H. Gabow, Z. Galil, T. Spencer and R.E. Tarjan, Efficient algorithms for finding minimum spanning trees in undirected and directed graphs, Combinatorica 6:2 (1986) 109-122.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....most nonzero entries in (MM ) the total time required to compute the graph G S is O (n ) Computing a minimum directed spanning tree with root r in a directed graph is generally referred to in the literature as a branching with root r. For information on branchings, see for example [6, 8, 10, 16]. Minimum spanning trees in directed graphs with x nodes and y edges can be found deterministically in time O(x log x y) 8] A simpler algorithm that runs in time O(y log x) is suitable for the case of sparse graphs [16, 6] which will generally be the case in our context. Since the total ....

....tree with root r in a directed graph is generally referred to in the literature as a branching with root r. For information on branchings, see for example [6, 8, 10, 16] Minimum spanning trees in directed graphs with x nodes and y edges can be found deterministically in time O(x log x y) [8]. A simpler algorithm that runs in time O(y log x) is suitable for the case of sparse graphs [16, 6] which will generally be the case in our context. Since the total number of edges in G S is at most n, the total time required to compute the minimum directed spanning tree in G S is O (n log n ....

H. N. Gabow, Z. Galil, T. Spencer, and R. E. Tarjan. Efficient algorithms for finding minimum spanning trees in undirected and directed graphs. Combinatorica 6(2):109122, 1986.

....The MST problem saw no new developments until the mid 1980s when Fredman and Tarjan [FT87] used 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 ....

....by one of several existing algorithms. Here density refers to the edge to vertex ratio. The procedure DenseCase(G; F ) takes as input a graph G and returns the MSF F of G. Our algorithm guarantees DenseCase will be called on graphs of density Omega Gamma 54 n) thus the algorithms presented in [FT87, GGST86, Chaz97] could be used as DenseCase since each runs in linear time for that density. 2.4 Soft Heap The main data structure used by our algorithm is the Soft Heap [Chaz98] The Soft Heap is a kind of priority queue that gives us an optimal tradeoff between accuracy and speed. It supports the following ....

H. N. Gabow, Z. Galil, T. Spencer, R. E. Tarjan. Efficient algorithms for finding minimum spanning trees in undirected and directed graphs. In Combinatorica 6 (1986), pp. 109--122.

.... second step can be achieved in time ### # time since ###### ## for # pair of vertices can be found in ### # time using the algorithm of Harel and Tarjan [15] Finally, the optimal augmentation for # in # # can be found in ### # time using the minimum weight branching algorithm from [16]. Theorem 2: The worst case time complexity of our algorithm for computing a 16 approximation for the optimal augmentation for VPN tree # in # is ### # # # # # ### ##. IV. CONCLUSIONS Given a VPN tree with a bandwidth reservation based on the hose model, we developed novel restoration ....

H. N Gabow, Z. Galil, T. Spencer, and R. E. Tarjan, "Efficient algorithms for finding minimum spanning trees in undirected and directed graphs," Combinatorica, vol. 6-2, pp. 109--122, 1986.

....for much of this century 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) ....

....e to f in T . Then g cannot be heavier than both e and f . 2.3 The Dense Case Algorithm The procedure DenseCase(G; F ) takes as input an n 1 node, m 1 edge graph G, where m 1 m and n 1 n= log (3) n, and returns the MSF F of G. It is not difficult to see that the algorithms presented in [FT87,GGST86,Chaz97,Chaz99,Pet99] will find the MSF of G in O(n m) time. 2.4 Soft Heap The main data structure used by our algorithm is the Soft Heap [Chaz98] The Soft Heap is a kind of priority queue that gives us an optimal tradeoff between accuracy and speed. It supports the following operations: ffl MakeHeap( returns ....

H. N. Gabow, Z. Galil, T. Spencer, R. E. Tarjan. Efficient algorithms for finding minimum spanning trees in undirected and directed graphs. In Combinatorica 6, pp. 109--122, 1986.

