e Disjoint Set Class

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equivalence relation

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A relation that is reflexive, symmetric, and transitive (p 845) forest A collection of trees (p 853) Kruskal's algorithm An algorithm used to select edges in increasing cost and that adds an edge to the tree if it does not create a cycle (p 851) minimum spanning tree A connected subgraph of G that spans all vertices at minimum total cost It is a fundamental graph theory problem (p 850) nearest common ancestor problem Given a tree and a list of pairs of nodes in the tree, find the nearest common ancestor for each pair of nodes Solution of this problem is important in graph algorithm and computational biology applications (p 853) offline algorithm An algorithm in which the entire sequence of queries are made visible before the first answer is required (p 847) online algorithm An algorithm in which an answer must be provided for each query before the next query can be viewed (p 847) path compression Makes every accessed node a child of the root until another union occurs (p 862) quick-find algorithm The unionlfind implementation in which find is a constant time operation (p 855) quick-union algorithm The unionlfind implementation in which union is a constant time operation (p 857) ranks In the disjoint set algorithm, the estimated heights of nodes ( p 863) relation Defined on a set if every pair of elements either is related or is not (p 845) spanning tree A tree formed by graph edges that connect all the vertices of an undirected graph (p 850) union-by-height Makes a shallower tree a child of the root of a deeper tree during a union operation (p 861 ) union-by-rank U n i o n - b y - h e i g h t w h e n p a t h c o m p r e s s i o n i s performed (p 863) union-by-size Makes a smaller tree a child of the root of a larger tree during a union operation (p 860) unionlfind algorithm An algorithm that is executed by processing union and find operations within a unionlfind data structure (p 847) uniodfind data structure A method used to manipulate disjoint sets (P 846)

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Common Errors

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1 In using union we often assume that its parameters are tree roots Havoc can result in code if we call such a union with non-roots as parameters 2 Although not often used, union is a keyword in C++

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On the Internet

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The disjoint sets class is available online The following are the filenames

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DisjSetsh DisjSetscpp TestDisjf3etsh

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Contains the disjoint sets class interface Contains the disjoint sets class implementation Contains a program to test the disjoint sets class

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Exercises

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In Short

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241 Show the result of the following sequence of instructions: union ( I 2), iinion(3, 4), union(3, 5): union(1: 7): union(3, 6) union(8, 9) union( 1 , 8), union(3, I O) urzion(3 1 1 ), iinion(3, 12), union(3 13), union( 14, 15), union( 16, O), uniorz(l4, 16) union( I, 3), and union( I 14) when the union operations are performed a arbitrarily b, by height c by size

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242 For each of the trees in Exercise 241: perform a find operation with path compression on the deepest node 243 Find the minimum spanning tree for the graph shown in Figure 2424

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Figure 2424 A graph G for Exercise 243

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m t -

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Set Class

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244 Show the operation of the NCA algorithm for the data given in Figure 248

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In Theory

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245 Prove that for the mazes generated by the algorithm in Section 2421 the path from the starting to ending points is unique 246 Design an algorithm that generates a maze that contains no path from start to finish but has the property that the removal of a prespecified wall creates a unique path 247 Prove that Kruskal's algorithm is correct In your proof do you assume that the edge costs are nonnegative 248 Show that, if a u n i o n operation is performed by height, the depth of any tree is logarithmic 249 Show that, if all the u n i o n operations precede the f i n d operations, then the disjoint set algorithm with path compression is linear, even if the unions are done arbitrarily Note that the algorithm does not change; only the performance changes 2410 Suppose that you want to add an extra operation, remove (x), which removes x from its current set and places it in its own Show how to modify the union/find algorithm so that the running time of a sequence of M union, f i n d , and remove operations is still O(Ma(M, N)) 2411 Prove that, if u n i o n operations are done by size and path compression is performed, the worst-case running time is still O(M log*N) 2412 Suppose that you implement partial path compression on f i n d ( i ) by changing the parent of every other node on the path from i to the root to its grandparent (where doing so makes sense) This process is called path halving Prove that, if path halving is performed on the f i n d s and either u n i o n heuristic is used, the worst-case running time is still O(M log*N)

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