A. Broder, Generating random spanning trees, Proceedings of 30th Annual Symposium on Foundations of Computer Science (Research Triangle Park, NC,  Home/Search   Document Not in Database   Summary   Related Articles   Check

....2, is part of a known circle of ideas relating spanning trees to Markov chain stationary distributions. Clearly Proposition 1 yields an algorithm for generating a uniform spanning tree in time O(EC) It is known [4] that EC , and [13] that for regular graphs EC 8n . Andrei Broder [7] has independently noted Proposition 1 and discusses these algorithmic aspects at greater length. Our emphasis is rather on theoretical properties of uniform spanning trees, a topic on which there seems no literature at all. Proposition 1 opens the door to their study via modern random walk ....

A. Broder. Generating random spanning trees. In Proc. 30'th IEEE Symp. Found. Comp. Sci., pages 442--447, 1989.

....sample from an irreducible N state Markov chain. Although not mentioned explicitly in their paper, the method described in Section 3 of Lovasz and Winkler can in fact be used to obtain an exact sample from the tree distribution of the Markov chain (as defined in Section 3. 1) Aldous [1] Broder [5], and Propp and Wilson [13] also describe algorithms for sampling from the tree distribution. Propp and Wilson [13] discuss and compare these and other methods of sampling from the tree distribution, and from the stationary distribution. Their discussion includes consideration of such issues as ....

Broder, A. Generating random spanning trees. In Proceedings of the 30th Annual Symposium on Foundations of Computer Science, pages 442--447, 1989.

....are used to determine global information. For example, random walks have been used in many applications in computer science, including on line algorithms [9] space efficient algorithms for undirected connectivity [7] approximation algorithms [10, 14] generation of random spanning trees [2, 6, 16], assigning processes to nodes in networks [4] and token management schemes and self stabilizing in distributed computing [13, 18] Frequently, we ask for the cover time of a graph, i.e. the expected time for a random walk to begin at a vertex and to visit the entire graph. While much work has ....

....are elements of E directed in both senses, a finite walk according to this view, is a word over A. The set of walks on G starting with vertex i will be denoted by W i . The generated tree (w) by the walk w is the set of all edges corresponding to the first entrance of the walk into a vertex of G [2, 6]; it is easy to see that (w) is an undirected tree. For a given vertex i and a tree T containing i, the language L i (T ) denotes the set of shortest finite walks (in the prefix order) starting with i and generating T , i.e. L i (T ) fw=w is a finite walk starting with i, such that (w) T ....

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A.Z. Broder. Generating random spanning trees. In Proc. 30th Ann. IEEE Symp. on Foundations of Computer Science, pages 442--453, October 1989.

....automates finis. This work has been supported by the ESPRIT Basic Research Working Group Computing by Graph Transformations II . Laboratoire associ e au CNRS 1 Introduction Random walks have been studied extensively, and have many applications such as generation of random spanning trees[1, 2, 12], token management schemes[8, 11] effective resistance of electrical networks[3, 10, 5] and on line algorithms[4] In this paper, we consider a connected simple undirected graph G = V; E) together with a positive real valued map w over E, w(e) being called the weight of e. A discrete time ....

....senses, a finite (infinite) walk according to this view, is a word (infinite word) over A. The set of infinite walks on G starting with vertex i will be denoted by W i . The generated tree (w) by the walk w is the set of all edges corresponding to the first entrance of the walk into a vertex of G [1, 2]; it is easy to see that (w) is an undirected tree. For a given vertex i and a tree T containing i, the language L i (T ) denotes the set of shortest finite walks (in the prefix order) starting with i and generating T , i.e. L i (T ) fw=w is a finite walk starting with i, such that (w) T ....

[Article contains additional citation context not shown here]

A.Z. Broder. Generating random spanning trees. In Proc. 30th Ann. IEEE Symp. on Foundations of Computer Science, pages 442--453, October 1989.

....and it requires time that is O(n ) Colbourne [48] modified the algorithm to require fewer determinant computations and reduced its time to O(n ) This might be acceptable for the initialization of an EA s population, but is too expensive to be the basis of a recombination operator. Broder [49] described a probabilistic method based on a random walk in G. A particle begins at an arbitrary node in G. At each step, it moves over a randomly chosen adjacent edge to one of its neighbors. When the particle visits a node for the first time, the edge it traverses joins the spanning tree. The ....

