B. Dixon, M. Rauch, and R. E. Tarjan. Verification and sensitivity analysis of minimum spanning trees in linear time. SIAM Journal on Computing, 21(6):1184--1192, 1992.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....that our lower bound is tight to within a factor of 2 in the t parameter. 1 Introduction The theoretically best minimum spanning tree algorithms [23, 10, 33] were made possible by even more fundamental algorithms and data structures, namely Komlos s minimum spanning tree verification algorithm [27, 17, 24, 5] and Chazelle s Soft Heap [11] It has been speculated by some (see, e.g. Chazelle [10, p. 1029] that the key to a faster MST algorithm is some interesting new data structure. In this paper we show that there are no linear solutions to the online minimum spanning tree verification problem, ....

.... the problem of answering interval maximum queries in a 1dimensional array can be done in constant time with linear preprocessing [27] contrast this with the superlinear lower bound in [13] for arbitrary semigroups) Solving MST verification offline on arbitrary trees can be done in linear time [27, 17, 24, 5], and the dual to this problem, MST sensitivity analysis, can be solved in randomized linear time [20, 17] or deterministic O(m log (m; n) time. All these problems have m (m;n) lower bounds when generalized to arbitrary semigroups [13] Given this history it is somewhat startling that the ....

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B. Dixon, M. Rauch, and R. E. Tarjan. Verification and sensitivity analysis of minimum spanning trees in linear time. SIAM J. Comput., 21(6):1184--1192, 1992.

....fact that it is just based on the computation of a standard minimum spanning tree. In a network with dynamic power control, the range assigned to the stations can be modified at any time: The algorithm can thus take advantage of all known techniques to dynamically maintain MSTs (see, for example, DRT92, Epp94, NPW00] Our results. In [WNE00] the performance of the above described heuristics has been evaluated by simulation and the worst case analysis of its quality in terms of the approximation ratio has been left open: The main result of this paper is the proof of an upper bound on the ....

B. Dixon, M. Rauch, and R.E. Tarjan. Verification and sensitivity analysis of minimum spanning trees in linear time. SIAM J. Comput., 21:1184--1192, 1992.

....Theorem. 7 Discussion An intriguing aspect of our algorithm is that we do not know its precise deterministic running time although we can prove that it is within a constant factor of optimal. Results of this nature have been obtained in the past for sensitivity analysis of minimum spanning trees [DRT92] and convex matrix searching [Lar90] Also, for the problem of triangulating a convex polygon, it was observed in [DRT92] that an alternate linear time algorithm could be obtained using optimal decision trees on small subproblems. However, these earlier algorithms make use of decision trees in ....

....although we can prove that it is within a constant factor of optimal. Results of this nature have been obtained in the past for sensitivity analysis of minimum spanning trees [DRT92] and convex matrix searching [Lar90] Also, for the problem of triangulating a convex polygon, it was observed in [DRT92] that an alternate linear time algorithm could be obtained using optimal decision trees on small subproblems. However, these earlier algorithms make use of decision trees in more straightforward ways than the algorithm presented here. As noted earlier, the construction of optimal decision trees ....

B. Dixon, M. Rauch, R. E. Tarjan. Verification and sensitivity analysis of minimum spanning trees in linear time. SIAM Jour. Comput., vol 21, pp. 11841192, 1992.

....E) with weight function w : E R and a MinimumSpanning Tree (MST) T = V; E T ) of G, the Sensitivity Analysis Problem is to find for each e 2 E, the maximum value by which its weight can be perturbed so that the MST remains the same. The best known solutions are by Tarjan[1] and by Dixon et al. [2], both having a worst case time complexity of O(m ff(m; n) where m and n are the number of edges and vertices of G respectively and ff( Gamma) is the functional inverse of Ackermann s function. The latter paper also presents a linear time algorithm for verification of minimum spanning trees. ....

B Dixon, M Rauch, R E Tarjan, Verification and Sensitivity Analysis of Minimum Spanning Trees in Linear Time, Siam J. of Comp. Vol 21 No 6, pp 1184-92, 1992.

.... in a graph was shown to be checkable in O(nA(n) m) time, where n is the number of vertices in the graph, m is the number of edges in the graph, and A(n) is the inverse of Ackermann s function [79] Later, this result was improved to O(n m) and this new algorithm is efficiently parallelizable [31]. Even more recently, it was shown that a minimum spanning tree of a graph could be found in randomized expected linear time [54] Checking the correctness of a sequence of n priority queue operations was shown to take O(n) time [76] In the same paper, a simple O(n) time algorithm was presented ....