....rooted at a vertex r is a spanning tree such that for any vertex v, the distance between r and v is the same as in the graph. Minimum spanning trees and shortest path trees are fundamental structures in the study of graph algorithms [10, 11, 15, 19] fast algorithms for finding each are known [12, 13]. Typically, the edge weighted graph G represents a feasible network. Each vertex represents a site. The goal is to install links between pairs of sites so that signals can be routed in the resulting network. Each edge of G represents a link that can be installed. The cost of the edge reflects ....

....tree. The running time is proportional to the number of relaxations. This is O(n) because each edge in TM or T S is relaxed at most twice by DFS and at most once by Add Path. If the shortest path tree and the minimum spanning tree are not given, they can be computed in O(m n log n) time [12, 13]. This establishes Theorem 1. Observation 1: In metric graphs (complete graphs with edge weights satisfying the triangle inequality, such as Euclidean graphs) the shortest path tree is trivial and can be found in O(n) time. For Euclidean graphs induced by points in the plane, the minimum spanning ....

[Article contains additional citation context not shown here]

H. N. Gabow, Z. Galil, T. Spencer and R. E. Tarjan, Efficient algorithms for finding minimum spanning trees in undirected and directed graphs, Combinatorica, 6 (2), pp. 109--122, (1986).

....T be a single vertex v from G while T has less than n vertices find the minimum edge connecting T to G Gamma T add it to T end end The precise time complexity of Prim s algorithm depends on the data structure used to organize the edges, but in any case O(n 2 log n) is an upper bound. see [10] for faster algorithms) Equipped with such a polynomial algorithm you can find minimum spanning trees with thousands of nodes within seconds on a personal computer. Compare this to exhaustive search According to our definition, MST is a tractable problem. 5 1.3.3 Intractable itineraries ....

H.N. Gabow, Z. Galil, T.H. Spencer, and R.E. Tarjan. Efficient algorithms for finding minimum spanning trees in undirected and directed graphs. Combinatorica, 6:109--122, 1986.

....problem we can hope for very fast algorithms. Here is a summary of the results in this direction. Yao [46] was the first to implement Boruvka Algorithm and obtained bound m log log n. This was further improved by Fredman and Tarjan [19] and finally 39 by Gabow, Galil, Spencer and Tarjan [20] and [21] to the bound m log fi(m; n) where fi(m; n) is a very slowly growing function defined as follows: fi(m; n) minfi; log log Delta Delta Delta log(n) m=ng Until recently this has been the best known deterministic algorithm for MST problem. This algorithm also involved an important new data ....

Gabov, H. N., Galil, Z., Spencer, T. H., Tarjan, R. E.: Efficient algorithms for finding minimum spanning trees in undirected and directed graphs. Combinatorica 6, 1986, pp. 109--122.

....in (MM T ) the total time required to compute the graph G S is O (n P n i=1 t GW (i) 2 ) Computing a minimum directed spanning tree with root r in a directed graph is generally referred to in the literature as a branching with root r. 1 For information on branchings, see for example [7, 9, 11, 19]. Minimum spanning trees in directed graphs with x nodes and y edges can be found deterministically in time O(x log x y) 9] A simpler algorithm that runs in time O(y log x) is suitable for the case of sparse graphs [19, 7] which will generally be the case in our context. Since the total number ....

....tree with root r in a directed graph is generally referred to in the literature as a branching with root r. 1 For information on branchings, see for example [7, 9, 11, 19] Minimum spanning trees in directed graphs with x nodes and y edges can be found deterministically in time O(x log x y) [9]. A simpler algorithm that runs in time O(y log x) is suitable for the case of sparse graphs [19, 7] which will generally be the case in our context. Since the total number of edges in G S is at most P n i=1 t GW (i) 2 n, the total time required to compute the minimum directed spanning ....

H. N. Gabow, Z. Galil, T. Spencer, and R. E. Tarjan. Efficient algorithms for finding minimum spanning trees in undirected and directed graphs. Combinatorica 6(2):109-122, 1986.