....expected time is O(n log n) for almost all graphs and O(n ) for a few special cases. Like PrimRST, RandWalkRST can honor constraints by disallowing edges that would violate them, though such a strategy will in general render non uniform the probabilities of valid spanning trees. Broder [49] also described a mechanism that walks randomly not through the graph but through the space of spanning trees on it. It begins with an arbitrary spanning tree, then repeatedly replaces a random edge with a new one that reconnects the tree. The distribution of spanning trees generated by this ....

Andrei Broder, "Generating random spanning trees," in IEEE 30th Annual Symposium on Foundations of Computer Science. 1989, pp. 442--447, IEEE.

.... seeing A at time t 1is # i#S # p j(i) k # cw(A) p k,i p j(i) k ## = cw(A) It follows that cw( is the stationary distribution for our tree process, but of course the stationary distribution for the roots is just # so we have that # i is proportional to w(# i ) Aldous [2] and Broder [11] use a closely related construction to design an elegant algorithm to generate a random spanning tree of a graph. Lemma 1 already provides a stopping rule, described below, that attains the stationary distribution. In contrast to the procedure described above, the stopping rule constructs a ....

A.Z. Broder, Generating random spanning trees, Proc.30thIEEESymp.onFound.of Computer Science (

....for arbitrary periodic lattices. The connection between simple random walks and spanning trees is documented in [Pem] the main result that will be used from there is that P(e 2 T) is determined by certain hitting probabilities, but the reader desiring more details may also consult [Al2] or [Bro]. Section 4 uses these lemmas to show that the Green s function is the unique limit of Green s functions on finite subgraphs and that it in fact determines the f.d.m. s for G via determinants of the transfer impedance matrix. Section 5 considers two examples. The first is the case G = ZZ 2 , ....

....of e 1 2 T. Now assume for induction that the theorem is true for k Gamma 1. The easy case to dispose of is when M 0 def = M(e 1 ; e k Gamma1 ) has zero determinant. Then by induction P(e 1 ; e k Gamma1 2 T) 0 and it follows (e.g. from the random walk construction of T in [Al2, Bro, Pem]) that the edges e 1 ; e k Gamma1 form some loop. Suppose without loss of generality that the loop is given by e 1 ; e r and that all edges are oriented forward along the loop. Then P r i=1 g e i = 0, hence M is singular and det(M) P(e 1 ; e k 2 T) 0. In the case ....

Broder, A. (1988). Generating random spanning trees. In: Symp. foundations of computer sci., Institute for Electrical and Electronic Engineers, New York 442 - 447.

....variables with distribution p. Or, assuming p a 0 for all a, the following scheme constructs R 1 with the distribution on R 1;n induced by p from a sequence of independent random variables W 0 ; W 1 ; with distribution p, by application of a general result for irreducible Markov chains [11] [33, x6.1] R 1 : f(Wm Gamma1 ; Wm ) Wm = 2 fW 0 ; Wm Gamma1 g; m 1g (40) 23 According to Theorem 11, no matter how a random tree R 1 is constructed with the distribution on R 1;n induced by p, if k Gamma 1 edges of R 1 are deleted at random, the result is a rooted forest of k ....

A. Broder. Generating random spanning trees. In Proc. 30'th IEEE Symp. Found. Comp. Sci., pages 442--447, 1989. 27

....that there is an intimate connection between exact or approximate enumeration of certain objects and between the problem of nding (exactly or approximately) a random element among them. What can be said about a random spanning tree of a graph G and how can you generate such an object Broder [148] and Aldous [145] proposed a very simple way to generate a random spanning tree in a nite graph: Start a random walk and add to the tree all edges in the walk which do not close a circle when rst traversed. Wilson [168] found a remarkable algorithm with superior performances and important ....

A. Broder, Generating random spanning trees, in 30th Annual Symp. Foundations Computer Sci., IEEE, New York, 1989, pp. 442-447.

....can be found in [Sin93, Vaz91, Kan94, MR95, JS95] A question of interest is the possibility of parallelizing the almost uniform generation and approximate counting problems. Consider the Markov Chain defined by Broder for almost uniform generation of perfect matchings in dense bipartite graphs ( Bro89] Teng has proved that the problem of computing the final node m 0 of a sequential walk, starting at a node m is P complete ( Ten95] This result does not exclude the possibility of generating in parallel an almost uniform perfect matching. The Teng result excludes the possibility of the ....

A. Broder. Generating random spanning trees. In 30th IEEE Symposium on Foundations of Computer Science, pages 442--447, 1989.