....the linear time off line union find algorithm [40] can be used to process these operations. Since delete operations only 30 create check edges, and check edges are only used during the final stage, the behavior of the algorithm is not altered by delaying this processing. Finally, Dixon et al. [31] show how the correctness of an MST as we generalized it can be checked in linear time. Thus we have the following lemma: Lemma 2.3.3: Let O be a sequence of N MPQ operations, and let A be a sequence of supposed answers to these operations. The correctness of the answers in A can be determined ....

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Dixon, B., Rauch, M., and Tarjan, R. E., "Verification and Sensitivity Analysis of Minimum Spanning Trees in Linear Time," SIAM Journal Computing, 21 (1992), 1184-1192.

....therefore the problem of finding a best swap edge with respect to a given optimization function for every edge in the network. In the past, the ABS problem has been solved both when the network is a MST and a SPT. In the first case, the fastest solution known to date is an O(m) time algorithm [2], while in the second case, an O(n 2 ) time algorithm has been presented in [9] However, in several applications, the used spanning tree is neither an MST nor a SPT. Rather, many network architectures look for minimizing the diameter of the spanning tree, that is the length of the longest path ....

B. Dixon, M. Rauch, and R. Tarjan. Verification and sensitivity analysis of minimum spanning trees in linear time. SIAM J. Computing, 21(6):1184-- 1192, 1992.

.... just based on the computation of a standard minimum spanning tree (shortly, MST) In a network with dynamic power control, the range assigned to the stations can be modified at any time: Our algorithm can thus take advantage of all known techniques to dynamically maintain MSTs (see, for example, [8, 10, 17]) MSTs have already been used in order to develop approximation algorithms for range assignment problems in wireless networks: However, we believe that the analysis of the performance of our algorithm (which is based on computational geometry techniques) is rather interesting by itself. Finally, ....

B. Dixon, M. Rauch, and R.E. Tarjan. Verification and sensitivity analysis of minimum spanning trees in linear time. SIAM J. Comput., 21:1184--1192, 1992.

....role has been played by the following subproblem of MST: MST VERIFICATION PROBLEM Given a weighted graph G = V; E) and its spanning tree T , decide whether T is minimal. 40 Building on the early work of Tarjan [44] and an algorithm of Koml os [30] it has been showed by Dixon, Rauch and Tarjan [15] that the MST Verification problem can be solved by a linear deterministic algoritm. Recently a simpler procedure has been found by King [25] King observed that the Koml os algorithm is simple and linear for balanced (full branched) trees. In order to apply this she transformed every tree to a ....

Dixon, B., Rauch, M., Tarjan, R.: Verification and sensitivity analysis of minimum spanning tress in linear time. SIAM J. of Computing 21, 6, 1992, pp. 1184--1192.

....has been played by the following sub problem of MST : MST VERIFICATION PROBLEM : Given a weighted graph G = V; E) and its spanning tree T decide whether T is a minimal. Building on an earlier work of Tarjan [Ta2] and an algorithm of Komlos [Ko] it has been showed by Dixon, Rauch and Tarjan [DRT] that the MST Verification problem can be solved by a linear deterministic algorithm. Recently a simpler procedure has been found by King [K] Valerie King observed that the Komlos algorithm is simple and linear for balanced (full branching) trees. In order to apply this she transformed every tree ....

B. Dixon, M. Rauch, R. Tarjan, Verification and sensitivity analysis of minimum spanning trees in linear time, SIAM J. of Computing 21, 6(1992), 1184--1192.

....sequential executions of balanced binary trees, priority queues, union find structures, and mergeable priority queues. In this approach the data structure code is modified to output additional information to assist in testing. Other work on sequential testing and related issues includes [HA84, DRT92, Ram92] Note that unlike testing sequential executions, testing parallel executions focuses on topological sorting since it does not assume a centralized serialization point or a central module implementing the data structure. The sequential trace work, in contrast, focuses on testing procedures ....

B. Dixon, M. Rauch, and R. E. Tarjan. Verification and sensitivity analysis of minimum spanning trees in linear time. SIAM Journal on Computing, 21(6):1184--1192, 1992.

