B. Chazelle, A Faster Deterministic Algorithm for Minimum Spanning Trees, FOCS'97, pp. 22-31.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....length is within a factor of (1 #) of the length of the true (#,L k ,Euclidean) MST. The problems of computing minimum spanning trees have been studied both in graph theoretic and geometric settings. The MST of a graph can be computed with cost about linear to the number of edges [Yao75, KKT95, Cha97] If applied to the complete graph, those algorithms would require quadratic time to compute the minimum spanning tree of a set of points. However, by taking advantage of geometric properties of L k metrics, there are algorithms that perform substantially better than working on the complete graph ....

Bernard Chazelle. A faster deterministic algorithm for minimum spanning trees. In Proc. 38th Annu. IEEE Sympos. Found. Comput. Sci., page To appear, 1997.

....Fibonacci heaps (presented in the same paper) to give an algorithm running in O(m fi(m; n) time ; in the worst case this algorithm runs in O(m log n) time . Soon thereafter Gabow et al. GGST86] refined this algorithm to obtain a running time of O(m log fi(m; n) Then recently Chazelle [Chaz97] presented an MST algorithm running in By definition, fi(m; n) minfi : log n m g; here log (1) n = log n; log (i 1) n = log log n. log n = minfi 1 : log n 1g. time O(mff(m; n) log ff(m; n) where ff is a certain inverse of Ackermann s function which grows extremely ....

....can be solved with no more than f(m; n) comparisons through any type of reasoning, including a nonconstructive proof, this would imply that our algorithm runs in O(f(m;n) time. Although our algorithm is optimal, its precise running time is not known at this time. In view of Chazelle s algorithm [Chaz97] we can state that the running time of our algorithm is O(m Delta ff(m; n) Delta log ff(m; n) Clearly, its running time is also Omega Gamma m) 2 Preliminaries The input is an undirected graph G = V; E) where each edge is assigned a distinct real valued weight. The minimum spanning forest ....

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B. Chazelle. A Faster Deterministic Algorithm for Minimum Spanning Trees. In FOCS '97, pp. 22--31, 1997.

....always rank algorithms correctly in terms of practical performance. Proofs and a more comprehensive discussion of these and other parallel MST algorithms can be found in [17] Known Results. The MST problem has been extensively studied in the sequential setting (see [19] for a survey up to 1984, [22, 8]) with the best result being the linear time randomized algorithm in [22] The MST problem has also a rich history in parallel computing and a number of PRAM algorithms have been proposed [9, 2, 12, 20, 10, 25] The best of these results implies the existence of a linear work BSP algorithm using ....

B. Chazelle. A faster deterministic algorithm for minimum spanning trees. In Proc. of Symp. on Foundations of Computer Science (FOCS), pages 22 --31, 1997.

....and yet despite its apparent simplicity, the problem is still not fully understood. Graham and Hell [GH85] give an excellent survey of results from the earliest known algorithm of Boruvka [Bor26] to the invention of Fibonacci heaps, which were central to the algorithms in [FT87,GGST86] Chazelle [Chaz97] presented an MST algorithm based on the Soft Heap [Chaz98] having complexity O(mff(m; n) log ff(m; n) where ff is a certain inverse of Ackermann s function. Recently Chazelle [Chaz99] modified the algorithm in [Chaz97] to bring down the running time to O(m Delta ff(m; n) Later, and in ....

....heaps, which were central to the algorithms in [FT87,GGST86] Chazelle [Chaz97] presented an MST algorithm based on the Soft Heap [Chaz98] having complexity O(mff(m; n) log ff(m; n) where ff is a certain inverse of Ackermann s function. Recently Chazelle [Chaz99] modified the algorithm in [Chaz97] to bring down the running time to O(m Delta ff(m; n) Later, and in independent work, a similar Part of this work was supported by Texas Advanced Research Program Grant 003658 0029 1999. Seth Pettie was also supported by an MCD Fellowship. algorithm of the same running time was presented in ....

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B. Chazelle. A faster deterministic algorithm for minimum spanning trees. In FOCS '97, pp. 22--31, 1997.

....deterministic algorithm for MST problem whose worst case complexity is bounded by Kmff(n) for a suitable constant K. His work seem to cast the problems related to the function ff in a new light. The description of Chazelle algorithm is beyond scope of this article, see B. Chazelle papers [6], 7] However fast (and almost linear) the Chazelle algorithms is still not linear and the following seems to be the most important problem in this area: PROBLEM: Does there exist a linear deterministic algorithm which solves MST Problem More precisely, does there exist a deterministic ....

B. Chazelle, B.: A faster Deterministic Algorithm for Minimum Spanning Trees. In: Proceedings of Thirty-Eight Annual IEEE Symposium on Foundations of Computer Science (FOCS), 1997, pp. 22--31.

....the ante to O(m log fi(m; n) a result which stood for a decade. Recently Chazelle described a non greedy approach to solving the MST problem which makes use of the soft heap [Chaz98a] a priority queue which is allowed to corrupt its own data in a controlled fashion. This led to an algorithm [Chaz97, Chaz98b] running in time O(mff log ff) where ff = ff(m; n) is a certain inverse of Ackermann s function. Pettie and Ramachandran [PR99] have just developed an optimal MST algorithm by breaking the larger MST problem into manageable subproblems and finding the MSTs on these subproblems using optimal ....

