Linear expected time of a simple union-find algorithm (original) (raw)

Worst-case and amortised optimality in union-find (extended abstract)

Proceedings of the thirty-first annual ACM symposium on Theory of Computing, 1999

We study the interplay between worst-case and amortised time bounds for the classic Disjoint Set Union problem (Union-Find). We ask whether it is possible to achieve optimal worst-case and amortised bounds simultaneously. Furthermore we would like to allow a tradeoff between the worst-case time for a query and for an update. We answer this question by first providing lower bounds for the possible worst-case time tradeoffs, as well as lower bounds which show where in this tradeoff range optimal amortised time is achievable. We then give an algorithm which tightly matches both lower bounds simultaneously. The lower bounds are provided in the cell-probe model as well as in the algebraic real-number RAM, and the upper bounds hold for a RAM with logarithmic word size and a modest instruction set. Our lower bounds show that for worst-case query and update time t, and t, respectively, one must have t, = a(log n/ log t,), and only for t, 2 cx(m, n) can this tradeoff be achieved simultaneously with the optimal amortised time of o(c~(m,n)). Our 'DIK", Camp. sci. Dept., ""ivorsity of Copenhagen, Denmark. E-mail: slephe"mdik".dk. Part of thin work W&S done while "isiting BRIGS and Lund University.

Memory management for union-find algorithms

1997

We provide a general tool to improve the real time performance of a broad class of Union-Find algorithms. This is done by minimizing the random access memory that is used and thus to avoid the well-known yon Neumann bottleneck of synchronizing CPU and memory. A main application to image segmentation algorithms is demonstrated where the real time performance is drastically improved.

Lower bounds for union-split-find related problems on random access machines

Proceedings of the twenty-sixth annual ACM symposium on Theory of computing - STOC '94, 1994

We prove fl(~i) lower bounds on the random access machine complexity of several dynamic, partially dynamic and static data structure problems, including the Union-Split-Find problem, dynamic prefix problems and one-dimensional range

A simple and efficient Union–Find–Delete algorithm

Theoretical Computer Science, 2011

The Union-Find data structure for maintaning disjoint sets is one of the best known and widespread data structures, in particular the version with constant-time Union and efficient Find. Recently, the question of how to handle deletions from the structure in an efficient manner has been taken up, first by Kaplan, Shafrir and Tarjan (2002) and subsequently by Alstrup et al. (2005). The latter work shows that it is possible to implement deletions in constant time, without affecting adversely the asymptotic complexity of other operations, even when this complexity is calculated as a function of the current size of the set. In this note we present a conceptual and technical simplification of the algorithm, which has the same theoretical efficiency, and is probably more attractive in practice.

A statistical peek into average case complexity

International Journal of Experimental Design and Process Optimisation, 2014

The present paper gives a statistical adventure towards exploring the average case complexity behavior of computer algorithms. Rather than following the traditional count based analytical (pen and paper) approach, we instead talk in terms of the weight based analysis that permits mixing of distinct operations into a conceptual bound called the statistical bound and its empirical estimate, the so called "empirical O". Based on careful analysis of the results obtained, we have introduced two new conjectures in the domain of algorithmic analysis. The analytical way of average case analysis falls flat when it comes to a data model for which the expectation does not exist (e.g. Cauchy distribution for continuous input data and certain discrete distribution inputs as those studied in the paper). The empirical side of our approach, with a thrust in computer experiments and applied statistics in its paradigm, lends a helping hand by complimenting and supplementing its theoretical counterpart. Computer science is or at least has aspects of an experimental science as well, and hence hopefully, our statistical findings will be equally recognized among theoretical scientists as well.

Average-case analysis of unification algorithms

Theoretical Computer Science, 1993

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Average-case analysis of Robinson's unification algorithm with two different variables

Information Processing Letters, 1989

We compute the average complexity of Robinson's unification algorithm in the case of binary functions with When restricting the input to unifiable instances, the resulting complexity is linear on the size of the input. two variables. Kkyworak Unificatioq average analysis * Work supported by CIRI'PEE86/2-14. 0020-0190/89/$3.50 0 1989, Elsevier Science Pub!ishers B