R. E. Tarjan. Sensitivity analysis of minimum spanning trees and shortest path problems. Inform. Proc. Lett., 14:30--33, 1982.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... a problem is to perturbations in its input [19, 26] A significant limitation of the current methods for sensitivity analysis in combinatorial optimization is that they measure only changes in the solution of a problem produced by perturbations in the value of a single element in its input [18, 25, 30, 31]. In some situations it is necessary to determine the maximum effect that bounded changes in the whole input of a problem can have over the value of its solution and, thus, in which sensitivity analysis does not suffice [11, 19, 28] In this paper we consider the important class of matroid ....

R.E. Tarjan, Sensitivity analysis of minimum spanning trees and shortest path trees, Information Processing Letters, 14 (1982), pp. 30--33.

....using m n units of storage. Chazelle and Rosenberg [10] strengthened the results of Yao [36] and Alon and Schieber [4] by showing that o ine interval sum is just as hard as online interval sum. In other words, there exist m queries that can only be answered using (m (m;n) operations. Tarjan [34] showed that the complexity of any o ine subset sum problem over an arbitrary semigroup is precisely the same as its dual. Tarjan s result was used to prove an O(m (m;n) upper bound on the MST sensitivity analysis problem, which is the dual to MST veri cation. Together with [10] it also ....

....the minimum weight element in each sequence. This peculiar data structure turns out to be useful in certain weighted matching algorithms [15] and several recent shortest path algorithms [35, 17, 27, 25, 26] It can also be used to solve the minimum spanning tree sensitivity analysis problem (see [34] for the de nition of MST sensitivity analysis. The fastest data structure to date [27] runs in O(m log (m; n) time, where m is the number of operations. It is a slight improvement over Gabow s data structure [15] which runs in O(m (m;n) time. The problem of nding row maxima in totally ....

R. E. Tarjan. Sensitivity analysis of minimum spanning trees and shortest path problems (see also corrigendum IPL 23(4), p. 219). Info. Proc. Lett., 14(1):30-33, 1982.

....the 1 dimensional offline version of the problem, where n is the size of the array and m the number of queries. This lower bound obviously extends to the online problem, and it relates to the MST verification problem because a 1 dimensional array is just a kind of tree. For general trees, Tarjan [34, 36] studied certain offline partial sums algorithms based on pathcompression. Online variants were studied in [6, 2] The lower bounds cited above assume that semigroup elements are only accessible via the semigroup operator . A consequence of this which is key to previous lower bounds is ....

.... useful in certain weighted matching [19] and shortest path algorithms [37, 21, 31, 29, 30] One application of split findmin not mentioned in [19, 31] is MST sensitivity analysis, which it solves in O(m log (m; n) time, an = log ) factor faster than Tarjan s path compression based algorithm [36]. inputs and queries, as described below. Assertions 2.2 and 2.3, given in Section 2.3, provide further restrictions on the input. 1. The input tree T is a full, rooted binary tree. 2. The query edge will connect a leaf to one of its ancestors. 3. The answer to the query e will be no, e 62 MST ....

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R. E. Tarjan. Sensitivity analysis of minimum spanning trees and shortest path problems (see also corrigendum IPL 23(4), p. 219). Info. Proc. Lett., 14(1):30--33, 1982.

....for the 1dimensional o ine version of the problem, where n is the size of the array and m the number of queries. This lower bound obviously extends to the online problem, and it relates to the MST veri cation problem because a 1 dimensional array is just a kind of tree. For general trees, Tarjan [Tar79a, Tar82] studied certain partial sums algorithms based on path compression. The lower bounds cited above assume that semigroup elements are only accessible via the semigroup operator . A consequence of this which is key to previous lower bounds is that any algorithm solving such a problem can be ....

....was known to be useful in certain weighted matching and shortest path algorithms. One application of split ndmin not mentioned in [Gab85, PR02a] is that it can solve MST sensitivity analysis in O(m log (m; n) time, an = log ) factor faster than Tarjan s path compression based algorithm [Tar82]. 2 Preliminaries The problem is to preprocess an edge weighted tree T so that given any query edge e, we can determine if e 2 MST (T [ feg) This is tantamount to deciding whether e is not the heaviest edge on the only cycle in T [ feg. For the sake of simpler notation we consider input trees ....

