R.E. Tarjan. Finding optimum branchings. Networks, 7:25--35, 1977. 15  Home/Search   Document Not in Database   Summary   Related Articles   Check

....R ) by techniques as shown in [AK88] As noted before this time bound can be improved to m R ) by allowing balancing arcs to be from V . A directed in tree of minimum weight in a graph with n vertices and m arcs can be computed in time O(min m log n, n ) by the algorithm from [Tar77] Thus Algorithm ALG PATH can be implemented to run in time R min (m R n) log n, n ) 6.4 An Approximation Algorithm for General Graphs In this section we present an approximation algorithm for the SOURCEDARP on general graphs. The algorithm uses ideas similar to the ones in ....

.... This means that ALG TSP R ) The balancing in ALG LA T can be accomplished in time R ) Completion of the graph by computing all pairs shortest paths can be done in E n log n) log n) CLR90, AMO93] All other steps can be carried out in time ) where again the algorithm from [Tar77] is employed for computing a minimum weight in tree. # 6.6 Hardness Results Since the SOURCE DARP generalizes the DARP, it follows from the hardness result in [FG93] that SOURCE DARP is NP hard to solve even on trees. We show that this hardness continues to hold even if the source order is a ....

R. E. Tarjan, Finding optimum branchings, Networks 7 (1977), 25--35.

....file per compressed file, this problem is equivalent to finding an optimum branching in a corresponding directed graph where each edge has a weight equal to size of the delta of with respect to reference file . This problem can be solved in time quadratic in the number of documents [12, 42], but the approach suffers from two drawbacks: First, the solution may contain very long chains of documents that have to be accessed in order to uncompress a particular file. Second, for large collections the quadratic time becomes unacceptable, particularly the cost of computing the appropriate ....

R. Tarjan. Finding optimum branchings. Networks, 7:25--35, 1977.

....LZ distance that is closely related to the performance of Lempel Ziv type compressing schemes. We also refer to [6] and the references therein for work on protocols for estimating file similarities over a communication link. Fast algorithms for the optimum branching problem are described in [4, 22]. While we are not aware of previous work that uses optimum branchings to compress collections of files, there are two previous applications that are quite similar. In particular, Tate [23] uses optimum branchings to find an optimal scheme for compressing multispectral images, while Adler and ....

....i.e. file is compressed by itself, files are compressed by computing a delta with respect to file , and file is compressed by computing a delta with respect to file . 2. 2 Experimental Results We implemented delta compression based on the optimal branching algorithm described in [4, 22], which for dense graphs takes time proportional to the number of edges. Table 1 shows compression results and times on several collections of web pages that we collected by crawling a limited number of pages from each site using a breadth first crawler. The results indicate that the optimum ....

R. Tarjan. Finding optimum branchings. Networks, 7:25--35, 1977.

....LZ distance that is closely related to the performance of Lempel Ziv type compressing schemes. We also refer to [6] and the references therein for work on protocols for estimating file similarities over a communication link. Fast algorithms for the optimum branching problem are described in [4, 22]. While we are not aware of previous work that uses optimum branchings to compress collections of files, there are two previous applications that are quite similar. In particular, Tate [23] uses optimum branchings to find an optimal scheme for compressing multispectral images, while Adler and ....

...., i.e. file is compressed by itself, files are compressed by computing a delta with respect to file , and file is compressed by computing a delta with respect to file . 2. 2 Experimental Results We implemented delta compression based on the optimal branching algorithm described in [4, 22], which for dense graphs takes time proportional to the number of edges. Table 1 shows compression results and times on several collections of web pages that we collected by crawling a limited number of pages from each site using a breadth first crawler. The results indicate that the optimum ....

R. Tarjan. Finding optimum branchings. Networks, 7:25--35, 1977.

....Moreover, we can order the vertices in U pre topologically. Then the first vertex in this order belongs to a strong component K so that each arc a entering K has l(a) 0. For more on algorithms for optimum branchings, see Edmonds [1967] Fulkerson [1974] Chu and Liu [1965] Bock [1971] Tarjan [1977]. 2. Matchings and covers 2.1. Matchings, covers, and Gallai s theorem Let G = V; E) be a graph. A coclique is a subset C of V such that e 6 C for each edge e of G. A vertex cover is a subset W of V such that e W 6= for each edge e of G. It is not difficult to show that for each U V : ....

R.E. Tarjan, Finding optimum branchings, Networks 7 (1977) 25--35.