.... (2n; n) Delta b(m; n)g) time, where TMST (m; n) is the time to compute a minimum spanning tree and b(m; n) is the worst case number of breakpoints of Z( It is known that b(m; n) O(m p n) Gus80, KaIb83] and b(m; n) Omega (mff(n) Epp95] and that TMST (m; n) O(m log fi(m; n) time [GGST86] (here fi(m; n) minfi : log (i) n m=ng) 3 . Our algorithm improves on the Eisner Severance method [EiSe76] which, when applied to parametric minimum spanning trees, takes O(TMST (m; n) Delta b(m; n) time. For example, if TMST (m; n) O(m log fi(m; n) and b(m; n) O(m p n) then our ....

H.N. Gabow, Z. Galil, T. Spencer, and R.E. Tarjan. Efficient algorithms for finding minimum spanning trees in undirected and directed graphs. Combinatorica, 6:109--122, 1986.

....fast algorithms. Here is a summary of the results in this direction mostly related to R. Tarjan : A. Yao [Ya] was the first to implement Boruvka Algorithm and obtained bound m log log n. This was further improved by Fredman and Tarjan [FT] and finally by Gabow, Galil, Spencer and Tarjan [GGS] and [GGST] to the bound mlog fi(m; n) where fi(m; n) is a very slowly growing function defined as follows : fi(m; n) minfi; loglog : log(n) z i m=ng Currently this is the best known deterministic algorithm for MST problem. This algorithm also involved an important new data structure ....

....showed that one can implement the Greedy Algorithm for graphs with presorted edge weights so that its complexity is mff(m; n) where ff(m; n) is the functional inverse to the Ackerman function. This function grows much slower than (already very slow) function fi. However for general weighted graphs [GGST] still presents the best deterministic algorithm for MST problem and the following seems to be the most important problem in this area : A FEW REMARKS ON THE HISTORY OF MST PROBLEM 19 PROBLEM : Does there exist a linear deterministic algorithm which solves MST Problem More precisely, does ....

H. N. Gabov, Z. Galil, T. H. Spencer, R. E. Tarjan, Efficient algorithms for finding minimum spanning trees in undirected and directed graphs, Combinatorica 6(1986), 109--122.

....the weights of all edges in G 0 . Let MST(G) denote the weight of a minimum spanning tree of G, and MS2T(G) denote the weight of a minimum spanning 2 tree (a spanning 2 tree with the minimum weight) of G. It is well known that a minimum spanning tree in a weighted graph can be found efficiently [8, 10]. On the other hand, it is NP hard to find a minimum spanning 2 tree, even for weighted complete graphs [1] and for weighted plane triangulations [5] We now consider approximation algorithms for the minimum Spanning 2 Trees 8 spanning 2 tree problem. Here we are concerned with finding a spanning ....

H.N. Gabow, Z. Galil, T.H. Spencer, and R.E. Tarjan. Efficient algorithms for finding minimum spanning trees in undirected and directed graphs. Combinatorica, 6 (2):109--122, 1986.

....m n g. The MST problem saw no new developments until the mid 1980s when Fredman and Tarjan [FT87] used Fibonacci heaps (presented in the same paper) to give an algorithm running in O(m fi(m; n) time 2 . In the worst case, fi(m; n) log n) Before the ink had dried on this result Gabow et al. [GGST86] upped 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 ....

....x in T represent those vertices of G h Gamma1 contracted to form v x . Building T efficiently is central to this algorithm but doing it bottom up (or equivalently, top down) does not give us the desired running time (though it is possible to match the performance of the algorithm of Gabow et al. [GGST86] using this method) We, however, will build T in post order: the children of a node x 2 T will be assembled from left to right in post order followed by node x. If only some of x s children are complete we say that x is under construction . At any given time there will be no more than one ....

H. N. Gabow, Z. Galil, T. Spencer, R. E. Tarjan. Efficient algorithms for finding minimum spanning trees in undirected and directed graphs. In Combinatorica 6 (1986), pp. 109--122.

....for much of this century 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) ....

