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A randomized algorithm for minimum cut
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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 double-counting 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)
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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 minimum-weight 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
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Figure 58 Prim s algorithm: the edges X form a tree, and S consists of its vertices
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growing to include the vertex v S of smallest cost: cost(v) = min w(u, v)
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This is strongly reminiscent of Dijkstra s algorithm, and in fact the pseudocode (Figure 59) is almost identical The only difference is in the key values by which the priority queue is ordered In Prim s algorithm, the value of a node is the weight of the lightest incoming edge from set S, whereas in Dijkstra s it is the length of an entire path to that node from the starting point Nonetheless, the two algorithms are similar enough that they have the same running time, which depends on the particular priority queue implementation Figure 59 shows Prim s algorithm at work, on a small six-node graph Notice how the nal MST is completely speci ed by the prev array
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Figure 59 Top: Prim s minimum spanning tree algorithm Below: An illustration of Prim s algorithm, starting at node A Also shown are a table of cost/prev values, and the nal MST procedure prim(G, w) Input: A connected undirected graph G = (V, E) with edge weights w e Output: A minimum spanning tree defined by the array prev for all u V : cost(u) = prev(u) = nil Pick any initial node u0 cost(u0 ) = 0 H = makequeue (V ) (priority queue, using cost-values as keys) while H is not empty: v = deletemin(H) for each {v, z} E: if cost(z) > w(v, z): cost(z) = w(v, z) prev(z) = v decreasekey(H, z)
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