....marginals are constrained by the data. To learn the tree structure with a fixed root X root ,we learn the conditionals P (X k Pa k ) andthenfindtheminimum spanning tree in the directed graph, whose edge weights are the appropriate conditional entropies. This tree can be found efficiently [8]. 4. Mixtures of trees The tree representation of an object allows for efficient learning and search. However, it is difficult to use a tree to model cases where some of the primitives constituting an object are missing due to occlusions, variations in aspect or failures of the local ....

R.E. Tarjan. Finding optimum branchings. Networks, 7(1):25--36, 1977.

....n Y i=1 P 0 (x i j i ) P 1 (x i j i ) n X i=1 X x P 0 (x) log P 0 (x i j i ) P 1 (x i j i ) n X i=1 X x i X pa i P 0 (x i ; i ) log P 0 (x i j i ) P 1 (x i j i ) 9) We show that the problem of maximizing eq. 9) is equivalent to the maximum branching problem [13]. In the maximum branching problem, a branching B of a directed graph G is a set of arcs such that: 1. if (x 1 ; y 1 ) and (x 2 ; y 2 ) are distinct arcs of B then y 1 6= y 2 . 2. B does not contain a cycle. Given a real value c(v; w) de ned for each arc of G, a maximum branching of G is a ....

....with node x i s plus a node x 0 with an arc from x 0 to all other nodes. W (i; j) P x i P x j P 0 (x i ; x j ) log P0 (x i jx j ) P1 (x i jx j ) is the weight associated with each arc in the graph. There are algorithms for solving the maximum branching problem in low order polynomial time [13]. To classify a pattern x, the Bayes decision rule eq. 1) is used. Similar to the method in [2] fast classi cation of a pattern can be achieved by constructing a table for all possible values of a variable and its parent. By using eq. 8) the log likelihood value in eq. 1) becomes: log P 0 ....

R. Tarjan. Finding optimum branchings. Networks, 7:25-35, 1977.

....are constrained by the data. To learn the tree structure with a fixed root ( we learn the conditionals 4132 5 and then find the minimum spanning tree in the directed graph, whose edge weights are the appropriate conditional entropies. This tree can be found efficiently [9]. 4. Mixtures of trees The tree representation of an object allows for efficient learning and search. However, it is difficult to use a tree to model cases where some of the primitives constituting an object are missing due to occlusions, variations in aspect or failures of the local ....

R.E. Tarjan. Finding optimum branchings. Networks, 7(1):25--36, 1977.

....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 ....

....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 tree in G S is O (n log n P n i=1 t GW (i) 2 ) All that remains is to perform the ....

R. E. Tarjan. Finding Optimum Branchings. Networks, 7:25-35, 1977.

....tree with all nodes except r having indegree one and r having indegree zero. A reverse arborescence rooted at r is one with edges directed towards r, i.e. all nodes except r have outdegree one and r has outdegree zero. Given a directed graph with edge weights and a root node r, the algorithms in [2, 3, 9] can be used to find a minimum weight arborescence. For a vertex v in a rooted tree T , let the components formed by deleting v be denoted by C 1 (v) C d(v) v) where d(v) is the degree of v in T . If v is not the root, we will assume that C 1 (v) is the component containing the root ....

R. E. Tarjan, "Finding optimum branchings", Networks, 7 (1977), pp. 25-35. 9

.... as shown in [3] As noted before this time bound can be improved to O(n mA ) by allowing balancing arcs to be from V V instead of just A(E) A rooted spanning tree of minimum weight in a graph with n vertices and m arcs can be computed in time O(minfm log n; n 2 g) by the algorithm from [12]. Thus Algorithm Alg Path can be implemented to run in time O(n mA minf(mA n) log n; n 2 g) 6 An Approximation Algorithm for General Graphs In this section we present our approximation algorithm for Fifo Darp on general graphs. The algorithm uses ideas similar to the ones in [8] In this ....

.... in the modi ed version of Alg Last Arcs can be accomplished in time O(n mA ) Completion of the graph by computing all pairs shortest paths can be done in time O(nmE n 2 log n) O(n 2 log n) 5, 1] All other steps can be carried out in time O(n 2 ) where again the algorithm from [12] is employed for computing a minimum weight directed spanning tree. ut ....

