## A linear-time algorithm for a special case of disjoint set union.(English)Zbl 0572.68058

This paper presents a linear-time algorithm for the special case of the disjoint set union problem in which the structure of the unions (defined by a ”union tree”) is known in advance. The algorithm executes an intermixed sequence of m union and find operations on n elements in $$O(m+n)$$ time and O(n) space. This is a slight but theoretically significant improvement over the fastest known algorighm for the general problem, which runs in $$O(m\alpha (m+n,n)+n)$$ time and O(n) space, where $$\alpha$$ is a functional inverse of Ackermann’s function.

### MSC:

 68R99 Discrete mathematics in relation to computer science 68Q25 Analysis of algorithms and problem complexity

union tree
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### References:

 [1] Aho, A.V.; Hopcroft, J.E.; Ullman, J.D., The design and annalysis of computer algorithms, (1974), Addison-Wesley Reading, Mass [2] Aho, A.V.; Hopcroft, J.E.; Ullman, J.D., On finding lowest common ancestors in trees, SIAM J. comput., 5, 115-132, (1976) · Zbl 0325.68018 [3] Dijkstra, E.W., A discipline of programming, (1976), Prentice-Hall Englewood Cliffs N.J · Zbl 0286.00013 [4] Doyle, J.; Rivest, R.L., Linear expected time of a simple union-find algorithm, Inform. process. lett., 5, 146-148, (1976) · Zbl 0345.68024 [5] Frederickson, G.N., Scheduling unit-time tasks with integer release times and deadlines, Inform. process. lett., 16, 171-173, (1983) · Zbl 0508.68023 [6] Gabow, H.N., An efficient implementation of edmonds’ algorithm for maximum matching on graphs, J. assoc. comput. Mach., 23, 221-234, (1976) · Zbl 0327.05121 [7] Gabow, H.N., A linear-time recognition algorithm for interval dags, Inform. process. lett., 12, 20-22, (1981) · Zbl 0454.68011 [8] Gabow, H.N., An almost-linear algorithm for two-processor scheduling, J. assoc. comput. Mach., 29, 766-780, (1982) · Zbl 0485.68034 [9] Gabow, H.N.; Tarjan, R.E., A linear-time algorithm for a special case of disjoint set union, (), 246-251 [10] Harel, D., A linear time algorithm for the least common ancestors problem, (), 308-319 [11] Harel, D.; Tarjan, R.E., Fast algorithms for finding nearest common ancestors, SIAM J. comput., 13, 338-355, (1984) · Zbl 0535.68022 [12] Havens, B., Experiments on an asymptotically optimum, special purpose set merging algorithm, () [13] Hopcroft, J.E.; Ullman, J.D., Set merging algorithms, SIAM J. comput., 2, 294-303, (1973) · Zbl 0253.68003 [14] Horowitz, E.; Sahni, S., Fundamentals of computer algorithms, (1978), Computer Science Potomac, Md · Zbl 0442.68022 [15] {\scH. Imai and T. Asano}, Efficient algorithms for geometric graph search problems, J. Algorithms, to appear. · Zbl 0591.68068 [16] Imai, H.; Asano, T., Dynamic segment intersection search with applications, (), 393-402 [17] Knuth, D.E.; Schönhage, A., The expected linearity of a simple equivalence algorithm, Theoret. comput. sci., 6, 281-315, (1978) · Zbl 0377.68024 [18] Lengauer, T.; Tarjan, R.E., A fast algorithm for finding dominators in a flowgraph, ACM trans. programming lang. systems, 1, 121-141, (1979) · Zbl 0449.68024 [19] Lipski, W.; Preparata, F., Efficient algorithms for finding maximum matchings in convex bipartite graphs and related problems, Acta inform., 15, 329-346, (1981) · Zbl 0445.68052 [20] Lipski, W., An O(n log n) Manhattan path algorithm, Inform. process. lett., 19, 99-102, (1984) · Zbl 0542.68052 [21] {\scS. Micali}, Private communication, May 1982. [22] Micali, S.; Vazirani, V.V., An O(√|V| · |E|) algorithm for finding maximum matching in general graphs, (), 17-27 [23] Papadimitriou, C.H.; Yannakakis, M., Scheduling interval-ordered tasks, SIAM J. comput., 8, 405-409, (1979) · Zbl 0421.68040 [24] Preparata, F.P.; Lipski, W., Three layers are enough, (), 350-357 [25] Sethi, R., Scheduling graphs on two processors, SIAM J. comput., 5, 73-82, (1976) · Zbl 0328.68057 [26] Tarjan, R.E., Testing flow graph reducibility, J. comput. system sci., 8, 355-365, (1974) · Zbl 0315.68018 [27] Tarjan, R.E., Efficiency of a good but not linear set union algorithm, J. assoc. comput. Mach., 22, 215-225, (1975) · Zbl 0307.68029 [28] Tarjan, R.E., Edge-disjoint spanning trees and depth-first search, Acta inform., 6, 171-185, (1976) · Zbl 0307.05104 [29] Tarjan, R.E., Applications of path compression on balanced trees, J. assoc. comput. Mach., 26, No.4, 690-715, (1979) · Zbl 0413.68063 [30] Tarjan, R.E., A class of algorithms which require non-linear time to maintain disjoint sets, J. comput. system-sci., 18, 110-127, (1979) · Zbl 0413.68039 [31] Tarjan, R.E.; Van Leeuwen, J., Worst-case analysis of set union algorithms, J. assoc. comput. Mach., 31, 245-281, (1984) · Zbl 0632.68043 [32] Harel, D., A linear algorithm for finding dominators in flow graphs and related problems, (), to appear
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