....n=ff) 8 Discussion An intriguing aspect of our algorithm is that we do not know its precise deterministic running time although we can prove that it is within a constant factor of optimal. Results of this nature have been obtained in the past for sensitivity analysis of minimum spanning trees [DRT92] and convex matrix searching [Lar90] Also, for the problem of triangulating a convex polygon, it was observed in [DRT92] that an alternate linear time algorithm could be obtained using optimal decision trees on small subproblems. However, these earlier algorithms make use of decision trees in ....

....although we can prove that it is within a constant factor of optimal. Results of this nature have been obtained in the past for sensitivity analysis of minimum spanning trees [DRT92] and convex matrix searching [Lar90] Also, for the problem of triangulating a convex polygon, it was observed in [DRT92] that an alternate linear time algorithm could be obtained using optimal decision trees on small subproblems. However, these earlier algorithms make use of decision trees in more straightforward ways than the algorithm presented here. As noted earlier, the construction of optimal decision trees ....

B. Dixon, M. Rauch, R. E. Tarjan. Verification and sensitivity analysis of minimum spanning trees in linear time. SIAM Jour. Comput., vol 21, pp. 1184-1192, 1992.

....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) Verification of a putative MST [4, 9] can be done in time linear in m. The randomized algorithm above uses this result. d) The MST (more generally, MSF) of a fully dynamic graph can be maintained in time O( p m) per change to the graph [6] While these algorithms are quite different in many respects, it is notable that they ....

....one pass and considers each edge only once. It is useful only for a multi pass approach like that of Fredman Tarjan or Bor uvka. Chapter x6 A linear time verification algorithm This section describes the O(m) MST verification algorithm of King [9] A previous such algorithm was due to Dixon [4]. The description falls into two parts. The first is a clever application of general principles that is due to Koml os [10] it shows how to perform verification using only O(m) edge weight comparisons. The second part is much more finicky. It shows how the additional bookkeeping required can ....

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B. Dixon, M. Rauch, and R. Tarjan. 1992. Verification and sensitivity analysis of minimum spanning trees in linear time. SIAM Journal of Computing, 21(6):1184--1192.

....traversal through e and undo the changes we made to # and A. The time for this method is O(log n) per binary search, or O(m log n) total. More precisely it is O(m log d) where d is the depth of T 0 . To improve this to the O(m n log n) bound claimed, we apply a sensitivity analysis algorithm [7, 18, 21]. Such algorithms find for each edge e = u, v)inG T 0 the heaviest edge e # on the corresponding path from u to v in T 0 . We use this information in two ways. First, e is in some MST i# w(e) w(e # ) so we can delete from EG all edges not part of some MST (this entails a modification to our ....

B. Dixon, M. Rauch, and R. E. Tarjan. Verification and sensitivity analysis of minimum spanning trees in linear time. SIAM J. Comput. 21 (1992) 1184--1192.

....m = o(n 2 =log 1=2 n) that is, for all but the most dense graphs. Let G Gamma e be the graph with edge e deleted. The All Edge Replacements problem (AER) is to determine the MST of G Gamma e for each edge e in E. AER is similar to but 4 simpler than ANR. For AER, Dixon, Rauch, and Tarjan [7] present an O(m) time sequential algorithm, and Dixon and Tarjan [8] present an O(log n) time parallel algorithm that uses Theta( m n) log n) processors on a CREW PRAM. These two algorithms are actually MST verification algorithms. Combined with a graph transformation of Tarjan [9, p. 713] ....

B. Dixon, M. Rauch, and R. E. Tarjan. "Verification and sensitivity analysis of minimum spanning trees in linear time." SIAM J. Comput. 21, no. 6 (Dec. 1992): 1184--1192.

.... the minimum only within a single interval of values of #, we perform this test by computing the values of all intra cluster edges at the parameter value #, and testing whether the given tree is still the minimum spanning tree at that parameter using a minimum spanning tree verification algorithm [8, 25]. # Lemma 10. We can find the first non positive intra cluster swap in a cluster of O(z) vertices in O(z log z) time. Proof: We apply parametric search with the decision oracle described above. For a simulated algorithm, we use sorting, since the sorted order is discontinuous at all swaps. Cole ....

B. Dixon, M. Rauch, and R. E. Tarjan. Verification and sensitivity analysis of minimum spanning trees in linear time. SIAM J. Computing, 21 (1992), 1184--1192.