R. E. Tarjan, Finding optimum branchings, Networks 7 (1977), 25-35.

....has a lower cost than the minimum spanning in tree for G. Hence either C(T ) C(T 0 ) and T is a communication tree with minimum communication cost or we have a contradiction. 2. There exist several efficient algorithms for finding the minimum spanning arborescences of a directed graph [5, 6, 21, 13]. The complexity for a dense graph with n vertices is O(n 2 ) 21] Given an expression E we construct the communication graph in O(n 2 ) steps and the communication tree with minimum communication cost can be found in O(n 2 ) steps. For example consider the array statement in Section 1 A(8 ....

....C(T ) C(T 0 ) and T is a communication tree with minimum communication cost or we have a contradiction. 2. There exist several efficient algorithms for finding the minimum spanning arborescences of a directed graph [5, 6, 21, 13] The complexity for a dense graph with n vertices is O(n 2 ) [21]. Given an expression E we construct the communication graph in O(n 2 ) steps and the communication tree with minimum communication cost can be found in O(n 2 ) steps. For example consider the array statement in Section 1 A(8 : 15) a 1 X1(0 : 7) a 2 X2(4 : 11) a 3 X3(8 : 15) a 4 ....

R.E. Tarjan. Finding optimum branchings. Networks, 7(1):25--36, 1977.

....the 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 ....

R.E. Tarjan. Finding optimum branchings. Networks, 7:25--35, 1977.

....that is, we assign a real w ij for each pair (i; j) of genetic events; w ij is in general asymmetric. We then find the rooted tree whose total weight (the sum of the weights of all edges in the tree) is maximized. The key algorithmic ingredient for this is a classical result due to Edmonds (see [Edm, Kar, Tar, Pap]) stating that the best rooted, directed tree can be found efficiently in fact, in O(n 2 ) time, where n is the number of genetic events. Notice that this is not the better known minimum spanning tree problem, which relates to undirected graphs, see [Pap] In the application of this method to ....

R.E. Tarjan. Finding Optimum Branchings. Networks 7(1977) 25--35.

.... Galil, Spencer and Tarjan [11] The K branchings of greatest weight can be generated in O(KE log V ) time and O(K E V ) space, as shown by Camerini, Fratta and Maffioli [3] Our method of generating branchings is similar to Camerini et al. which applies the branchings algorithm of Tarjan [37], but has some differences. These differences are due to our particular application, namely generating 19 branchings to meet a dovetail chain constraint, which allows us to apply the algorithm of Gabow et al. 11] to generate K branchings in O(K(E V log V ) time. 4.2.1 Forming constraints ....

....property if B has no in edge and (A;B) does not create a cycle. Since (A;B) forms a cycle if and only if A and B are members of the same arborescence, we can test for cycle creation in essentially constant time by maintaining a partition of fragments into arborescences with disjoint sets [37]. Thus the dominant step is sorting the edges. In short greedy repair can be performed in O(E log V ) time worst case. Interestingly it is asymptotically more expensive to greedily repair a branching than to compute one of maximum weight. This is in the worst case, however. For the sparse graphs ....

Tarjan, Robert. Finding optimum branchings. Networks 7, 25--35, 1977.

....of any band ordering, or the optimal compression order of the bands of the multispectral image. It is known that a maximum weight directed spanning forest (also known as an optimal branching ) can be found in O(jV j log jEj jEj) time on sparse graphs and O(jV j 2 ) time on dense graphs [7, 8], so it follows that an optimal compression order can be found in O(n 2 ) time. The graph G that is derived from the sample A and B matrices of Figure 1 is shown in Figure 2, where dashed and solid lines both represent graph edges, and the solid lines are the edges that are part of the maximum ....

R. E. Tarjan, "Finding optimum branchings," Networks, vol. 7, pp. 25--35, 1977.

....of any band ordering, or the optimal compression order of the bands of the multispectral image. It is known that a maximum weight directed spanning forest (also known as an optimal branching ) can be found in O(jV j log jEj jEj) time on sparse graphs and O(jV j 2 ) time on dense graphs [13, 2], so it follows that an optimal compression order can be found in O(n 2 ) time. The graph G that is derived from the sample A and B matrices of Figure 1 is shown in Figure 2, where dashed and solid lines both represent graph edges, and the solid lines are the edges that are part of the maximum ....

R. E. Tarjan. "Finding Optimum Branchings," Networks, Vol. 7, pp. 25--35, 1977.

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R.E. Tarjan. Finding optimum branchings. Networks, 7:25--35, 1977. 15

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R.E. Tarjan. Finding optimum branchings. Networks, 7(1):25--36, 1977.

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R. Tarjan. Finding optimum branchings. Networks, 7:25--35, 1977.

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