Computational Mathematicsematics

Algorithms and Theory of Computation Handbook by Richard E. Klima, Neil Sigmon, Ernest Stitzinger

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By Richard E. Klima, Neil Sigmon, Ernest Stitzinger

As well as conventional themes, this finished compendium of algorithms, information constructions, and thought of computation covers:oapplications parts the place algorithms and knowledge structuring strategies are of exact significance ograph drawingorobot algorithmsoVLSI layoutovision and picture processing algorithmsoschedulingoelectronic cashodata compressionodynamic graph algorithmsoon-line algorithmsomultidimensional information structuresocryptographyoadvanced themes in combinatorial optimization and parallel/distributed computingUnique insurance of Algorithms and idea of Computation instruction manual makes it a necessary reference for researchers and practitioners in those functions parts.

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By the triangle inequality, Cl,k ≤ Cl,m + Cm,k , and by symmetry we can combine these two inequalities to get Cl,k ≤ Cl,m + Cx,y . Adding this last inequality to the first one above, Cl,k + Ck,m ≤ Cl,m + 2Cx,y , that is, Cl,k + Ck,m − Cl,m ≤ 2Cx,y . Thus adding city k between cities l and m adds no more to In than 2Cx,y . Summing these incremental amounts over the cost of the entire algorithm tells us |In | ≤ 2 |On | , as we claimed. 3 we saw that we could sort faster than na¨ıve (n2 ) worst-case behavior algorithms: we designed more sophisticated (n log n) worst-case algorithms.

If we have many persons (more precisely k > log n), we can use binary search. In both cases, the solution is optimal in the worst case. If we have two persons, a first solution would be to start using binary search with the first person, and then use the second sequentially in the remaining segment. In the worst case, the first person fails in the first jump, giving a n/2 jumps algorithm. The problem is that both persons do not perform the same amount of work. We can balance the work by using the following algorithm: the first person tries sequentially every n/p floors for a chosen p, that is n/p, 2n/p, etc.

This observation follows by examining the correspondence between permutations and outcome boxes. Since the decision tree arose by tracing through the algorithm for all © 1999 by CRC Press LLC possible input sequences (that is, permutations), an outcome box must have occurred as the result of some input permutation or it would not be in the decision tree. Moreover, it is impossible that there are two different permutations corresponding to the same outcome box—such an algorithm cannot sort all input sequences correctly.

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