....between the values of OE, so it does not contribute to the information theory cost of the algorithm. Its cost is only considered with regard to implementation. Koml os only achieves linear time implementation in the setting of strings. A variation on the Koml os algorithm, presented in [DRT], achieves linear time implementation in all settings, but that result is not within the scope of this paper. 4.2 The Full Branching Tree The second extreme example is the case of a full branching tree. Recall that a full branching tree is a rooted tree in which all the leaves are on the same ....

B. Dixon, M. Rauch, and R. E. Tarjan, "Verification and sensitivity analysis of minimum spanning trees in linear time," SIAM J. of Computing 21, 1992, pp. 1184-1192.

....running in O(m ff(m; n) time, where ff is a functional inverse of Ackerman s function. Later, Koml os [19] showed that a minimum spanning tree can be verified in O(m) binary comparisons of edge weights, but with nonlinear overhead to decide which comparisons to make. Dixon, Rauch and Tarjan [7] combined these algorithms with a table lookup technique to obtain an O(m) time verification algorithm. King [17] recently obtained a simpler O(m) time verification algorithm that combines ideas of Boruvka, Koml os, and Dixon, Rauch, and Tarjan. In this paper we describe a randomized algorithm ....

....edge can be in the minimum spanning forest of G. This is a consequence of the cycle property. Given a forest F in G, the F heavy edges of G can be computed in time linear in the number of edges of G, using an adaptation of the verification algorithm of Dixon, Rauch, and Tarjan (page 1188 in [7] describes the changes needed in the algorithm) or of that of King. Lemma 1 Let H be a subgraph obtained from G by including each edge independently with probability p, and let F be the minimum spanning forest of H. The expected number of F light edges in G is at most n=p where n is the number ....

B. Dixon, M. Rauch, and R. E. Tarjan, "Verification and sensitivity analysis of minimum spanning trees in linear time," SIAM J. on Computing 21, 1992, pp. 1184-1192.

....G. The choice of edges to delete in Step 2 is based on a simpler condition than that used in Filter. For an edge vw of G, if there is a path in the MSF of G 0 that connects v to w, and every edge on this path is cheaper than vw, then vw does not belong to the MSF of G. Dixon, Rauch, and Tarjan [7] have given a parallel algorithm to implement this check for all edges of G in logarithmic time and linear work. As in the main algorithm, we let m denote the number of edges in the original input graph G0 . Then at the beginning of FinishUp(G) the graph G certainly has at most m edges, so the ....

B. Dixon, M. Rauch, and R. E. Tarjan, "Verification and sensitivity analysis of minimum spanning trees in linear time," SIAM J. on Computing 21, 1992, pp. 1184-1192.

....any forest F , we say an edge e not in the forest is F heavy if the endpoints of e are connected by a path in F and every edge on that path has weight less than that of e. An F heavy edge is not in the minimum spanning forest (see, e.g. 21] and hence can be discarded. Dixon, Rauch, and Tarjan [6] give a linear time algorithm that can be used to determine the set of F heavy edges. King [14] has recently given a simpler linear time algorithm for this problem. Klein and Tarjan [16] show that the number of edges that are not F heavy is probably not much more than n=p, where p is the sampling ....

B. Dixon, M. Rauch, and R. E. Tarjan, "Verification and sensitivity analysis of minimum spanning trees in linear time," SIAM J. on Computing 21, 1992, pp. 1184-1192.

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B. Dixon, M. Rauch, and R. E. Tarjan. Verification and sensitivity analysis of minimum spanning trees in linear time. SIAM Journal on Computing, 21(6):1184--1192, 1992.

No context found.

B. Dixon, M. Rauch, and R.E. Tarjan. Verification and sensitivity analysis of minimum spanning trees in linear time. SIAM J. Comput., 21:1184--1192, 1992.

No context found.

B. Dixon, M. Rauch, and R. E. Tarjan. Verification and sensitivity analysis of minimum spanning trees in linear time. SIAM J. Comput., 21(6):1184--1192, 1992.

No context found.

B. Dixon, M. Rauch, and R. E. Tarjan. Verification and sensitivity analysis of minimum spanning trees in linear time. SIAM J. Comput., 21(6):1184--1192, 1992.

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B. Dixon, M. Rauch, and R. E. Tarjan. Verification and sensitivity analysis of minimum spanning trees in linear time. SIAM J. Comput., 21(6) (1992), 1184--1192.

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B. Dixon, M. Rauch, R. Tarjan, Verification and sensitivity analysis of minimum spanning trees in linear time, SlAM Journal of Computing, vol.21. Nov. 6, pp.1184- 1192, (Dec. 1992).

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