R. E. Tarjan. Sensitivity analysis of minimum spanning trees and shortest path problems (see also corrigendum IPL 23(4), p. 219). Info. Proc. Lett., 14(1):30-33, 1982.

....the usefulness of set maxima in actually nding an optimal base. Many other concrete problems are instances of set maxima (or are reducible to it) These include verifying a partial order [GKKS93] sensitivity analysis (including veri cation) of minimum spanning trees and shortest path trees [Tar82, Kom85] and orienting the edges of an undirected, node weighted graph from the lesser to greater endpoint. This last problem was dubbed local sorting in [GKKS93] Besides the simple set maxima algorithm of [GYY80] and the trivial algorithm, there are really only two results to speak of for the ....

.... MST and [DRT92] for MST SSSP sensitivity analysis) determin 2 This is the same function as the one de ned in Fredman Tarjan s minimum spanning tree algorithm [FT87] istic algorithms which take time O(m (m;n) where (m; n) is Tarjan s inverse Ackermann function (see [Cha00a] for MST and [Tar82] for MST SSSP sensitivity analysis) and expected linear time algorithms which use a linear number of random bits [KKT95, DRT92] For these two problems we give expected linear time algorithms which use just log n random bits. 1.1.4 Selection The problem of selection ( nding order ....

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R. E. Tarjan. Sensitivity analysis of minimum spanning trees and shortest path problems (see also corrigendum ipl 14(1), pp. 30-33). Info. Proc. Lett., 14(1):30-33, 1982.

....Given a graph G = V; 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 ....

R E Tarjan, Sensitivity Analysis of Minimum Spanning Trees and Shortest Path Trees, Inform. Process. Lett. 14(

....the usefulness of set maxima in actually nding an optimal base. Many other concrete problems are instances of set maxima (or are reducible to it) These include verifying a partial order [KMK89] sensitivity analysis (including veri cation) of minimum spanning trees and shortest path trees [Tar82, Kom85] and orienting the edges of an undirected, node weighted graph from the lesser to greater endpoint. This last problem was dubbed local sorting by Goddard et al. GKKS93] Besides the simple set maxima algorithm of [GYY80] and the trivial algorithm, there are really only two results to ....

.... each of these problems there exist optimal deterministic algorithms with unknown complexities (see [PR00] for MST and [DRT92] for MST SSSP sensitivity analysis) deterministic algorithms which take time O(m (m;n) where (m; n) is Tarjan s inverse Ackermann function (see [Cha00a] for MST and [Tar82] for MST SSSP sensitivity analysis) and expected linear time algorithms which use a linear number of random bits [KKT95, DRT92, GKKS93] For both problems we give expected linear time algorithms which use just log n random bits. 1.2 Organization The rest of the paper is organized as ....

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R. E. Tarjan. Sensitivity analysis of minimum spanning trees and shortest path problems (see also corrigendum ipl 14(1), pp. 30-33). Info. Proc. Lett., 14(1):30-33, 1982. 21

....to occur, what change to the tree would follow, without going to the expense of making that change. The persistent version of this problem is used to find the k best spanning trees of a graph [11, 15, 17, 19] The problem can be solved by known dynamic MST algorithms, or by sensitivity analysis [23, 24]. The latter approach, if made persistent, uses almost linear time per update, constant time per query, and O(n) space per update. We solve the problem in linear time per update, and polylogarithmic query time and update space. 2 Reduction to O#ine Problem Fix a problem in which updates consist ....

....or by re tracing paths through the history tree with non persistent algorithms [15] but a better approach for the application is to calculate, after each update, the e#ects of all possible queries. This method is known as sensitivity analysis; it can be performed in almost linear time per update [23, 24], after which each query can be looked up in constant time. If made persistent, sensitivity analysis requires O(n) space per update to store the replacement edges for each MST edge. The non MST replacements would seem to take O(m) space, but this can be reduced to O(n) using an algorithm based on ....

R.E. Tarjan. Sensitivity analysis of minimum spanning trees and shortest path trees. Inf. Proc. Lett. 14 (1982) 30--33.

