14.Computers by John G. Webster (Editor)

By John G. Webster (Editor)

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Flow networks model many real-life situations, like communication networks and electrical networks. A key problem in this model is to determine how many units of flow the network can support. A simple greedy algorithm to accomplish this is the Ford–Fulkerson algorithm. It repeatedly finds a path from s to t (called an augmenting path) that can admit more flow and increases the flow along this path to the extent possible. It terminates when no augmenting path can be found. It can be shown that this procedure can be inefficient if the choice of the augmenting path is not done prop- 418 ALGORITHM THEORY erly.

Unlike mergesort, which has a running time of ⌰(nlogn) in the worst case, quicksort has a worst-case running time of ⌰(n2). However, on the average, its running time is ⌰(nlogn), and it sorts in place, that is, the amount of extra storage it requires does not depend on n. Good implementations of quicksort are often significantly faster than mergesort. n] be the array to be sorted. n]. These two subarrays are then recursively sorted (in place). An array with only one element is the base case of the recursion and is already sorted.

Otherwise, consider the triangle vivkvj for some k. This divides the remaining part of the polygon ͗vi, . , vj͘ into two subpolygons, ͗vi, . , vk͘ and ͗vk, . , vj͘. Thus, Tij ϭ minjϪ1 kϭiϩ1(Tik ϩ Tkj ϩ perimeter of ⌬vivkvj). The quantities Tik and Tkj have already been computed, so Tij can be evaluated in time ␪( j Ϫ i Ϫ 1). Therefore, this is an O(n3)-time algorithm, but one that computes only the weight of an optimal triangulation. However, the triangulation itself can also be easily constructed by storing, for all i, j, the value of k that minimizes the previous expression for Tij.

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