....other edge is in P then by our choice of g it must be lighter than g. If the other edge is either e or f then by our assumption it must be lighter than g. In both cases g could not be chosen next by the DJP algorithm, a contradiction. 2 2. 3 The Dense Case Algorithm The algorithms presented in [FT87, GGST86, Chaz97, Chaz99, Pet99] will find the MSF of a graph in linear time if the graph is sufficiently dense, i.e. has a sufficiently large edge to vertex ratio. For our purposes, sufficiently dense will mean Omega Gammaan (3) n) where n is the number of vertices in the graph. All of the above algorithms run in linear ....

H. N. Gabow, Z. Galil, T. Spencer, R. E. Tarjan. Efficient algorithms for finding minimum spanning trees in undirected and directed graphs. In Combinatorica 6, pp. 109--122, 1986. 16

....[11] previously invented by Varn ik in 1930) and Prim [13] as well as the 1926 algorithm of Bor uvka [2] With this foundation, the paper proceeds with its main task: to explicate four algorithms from the more recent literature. a) The asymptotically fastest deterministic algorithm known is [7], an improvement on [5] These algorithms are all but linear in m, i.e. linear times a log type factor. b) A randomized Las Vegas algorithm [8] achieves expected time that is truly linear in m, and in fact guarantees linear time performance with all but exponentially small probability. c) ....

....bad edges into the right order, only to discard them as soon as it sees them x3.14 Greedy discipline. Let us close this section with an unoriginal observation that The key idea shared by Kruskal s and Prim s will come in handy later. The algorithms of Kruskal and Prim are instances of what [7] calls the generalized greedy algorithm for constructing MSTs. The generalized greedy algorithm initializes a forest F to (V; and adds edges one at a time until F is connected. Every edge that it adds is the lightest edge leaving some component T of F . An algorithm that behaves in this way ....

[Article contains additional citation context not shown here]

H. N. Gabow, Z. Galil, T. Spencer, and R. E. Tarjan. 1986. Efficient algorithms for finding minimum spanning trees in undirected and directed graphs. Combinatorica 6:109--122.

....difference is that the widely known MST algorithms of Prim and Kruskal were developed for undirected graphs. Our progressive and two phase techniques can build upon these techniques if the heterogeneous network is symmetric. For asymmetric networks, MST algorithms for directed graphs can be used [8]. In designing a heuristic, we must give special attention to two kinds of nodes: a) Nodes which are hard to reach from every other node, and are also unable to reach other nodes quickly. The message to such a node should be sent early in the schedule, so that this communication event does not ....

H. N. Gabow, Z. Galil, T. Spencer, and R. E. Tarjan. Efficient algorithms for finding Minimum Spanning Trees in undirected and directed graphs. Combinatorica, 6(2):109-- 122, 1986.

....total weight. Many graph minimum spanning tree algorithms have been published, with the best taking linear time in either a randomized expected case model [71] or with the assumption of integer edge weights [61] For deterministic comparison based algorithms, slightly superlinear bounds are known [62]. Therefore, by constructing a complete geometric graph, one can find a geometric minimum spanning tree in time O(n 2 ) in any metric space for which distances can be computed quickly. Faster algorithms for most geometric minimum spanning tree problems are based on a simple idea: quickly find ....

H. N. Gabow, Z. Galil, T. Spencer, and R. E. Tarjan. Efficient algorithms for finding minimum spanning trees in undirected and directed graphs. Combinatorica, vol. 6, 1986, pp. 109--122.

....following theorem. Theorem 3.4 For the class of weighted bi horn logic programs and the reusability measure cst reu we have: 1. Set of atoms derivation problem is NP complete 2. All atoms derivations is in P. Proof: Part (2) follows from the well known results for the DMCST problem [Tar77, GGST86] We will now prove part (1) by showing that DMST problem is NP complete. It is clear that the problem is in NP. To prove NP hardness, we will use the fact that the undirected version of Minimum Steiner Tree problem is known to be NP complete [GJ79] Let us recall that the Minimum Steiner Tree ....

H.N. Gabow, Z. Galil, T. Spencer, and R.E. Tarjan. Efficient algorithms for finding minimum spanning trees in undirected and directed graphs. Combinatorica, 6:109--122, 1986.

