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how to generate a barcode using asp.net c# A randomized algorithm for minimum cut in Software
A randomized algorithm for minimum cut Encoding QR Code JIS X 0510 In None Using Barcode generator for Software Control to generate, create QRCode image in Software applications. QRCode Decoder In None Using Barcode reader for Software Control to read, scan read, scan image in Software applications. We have already seen that spanning trees and cuts are intimately related Here is another connection Let s remove the last edge that Kruskal s algorithm adds to the spanning tree; this breaks the tree into two components, thus de ning a cut (S, S) in the graph What can we say about this cut Suppose the graph we were working with was unweighted, and that its edges were ordered uniformly at random for Kruskal s algorithm to process them Here is a remarkable fact: with probability at least 1/n 2 , (S, S) is the minimum cut in the graph, where the size of a cut (S, S) is the number of edges crossing between S and S This means that repeating the process O(n 2 ) times and outputting the smallest cut found yields the minimum cut in G with high probability: an O(mn 2 log n) algorithm for unweighted minimum cuts Some further tuning gives the O(n 2 log n) minimum cut algorithm, invented by David Karger, which is the fastest known algorithm for this important problem So let us see why the cut found in each iteration is the minimum cut with probability at least 1/n2 At any stage of Kruskal s algorithm, the vertex set V is partitioned into connected components The only edges eligible to be added to the tree have their two endpoints in distinct components The number of edges incident to each component must be at least C, the size of the minimum cut in G (since we could consider a cut that separated this component from the rest of the graph) So if there are k components in the graph, the number of eligible edges is at least kC/2 (each of the k components has at least C edges leading out of it, and we need to compensate for the doublecounting of each edge) Since the edges were randomly ordered, the chance that the next eligible edge in the list is from the minimum cut is at most C/(kC/2) = 2/k Thus, with probability at least 1 2/k = (k 2)/k, the choice leaves the minimum cut intact But now the chance that Kruskal s algorithm leaves the minimum cut intact all the way up to the choice of the last spanning tree edge is at least n 2 n 3 n 4 2 1 1 = n n 1 n 2 4 3 n(n 1) Denso QR Bar Code Maker In C# Using Barcode printer for Visual Studio .NET Control to generate, create QR Code JIS X 0510 image in VS .NET applications. QR Code 2d Barcode Generation In .NET Framework Using Barcode creator for ASP.NET Control to generate, create QR Code image in ASP.NET applications. Prim s algorithm
Print QR Code 2d Barcode In .NET Using Barcode drawer for .NET Control to generate, create QR image in VS .NET applications. Encoding QR Code 2d Barcode In VB.NET Using Barcode creation for .NET framework Control to generate, create QR Code image in .NET framework applications. Let s return to our discussion of minimum spanning tree algorithms What the cut property tells us in most general terms is that any algorithm conforming to the following greedy schema is guaranteed to work X = { } (edges picked so far) repeat until X = V  1: pick a set S V for which X has no edges between S and V S let e E be the minimumweight edge between S and V S X = X {e} A popular alternative to Kruskal s algorithm is Prim s, in which the intermediate set of edges X always forms a subtree, and S is chosen to be the set of this tree s vertices On each iteration, the subtree de ned by X grows by one edge, namely, the lightest edge between a vertex in S and a vertex outside S (Figure 58) We can equivalently think of S as 139 UPCA Creator In None Using Barcode drawer for Software Control to generate, create UPCA Supplement 2 image in Software applications. Draw Code 128 Code Set B In None Using Barcode creator for Software Control to generate, create ANSI/AIM Code 128 image in Software applications. Figure 58 Prim s algorithm: the edges X form a tree, and S consists of its vertices
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