By Wolfgang Hackbusch, Ulrich Trottenberg
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Additional info for Multigrid Methods II
And let ei be the ﬁrst mistake. In other words, there is a minimum spanning tree Tmin containing e1 , e2 , . . , ei−1 , but no minimum spanning tree containing e1 , e2 , . . , ei . Imagine what happens if we add the edge ei to Tmin : since Tmin is a spanning tree, the addition of ei causes a cycle containing ei . Let emax be the highest-cost edge on that cycle not among e1 , e2 , . . , ei . There must be such an emax because e1 , e2 , . . , ei are acyclic, since they are in the spanning tree constructed by Prim’s algorithm.
The probe sequence is then generated as h0 (x) = h(x); hi+1 (x) = (hi (x) + s(x)) mod m. 3 shows the result of inserting keys A, B, . . m − 1], with replacement. For both models, it can be shown that 1 1 +Θ 1−α m 1 1 1 ESn = ln +Θ α 1−α m EUn = Again, for a full table, the above expressions are useless, but we can prove that the search costs are Θ(m) and Θ(log m), respectively. Deletions cannot be done by simply erasing the given element, because searches would stop there and miss any element located beyond that point in its probe sequence.
Combinatorial Algorithms, Theory and Practice, PrenticeHall, Englewood Cliﬀs, NJ, 1977. 11. , An analysis of several heuristics for the traveling salesman problem, SIAM J. , 6, 563–581, 1977. 12. , Data Structures and Network Algorithms, Society of Industrial and Applied Mathematics, Philadelphia, PA, 1983. 2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . Sequential Search . . . . . . . . . . . . . . . . . . . . . . . . 3 Sorted Array Search .