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By Mehlhorn K., Sanders P.

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En if and only if the polynomial identity (z − e1 ) · (z − en ) − (z − e1 ) · (z − en ) = 0 holds for all z. For any ε > 0 let p denote a prime such that p > max{n/ε, e1 , . . , en , e1 , . . , en }. p − 1. Show that if e 1 , . . , en is not a permutation of e1 , . . , en then the result of the evaluation is zero with probability at most ε. Hint: A nonzero polynomial of degree n has at most n zeroes. 1 Simple Sorters Perhaps the conceptually simplest sorting algorithm is selection sort: Start with an empty output sequence.

Therefore, we only discuss some qualitative issues without detailed proof. An advantage of linear probing is that, in each table access, a contiguous piece of memory is accessed. The memory subsystems of modern processors are optimized for this kind of access pattern, whereas they are quite slow at chasing pointers when the data does not fit in cache memory. A disadvantage of linear probing is that search times become very high when the number of elements approaches the table size. For chaining, the expected access time remains very small.

The execution time in the more detailed model can be recovered by adding the time for one hash function evaluation and for a constant number of element moves to insert and remove. Hash Tables 61 optimistic assumptions about the properties of the hash function. In this model, we achieve constant expected time for dictionary operations. 2 we drop these assumptions and explain how to construct hash functions that come with (probabilistic) performance guarantees. Note that our simple examples already show that finding good hash functions is nontrivial.

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Concise Algorithmics: The Basic Toolbox by Mehlhorn K., Sanders P.

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