....time, even if edge costs are only subject to comparisons. It is still unknown whether these more powerful models are necessary to compute the MST in linear time. In this paper we present a deterministic minimum spanning tree algorithm running in time O(mff(m; n) The increase in speed over [Chaz97, Chaz98b] is the result of dealing with bad edges 4 more intelligently, which also calls for changes to the recursive structure of the 1997 algorithm. In addition, we believe our exposition highlights the underlying elegance of the algorithm. We would like to give due credit to Chazelle on two ....

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B. Chazelle. A Faster Deterministic Algorithm for Minimum Spanning Trees. In FOCS '97, pp. 22--31, 1997.

....and yet despite its apparent simplicity, the problem is still not fully understood. Graham and Hell [GH85] give an excellent survey of results from the earliest known algorithm of Boruvka [Bor26] to the invention of Fibonacci heaps, which were central to the algorithms in [FT87, GGST86] Chazelle [Chaz97] presented an MST algorithm based on the Soft Heap [Chaz98] having complexity O(mff(m; n) log ff(m; n) where ff is a certain inverse of Ackermann s function. Recently Chazelle [Chaz99] modified the algorithm in [Chaz97] to bring down the running time to O(m Delta ff(m; n) Later, and in ....

....heaps, which were central to the algorithms in [FT87, GGST86] Chazelle [Chaz97] presented an MST algorithm based on the Soft Heap [Chaz98] having complexity O(mff(m; n) log ff(m; n) where ff is a certain inverse of Ackermann s function. Recently Chazelle [Chaz99] modified the algorithm in [Chaz97] to bring down the running time to O(m Delta ff(m; n) Later, and in independent work, a similar algorithm of the same running time was presented in Pettie [Pet99] which gives an alternate exposition of the O(m Delta ff(m; n) result. This is the tightest time bound for the MST problem to ....

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B. Chazelle. A faster deterministic algorithm for minimum spanning trees. In FOCS '97, pp. 22--31, 1997.

....to use this method to bound KF (n) by P F (n) 1.2. History and New Results Study of minimum spanning trees has a long rich history [18] Currently, it is known how to compute the minimum spanning tree in randomized linear expected time [22] or deterministically in time O(m#(m, n) log #(m, n) [5].Effi cient algorithms have been developed for maintaining the minimum spanning tree of a graph as edges are inserted into or deleted from the graph [11,15,20,21] The parametric minimum spanning tree problem has also been previously studied, most recently by Fern andez Baca et al. 14] In that ....

B. Chazelle. A faster deterministic algorithm for minimum spanning trees. In Proc. 38th Symp. Foundations of Computer Science, 1997, 22--31.

....The literature also contains the alternative related terminology (which we will avoid) of BPP and RP algorithms. 4 With one exception, the Monte Carlo matching algorithm of x4.4. 5 Actually [9] did not prove this for spanners and banyans. However by applying any o(N log N) MST algorithm [6] to either of these graphs we would recover an approximate MST, hence by [9] s Omega Gamma N log N) lower bound for approximate MSTs, the result follows. They also did not prove this for MM, M2M, EC, ANN, but we do claim this (although we omit the proof) We also claim that approximating the ....

B.M. Chazelle. A faster deterministic algorithm for minimum spanning trees. In Proceedings of the 38th Annual Symposium on Foundations of Computer Science, pages 22--34, 1997.

....is minimized. This problem has a long history. Sequential MST algorithms running in O(m log n) time are well known (see [17] for a survey) A more involved algorithm, developed by Gabow et al. 8] runs in O(m log fi(m; n) time, where fi(m; n) minfi j log (i) n m=ng. Very recently, Chazelle [2] further improved the time complexity to O(mff(m; n) log ff(m; n) where ff(m; n) is the inverse Ackerman function. On the other hand, Karger et al. 13] designed a randomized algorithm running in expected linear time. In the parallel context, the MST problem is closely related to the connected ....

B. Chazelle, A Faster Deterministic Algorithm for Minimum Spanning Trees, FOCS'97, pp. 22-31.

....area of MAX SNP hardness. 1.6 Organization of the paper In x2 we describe how to approximate the traveling salesman tour in the plane; x3 describes banyans and their use for 5 Actually [6] did not prove this for spanners and banyans. However by applying any o(N log N) MST algorithm [4] to either of these graphs we would recover an approximate MST, hence by [6] s Omega Gamma N log N) lower bound for approximate MSTs, the result follows. 6 Although our algorithms are most easily expressed in a way that use the floor function (so they are not pure algebraic, and hence ....

B.M. Chazelle. A faster deterministic algorithm for minimum spanning trees. In Proceedings of the 38th Annual Symposium on Foundations of Computer Science, pages 22--34, 1997.

....LCA queries. ff(m; n) is the standard functional inverse of the Ackermann function. Problem Previous Pointer Machine Bound Previous RAM Bound Off line LCAs O(pff(p; n) n) 1] O(n p) 16, 25] MST Verification O(mff(m; n) n) 27] O(n m) 9, 18] MST Construction O(mff(m; n) log ff(m; n) n) [7] O(n m) 13, 17] Dominators O(mff(m; n) n) 20] O(n m) 3, 15] level microtrees, and pointer based radix sort to the MST verification (and construction) and dominators problems. 2 Least Common Ancestors Let T = V; E) be a tree with root r, and let P V Theta V be a set of pairs of ....

B. Chazelle. A faster deterministic algorithm for minimum spanning trees. In Proc. 38th IEEE FOCS, pages 22--31, 1997.

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B. Chazelle, A Faster Deterministic Algorithm for Minimum Spanning Trees, FOCS'97, pp. 22-31.

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