move of elements only allowed operation • Note: • We can allow copies (and destruction of elements) • Bound also a lower bound on sorting • Proof: • View memory and disk as array of N tracks of B elements • Assume all I/Os track aligned (assumption can be removed) D M

  2. I/O-efficient Algorithms and Data Structures Permuting Lower Bound • Array contains permutation of N elements at all times • We will count how many permutations can be reached (produced) with t I/Os • Input: • Choose track: N possibilities • Rearrange ≤ B element in track and place among ≤ M-B elements in memory: • possibilities if “fresh” track • otherwise  at most permutations after t inputs • Output: Choose track: N possibilities D M

  3. I/O-efficient Algorithms and Data Structures Permuting Lower Bound • Permutation algorithm needs to be able to produce N! permutations (using Stirlings formula and ) • If we have • If we have and thus    

  4. I/O-efficient Algorithms and Data Structures Sorting lower bound • Similar argument but assuming comparison model in internal memory • Initially N! possible orderings • Count how may possible after t I/Os  Sorting N elements takes I/Os in comparison model   

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