Dynamic Equivalence and Two Applications

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Figure 243 At some point in the algorithm, several walls have been knocked down and sets have been merged At this point, if we randomly select the wall between 8 and 13, this wall is not knocked down because 8 and 13 are already connected

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Figure 244 We randomly select the wall between squares 18 and 13 in Figure 243; this wall has been knocked down because 18 and 13 were not already connected, and their sets have been merged

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trivialize the maze Suppose that we randomly target cells 18 and 13 next By performing two f i n d operations we determine that these cells are in different sets: thus I 8 and 13 are not already connected Therefore we knock down the wall that separates them, as shown in Figure 244 As a result of this operation, the sets containing cells 18 and 13 are combined by a u n i o n operation The reason is that all the cells previously connected to 18 are now connected to all the cells previously connected to 13 At the end of the algorithm, as depicted in Figure 245 all the cells are connected, and we are done

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The D~sjoint Set Class

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Figure 245 Eventually, 24 walls have been knocked down, and all the elements are in the same set

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The running time of the algorithm is dominated by the unionlfind costs The size of the unionlfind universe is the number of cells The number of f i n d operations is proportional to the number of cells because the number of removed walls is 1 less than the number of cells If we look carefully, however, we can see that there are only about twice as many walls as cells in the first place Thus, if N is the number of cells and as there are two f i n d s per randomly targeted wall, we get an estimate of between (roughly) 2N and 4N f i n d operations throughout the algorithm Therefore the algorithm's running time depends on the cost of O ( N )u n i o n and O ( N ) f i n d operations

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2422 Application: Minimum Spanning Trees

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The minimum spanning tree is a connected subgraph of G that spans all vertices at minimum total cost

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A spanning tree of an undirected graph is a tree formed by graph edges that connect all the vertices of the graph Unlike the graphs in 15, an edge (u, v) in a graph G is identical to an edge (v, u) The cost of a spanning tree is the sum of the costs of the edges in the tree The minimum spanning tree is a connected subgraph of G that spans all vertices at minimurn cost A

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minimum spanning tree exists only if the subgraph of G is connected As we show shortly, testing a graph's connectivity can be done as part of the minimum spanning tree computation In Figure 246(b), the graph is a minimum spanning tree of the graph in Figure 246(a) (it happens to be unique, which is unusual if the graph has many edges of equal cost) Note that the number of edges in the minimum spanning tree is 1 V1 - 1 The minimum spanning tree is a tree because it is acyclic, it is spanning because it covers every vertex, and it is minimum for the obvious reason Suppose that we need to connect several towns with roads, minimizing the total construction cost, with the provision that we can

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Dynamic Equivalence and Two ~

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(a) A graph G and (b) its minimum spanning tree

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transfer to another road only at a town (in other words, no extra junctions are allowed) Then we need to solve a minimum spanning tree problem, where each vertex is a town, and each edge is the cost of building a road between the two cities it connects A related problem is the minimum Sreiner tree problem, which is like the minimum spanning tree problem, except that junctions can be created as part of the solution The minimum Steiner tree problem is much more difficult to solve However, it can be shown that if the cost of a connection is proportional to the Euclidean distance, the minimum spanning tree is at most 15 percent more expensive than the minimum Steiner tree Thus a minimum spanning tree, which is easy to compute, provides a good approximation for the minimum Steiner tree, which is hard to compute A simple algorithm, commonly called Kruskal's algorithm, is used to select edges continually in order of smallest weight and to add an edge to the tree if it does not cause a cycle Formally, Kruskal's algorithm maintains a forest-a collection of trees Initially, there are (VI single-node trees Adding an edge merges two trees into one When the algorithm terminates, there is only one tree, which is the minimum spanning tree2 By counting the number of accepted edges, we can determine when the algorithm should terminate Figure 247 shows the action of Kruskal's algorithm on the graph shown in Figure 246 The first five edges are all accepted because they do not create cycles The next two edges, (v,, v3) (of cost 3) and then (v,,, v2) (of cost 4), are rejected because each would create a cycle in the tree The next edge considered is accepted, and because it is the sixth edge in a seven-vertex graph, we can terminate the algorithm

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Kruskal'salgorithmis select edges in order of increasing cost and adds an edge to the tree if it doesnotcreatea cycle

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2 If the graph is not connected, the algorithm will terminate with more than one tree Each tree then represents a minimum spanning tree for each connected component of the graph

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