# Optimal Parallel Algorithms on Circular-Arc Graphs

19 Dec 1989-Vol. 405, pp 44-55

TL;DR: It would be interesting to investigate whether the techniques used in this paper can be extended to obtain efficient sequential and parallel algorithms for the weighted versions of the MiS, MCC, and MDS problems.

Abstract: We have presented in a unified way optimal parallel algorithms for the unweighted versions of the MiS, MCC, and MDS problem on circular-arc graphs using greedy methods. It would be interesting to investigate whether our techniques can be extended to obtain efficient sequential and parallel algorithms for the weighted versions of these problems.

Topics: Parallel algorithm (61%), Clique cover (55%)

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TL;DR: It is shown that if G is a polygon-circle graph, then so is [ L ( G )] 2 , and the same holds for asteroidal triple-free and interval-filament graphs, and it follows that the induced matching problem is polytime-solvable in these classes.

Abstract: An induced matching in a graph G is a set of edges, no two of which meet a common node or are joined by an edge of G ; that is, an induced matching is a matching which forms an induced subgraph. Induced matchings in graph G correspond precisely to independent sets of nodes in the square of the line-graph of G , which we denote by [ L ( G )] 2 . Often, if G has a nice representation as an intersection graph, we can obtain a nice representation of [ L ( G )] 2 as an intersection graph. Then, if the independent set problem is polytime-solvable in [ L ( G )] 2 , the induced matching problem is polytime-solvable in G . In particular, we show that if G is a polygon-circle graph, then so is [ L ( G )] 2 , and the same holds for asteroidal triple-free and interval-filament graphs. It follows that the induced matching problem is polytime-solvable in these classes. Gavril's interval-filament graphs include cocomparability and polygon-circle graphs, and the latter include circle graphs, circular-arc graphs, chordal graphs, and outerplanar graphs.

81 citations

### Cites methods from "Optimal Parallel Algorithms on Circ..."

...Similarly, NC algorithms for maximum independent set [1,29], maximum clique [1], and minimum clique cover [29] in circular-arc graphs provide NC algorithms for maximum induced matching, maximum neighbourly set, and minimum neighbourly set cover in this class....

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18 Dec 2000-

TL;DR: Efficient (sometimes improved) solutions to the following three problems are obtained: the shooter location problem, computing a minimum piercing set for arcs on a circle, and dynamically maintaining a box cover for a d-dimensional point set.

Abstract: We show how to efficiently maintain a minimum piercing set for a set S of intervals on the line, under insertions and deletions to/from S. A linear-size dynamic data structure is presented, which enables us to compute a new minimum piercing set following an insertion or deletion in time O(c(S) log |S|), where c(S) is the size of the new minimum piercing set. We also show how to maintain a piercing set for S of size at most (1 + Ɛ)c(S), for 0 < Ɛ ≤ 1, in O(log |S|/Ɛ) amortized time per update. We then apply these results to obtain efficient (sometimes improved) solutions to the following three problems: (i) the shooter location problem, (ii) computing a minimum piercing set for arcs on a circle, and (iii) dynamically maintaining a box cover for a d-dimensional point set.

5 citations

### Cites background from "Optimal Parallel Algorithms on Circ..."

...Hsu and Tsai [7] and Rao and Rangan [12] showed that if A itself is not a clique, then it suffices to consider only linear cliques....

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##### References

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01 Jan 1979-

42,654 citations

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01 Jan 1979-

Abstract: This is the second edition of a quarterly column the purpose of which is to provide a continuing update to the list of problems (NP-complete and harder) presented by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NP-Completeness,’’ W. H. Freeman & Co., San Francisco, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed. Readers having results they would like mentioned (NP-hardness, PSPACE-hardness, polynomial-time-solvability, etc.), or open problems they would like publicized, should send them to David S. Johnson, Room 2C355, Bell Laboratories, Murray Hill, NJ 07974, including details, or at least sketches, of any new proofs (full papers are preferred). In the case of unpublished results, please state explicitly that you would like the results mentioned in the column. Comments and corrections are also welcome. For more details on the nature of the column and the form of desired submissions, see the December 1981 issue of this journal.

39,992 citations

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01 Jan 1980-

TL;DR: This new Annals edition continues to convey the message that intersection graph models are a necessary and important tool for solving real-world problems and remains a stepping stone from which the reader may embark on one of many fascinating research trails.

Abstract: Algorithmic Graph Theory and Perfect Graphs, first published in 1980, has become the classic introduction to the field. This new Annals edition continues to convey the message that intersection graph models are a necessary and important tool for solving real-world problems. It remains a stepping stone from which the reader may embark on one of many fascinating research trails. The past twenty years have been an amazingly fruitful period of research in algorithmic graph theory and structured families of graphs. Especially important have been the theory and applications of new intersection graph models such as generalizations of permutation graphs and interval graphs. These have lead to new families of perfect graphs and many algorithmic results. These are surveyed in the new Epilogue chapter in this second edition. New edition of the "Classic" book on the topic Wonderful introduction to a rich research area Leading author in the field of algorithmic graph theory Beautifully written for the new mathematician or computer scientist Comprehensive treatment

4,086 citations

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TL;DR: A recurstve construction is used to obtain a product circuit for solving the prefix problem and a Boolean clrcmt which has depth 2[Iog2n] + 2 and size bounded by 14n is obtained for n-bit binary addmon.

Abstract: The prefix problem is to compute all the products x t o x2 . . . . o xk for i ~ k .~ n, where o is an associative operation A recurstve construction IS used to obtain a product circuit for solving the prefix problem which has depth exactly [log:n] and size bounded by 4n An application yields fast, small Boolean ctrcmts to simulate fimte-state transducers. By simulating a sequentml adder, a Boolean clrcmt which has depth 2[Iog2n] + 2 and size bounded by 14n Is obtained for n-bit binary addmon The size can be decreased significantly by permitting the depth to increase by an addmve constant

1,133 citations

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TL;DR: A parallel implementation of merge sort on a CREW PRAM that uses n processors and O(logn) time; the constant in the running time is small.

Abstract: We give a parallel implementation of merge sort on a CREW PRAM that uses n processors and $O(\log n)$ time; the constant in the running time is small. We also give a more complex version of the algorithm for the EREW PRAM; it also uses n processors and $O(\log n)$ time. The constant in the running time is still moderate, though not as small.

821 citations