....Steiner tree problem is NP complete [13] and it remains NP complete even after the delay constraint is removed [16] We prove in this paper that the delay constrained MST (DCMST) problem is also NP complete. However, several polynomial time algorithms exist for the unconstrained MST problem [17, 18]. In the future, many real time applications will involve all nodes in a given network. Some distributed real time control applications and the broadcasting of critical network state information are just a few examples. For such applications, DCMSTs are needed to broadcast the real time traffic ....

H. Gabow, Z. Galil, T. Spencer, and R. Tarjan, "Efficient Algorithms for Finding Minimum Spanning Trees in Undirected and Directed Graphs," Combinatorica, vol. 6, no. 2, pp. 109-- 122, 1986.

....Mathematics and Theoretical Computer Science) a National Science and Technology Center, Grant No. NSF STC88 09648. history of the problem up to 1985. In the last two decades faster and faster algorithms were found, the fastest being an algorithm of Gabow, Galil, and Spencer [10] see also [11]) with a running time of O(m log fi(m; n) on a graph of n vertices and m edges. Here fi(m; n) minfi j log (i) n m=ng. This and earlier algorithms used as a computational model the sequential unit cost randomaccess machine with the restriction that the only operations allowed on the edge ....

H. N. Gabow, Z. Galil, T. Spencer, and R. E. Tarjan, "Efficient algorithms for finding minimum spanning trees in undirected and directed graphs," Combinatorica 6, 1986, pp. 109-122.

.... There are many efficient algorithms for finding a minimum spanning tree, given only the graph G and the edge costs; see the survey paper by Graham and Hell [10] or the monograph by Tarjan [19, Chapter 6] The fastest known algorithm for finding a minimum spanning tree is that of Gabow, et al. [7], which runs in O(m log fi(m; n) time, where fi(m; n) minfi j log (i) n m=ng, and log (i) n is defined recursively by log (0) n = n ; log (i 1) n = log log (i) n. The verification problem was considered by Tarjan [17] and subsequentlyby Koml os [14] Tarjan proposed a ....

H.N. Gabow, Z. Galil, T. Spencer, and R.E. Tarjan, Efficient Algorithms for Finding Minimum Spanning Trees in Undirected and Directed Graphs, Combinatorica 6(2) (1986) pp. 109-122.

....algorithm was devised by Boruvka in the 1920 s [3] An informative survey paper by Graham and Hell [10] describes the history of the problem up to 1985. In the last two decades faster and faster algorithms were found, the fastest being an algorithm of Gabow, Galil, and Spencer [8] see also [9]) with a running time of O(m log fi(m; n) on a graph of n vertices and m edges. Here fi(m; n) minfi j log (i) n m=ng. This and earlier algorithms used as a computational model the sequential unit cost random access machine with the restriction that the only operations allowed on the edge ....

H. N. Gabow, Z. Galil, T. Spencer, and R. E. Tarjan, "Efficient algorithms for finding minimum spanning trees in undirected and directed graphs," Combinatorica 6, 1986, pp. 109-122.

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H. Gabow, Z. Galil, T. Spencer and R.E. Tarjan, Efficient algorithms for finding minimum spanning trees in undirected and directed graphs, Combinatorica 6:2 (1986) 109-122.

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Gabow, H. N., Galil, Z., Spencer, T. H., and Tarjan, R. E., "Efficient Algorithms for Finding Minimum Spanning Trees in Undirected and Directed Graphs", Combinatorica 6 (1986), pp. 109-122. 162

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H. N. Gabow, Z. Galil, T. Spencer and R. E. Tarjan, Efficient algorithms for finding minimum spanning trees in undirected and directed graphs, Combinatorica, 6 (2), pp. 109--122, (1986).

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H. N. Gabow, Z. Galil, T. Spencer and R. E. Tarjan, Efficient algorithms for finding minimum spanning trees in undirected and directed graphs, Combinatorica, 6 (2), pp. 109-- 122, (1986).

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