....spanning tree problem [15, 16, 18, 21] In [21] it is shown that when the edges of a graph have arbitrary destruction costs, the problem of computing the maximum increase in the weight of the minimum spanning trees achievable by removing edges of a certain total destruction cost is NP hard. In [15, 16, 18, 27] algorithms are given for finding the single most vital edge for a minimum spanning tree. The continuous version of the robustness problem for minimum spanning trees can be seen as a generalization of the sensitivity analysis for minimum spanning trees [27] since we need to consider simultaneous ....

....cost is NP hard. In [15, 16, 18, 27] algorithms are given for finding the single most vital edge for a minimum spanning tree. The continuous version of the robustness problem for minimum spanning trees can be seen as a generalization of the sensitivity analysis for minimum spanning trees [27], since we need to consider simultaneous changes in the weights of several edges. Other properties of a network that have been studied in the context of link failures or degeneration of the performance of links are maximum flow [23] minimum cost flow [26] and shortest distance between two ....

R.E. Tarjan, Sensitivity analysis of minimum spanning trees and shortest path trees, Information Processing Letters 14 (1982), 30--33.

....for constructing a minimum spanning tree. Our algorithm is based on an efficient method of finding the replacement edge for each edge in a minimum spanning tree in O(n) sequential time. It applies existing parallel algorithm design techniques for fast parallelization of the method. As shown in [3, 11, 14, 15], finding minimum replacement edges for all edges in a minimum spanning tree can lead directly to the solutions to verification of a minimum spanning tree and sensitivity analysis of minimum spanning trees. It is not hard to see that our algorithm can be easily modified to the algorithms for ....

R. E. Tarjan. Sensitivity analysis of minimum spanning trees and shortest path trees. Information Processing Letters, 14, 1982.

....for constructing a minimum spanning tree. Our algorithm is based on an efficient method of finding the replacement edge for each edge in a minimum spanning tree in O(n) sequential time. It applies existing parallel algorithm design techniques for fast parallelization of the method. As shown in [4, 10, 14, 15], finding the minimum replacement edges for all edges in a minimum spanning tree can lead directly to the solutions to verification of a minimum spanning tree and sensitivity analysis of minimum spanning trees. It is not hard to see that our algorithm can be easily modified to the algorithms for ....

R. E. Tarjan. Sensitivity analysis of minimum spanning trees and shortest path trees. Information Processing Letters, 14, 1982.

....of a shortest path tree. ffl Performing sensitivity analysis of a minimum cost network flow. Sensitivity analysis measures the robustness of a minimum spanning tree or shortest path tree by determining how much the cost of each individual edge can be perturbed before the tree is no longer minimal [15, 19]. Let e be some edge in a minimum spanning tree of G. The replacement edge for e is the non tree edge that replaces e in the minimum spanning tree of G 0 = V; E Gamma feg) Finding replacement edges is an important component of an algorithm for determining the k smallest spanning trees of a ....

....of replacement edges, we may verify the minimality of a spanning tree. Sensitivity analysis of shortest paths and network flows has been studied by Shier and Witzgall [15] and Gusfield [11] The fastest known algorithms for these problems and also for the replacement edge problem are due to Tarjan [19, 18] and run in time and space O(mff(m; n) where ff is the functional inverse of Ackermann s function. Gabow [8] also achieves these bounds. Here we show that in the special case of planar graphs, these problems can be solved in O(n) time and space. Our results also remedy a lacuna in Fredrickson s ....

[Article contains additional citation context not shown here]

R. E. Tarjan. Sensitivity analysis of minimum spanning trees and shortest path trees. Inf. Process. Lett., 14:30--33, 1982.

.... a problem is to perturbations in its input [19, 26] A significant limitation of the current methods for sensitivity analysis in combinatorial optimization is that they measure only changes in the solution of a problem produced by perturbations in the value of a single element in its input [18, 25, 30, 31]. In some situations it is necessary to determine the maximum effect that bounded changes in the whole input of a problem can have over the value of its solution and, thus, in which sensitivity analysis does not suffice [11, 19, 28] In this paper we consider the important class of matroid ....

R.E. Tarjan, Sensitivity analysis of minimum spanning trees and shortest path trees, Information Processing Letters, 14 (1982), pp. 30--33.

