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A greedy algorithm in .NET Writer gs1 datamatrix barcode in .NET A greedy algorithm

A greedy algorithm generate, create 2d data matrix barcode none with .net projects GS1 RSS At this point, the read datamatrix 2d barcode for .NET er might be forgiven for feeling a slight sense of futility: we have examined several techniques for designing approximation algorithms for the set cover problem, and they have all led to the same result, an approximation algorithm with performance guarantee f . But, as in life, perseverance and some amount of cleverness often pay dividends in designing approximation algorithms.

We show in this section that a type of algorithm called a greedy algorithm gives an approximation algorithm with a performance guarantee that is often signi cantly better than f . Greedy algorithms work by making a sequence of decisions; each decision is made to optimize that particular decision, even though this sequence of locally optimal (or greedy ) decisions might not lead to a globally optimal solution. The advantage of greedy Electronic web edition.

Copyright 2010 by David P. Williamson and David B. Shmoys.

To be published by Cambridge University Press. A greedy algorithm I Sj Sj j while VS .NET data matrix barcodes I is not a set cover do w l arg minj:Sj = j I I {l} Sj Sj Sl. Sj Algorithm 1.2: A greedy Visual Studio .NET DataMatrix algorithm for the set cover problem.

algorithms is that they are typically very easy to implement, and hence greedy algorithms are a commonly used heuristic, even when they have no performance guarantee. We now present a very natural greedy algorithm for the set cover problem. Sets are chosen in a sequence of rounds.

In each round, we choose the set that gives us the most bang for the buck ; that is, the set that minimizes the ratio of its weight to the number of currently uncovered elements it contains. In the event of a tie, we pick an arbitrary set that achieves the minimum ratio. We continue choosing sets until all elements are covered.

Obviously, this will yield a polynomial-time algorithm, since there can be no more than m rounds, and in each we compute O(m) ratios, each in constant time. A formal description is given in Algorithm 1.2.

Before we state the theorem, we need some notation and a useful mathematical fact. Let 1 Hk denote the kth harmonic number: that is, Hk = 1 + 1 + 1 + + k . Note that Hk ln k.

2 3 The following fact is one that we will use many times in the course of this book. It can be proven with simple algebraic manipulations. Fact 1.

10: Given positive numbers a1 , . . .

, ak and b1 , . . .

, bk , then k ai ai ai i=1 max . min k i=1,..

.,k bi i=1,..

.,k bi i=1 bi Theorem 1.11: Algorithm 1.

2 is an Hn -approximation algorithm for the set cover problem. Proof. The basic intuition for the analysis of the algorithm is as follows.

Let OPT denote the value of an optimal solution to the set cover problem. We know that an optimal solution covers all n elements with a solution of weight OPT; therefore there must be some subset that covers its elements with an average weight of at most OPT /n. Similarly, after k elements have been covered, the optimal solution can cover the remaining n k elements with a solution of weight OPT, which implies that there is some subset which covers its remaining uncovered elements with an average weight of at most OPT /(n k).

So in general the greedy algorithm pays about OPT /(n k + 1) to cover the kth uncovered element, giving a performance guarantee 1 of n n k+1 = Hn . k=1 We now formalize this intuition. Let nk denote the number of elements that remain uncovered at the start of the kth iteration.

If the algorithm takes iterations, then n1 = n, and we set n +1 = 0. Pick an arbitrary iteration k. Let Ik denote the indices of the sets chosen j in iterations 1 through k 1, and for each j = 1, .

. . , m, let S denote the set of uncovered elements in Sj at the start of this iteration; that is, Sj = Sj p Ik Sp .

Then we claim that for the set j chosen in the kth iteration, wj nk nk+1 OPT . nk (1.5).

Electronic web edition. Copyright 2010 by David P. Williamson and David B.

Shmoys. To be published by Cambridge University Press.
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