....circuits. If M is a graphic matroid with r(M) n and jE(M)j = m, Koml os algorithm finds the fundamental circuit maxima with O(m n) comparisons [13] Any algorithm for verifying the optimum basis of a graph can in fact be represented as a transmuter, for whose definition we refer to [18]. Let m 0 be the number of edges in the transmuter and n 0 the number of vertices of the transmuter that do not correspond to elements of M; the number of comparisons is m 0 Gamma n 0 Gamma (m Gamma n 1) Then, Koml os result can be restated by saying that there is a transmuter ....

Tarjan, R. E. Sensitivity analysis of minimum spanning trees and shortest path trees. Information Processing Letters 14, 1 (Mar. 1982), 30--33.

....some more definitions to make. If M l [x; y] D(G h ; x; y) we say that x and y are separable. 10 Thus, x and y are separable iff M l [x; y] D(T; x; y) A link edge, LINK(a; b) is an edge in P T (a; b) with weight M h (LINK(a; b) D(T; a; b) cf. sensitivity analysis a la Tarjan [36]) For each edge e in T , cut weight CW (e) maxfM l (x; y)jLINK(x; y) e.g. Theorem 3 There exists an ultrametric tree U 2 [M l ; M h ] iff each pair of vertices is separable. Proof: Suppose there is a pair (x; y) that is not separable. This means that M l [x; y] M h [LINK(x; y) If e is an ....

R.E. Tarjan, Sensitivity Analysis of Minimum Spanning Trees and Shortest Path Trees, Information Processing Letters 14(1):30-33, 1982.

.... remainder of the paper we restrict our attention to those scheduling problems, which may be represented in terms of linear binary programming [3, 6, 11] or, similarly, in terms of a linear trajectory problem [4, 5, 19] A concept of stability analysis for the latter problem has been developed in [4, 5, 10, 11, 12, 24] and in some other papers (see [20] for the extensive survey) It should be noted that most results have been obtained for the stability radius of the whole set of optimal trajectories, i.e. for the largest radius of an open ball in the space of the numerical input data such that a new optimal ....

Tarjan, R.E.: Sensitivity Analysis of Minimum Spanning Trees and Shortest Path Trees, Inform. Processing Letters, Vol. 14 (1982) 30 - 33.

....deletes m n k edges that will be in none of the k smallest spanning trees, and if k n, it identifies and contracts n k edges that will be in all of these trees. Identifying these edges uses an algorithm for the sensitivity analysis of minimum spanning trees, either Tarjan s algorithm [T1] [T2] for general graphs or the algorithm of Booth and Westbrook [BW] for planar graphs. Also used is the linear time selection algorithm [BFPRT] We call the resulting graph the contracted graph. Note that the k smallest spanning trees of the contracted graph are in one to one correspondence with the ....

....four levels, noting that all nodes shown have children except the lower rightmost one. The time required by the algorithm will be the following. From [GGST] E] T1] 518 GREG N. FREDERICKSON Fig. 9. The first four levels of inclusion exclusion for Fig. 1, with spanning tree costs indicated. [T2], BFPRT] and [F1] finding the contracted graph and transforming it into one with maximum degree 3 will take O(m log #(m, n) time and O(m) space. From [CT] E] BW] BFPRT] and [F1] finding the contracted graph of a planar graph and transforming it into one with maximum degree 3 will take ....

R. E. Tarjan, Sensitivity analysis of minimum spanning trees and shortest path trees, Inform. Process. Lett., 14 (1982), pp. 30--33.

....can occur: 1. e is in T and Delta is negative. 2. e is not in T (e is in T ) and Delta is positive. 3. e is not in T and Delta is negative. 4. e is in T and Delta is positive. Clearly, Cases 1 and 2 have no effect on the spanning trees. Now consider Case 3. It is well known (e.g. see [10, 30]) that in this case T is no longer minimum if the weight of e is less than the weight of the maximum cost edge d in the cycle formed by adding e to T . Edge d is found by executing evert(orig(e 0 ) followed by find max(orig(e 2 ) In the special case where the find max operation returns ....

R. E. Tarjan. Sensitivity analysis of minimum spanning trees and shortest path trees. Inf. Process. Lett., 14:30--33, 1982.

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R. E. Tarjan. Sensitivity analysis of minimum spanning trees and shortest path problems. Inform. Proc. Lett., 14:30--33, 1982.

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R. E. Tarjan. Sensitivity analysis of minimum spanning trees and shortest path problems. Inform. Proc. Lett., 14:30--33, 1982.

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