....are three basic approaches for sampling random spanning trees. The first is to select one edge at a time to be in or out of the tree, using the the matrix tree theorem to find the appropriate probabilities; this gives an exactly uniform choice in O(mn 2. 376 ) arithmetic operations [16] Broder [5] cites two papers with more sophisticated versions of this approach that reduce the time to that for a single determinant. The second approach to spanning tree generation can be found in papers by Aldous and Broder [1, 5] Their method takes a random walk on the vertices of given graph, and ....

....exactly uniform choice in O(mn 2.376 ) arithmetic operations [16] Broder [5] cites two papers with more sophisticated versions of this approach that reduce the time to that for a single determinant. The second approach to spanning tree generation can be found in papers by Aldous and Broder [1, 5]. Their method takes a random walk on the vertices of given graph, and selects for each vertex (except the start point) the edge by which that vertex was first visited. These n 1 edges form a spanning tree which can be shown to be selected exactly uniformly at random. The expected time of this ....

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A. Z. Broder. Generating random spanning trees. 30th IEEE Symp. Foundations of Computer Science (1989) 442--447.

.... in an undergraduate thesis [36] of one of his students) but was apparently not published until the paper of Anantharam and Tsoucas [7] The fact that the Markov chain tree theorem can be interpreted as an algorithm for generating uniform random spanning trees was observed by Aldous [2] and Broder [11], both deriving from conversations with Diaconis. 2] initiated study of theoretical properties of uniform random spanning trees, proving e.g. the following bounds on the diameter Delta of the random tree in a regular n vertex graph. n 1=2 K 1 2 log n E Delta K 2 1=2 2 n 1=2 log n ....

A. Broder. Generating random spanning trees. In Proc. 30'th IEEE Symp. Found. Comp. Sci., pages 442--447, 1989.

....distribution of the random walk) For every connected non bipartite graph G RG is ergodic and, for every vertex v 2 V , v = d(v) 2m . Proof: Subsumed by Lemma 2 below. One can use random walks on graphs, for instance, for sampling spanning trees of a graph. This was accomplished by Broder [2] by means of the following algorithm: start at a fixed node, and perform a random walk, marking an edge whenever it visits a previously unvisited node. Stop when all nodes are visited and output the set of marked nodes. 3.1.1 Reversible MC and random walks on weighted graphs Question: When is it ....

Andrei Broder. Generating random spanning trees. In 30th Annual Symposium on Foundations of Computer Science, pages 442--447, Research Triangle Park, North Carolina, 30 October--1 November 1989. IEEE.

....automates finis. This work has been supported by the ESPRIT Basic Research Working Group Computing by Graph Transformations II . y Laboratoire associ e au CNRS 1 Introduction Random walks have been studied extensively, and have many applications such as generation of random spanning trees[1, 2, 12], token management schemes[8, 11] effective resistance of electrical networks[3, 10, 5] and on line algorithms[4] In this paper, we consider a connected simple undirected graph G = V; E) together with a positive real valued map w over E, w(e) being called the weight of e. A discrete time ....

....a finite (infinite) walk according to this view, is a word (infinite word) over A. The set of infinite walks on G starting with vertex i will be denoted by W i . The generated tree (w) by the walk w is the set of all edges corresponding to the first entrance of the walk into a vertex of G [1, 2]; it is easy to see that (w) is an undirected tree. For a given vertex i and a tree T containing i, the language L i (T ) denotes the set of shortest finite walks (in the prefix order) starting with i and generating T , i.e. L i (T ) fw=w is a finite walk starting with i, such that (w) ....

[Article contains additional citation context not shown here]

A.Z. Broder. Generating random spanning trees. In Proc. 30th Ann. IEEE Symp. on Foundations of Computer Science, pages 442--453, October 1989.

....random walks on G, where at each step the random walk moves from the current vertex to a neighbour chosen with probability proportional to the weight of traversed edge. Random walks have been studied extensively, and have numerous applications, including generation of random spanning trees [1, 6, 16, 18], online algorithms[8] space efficient algorithms for undirected connectivity[7] approximation algorithms [9, 14] assigning processes to nodes in networks [4] and token management schemes and self stabilizing in distributed computing [13, 17] The cover time of a random walk on a graph is the ....

....to random walks on graphs, such as mean cover time, is due to the syntactic complexity of the tour languages. 5 Hitting time and cover time As we saw in the previous section, sets of walks over a connected undirected graph may be identified with a rational language over an alphabet. Many authors [1, 6, 5] have studied uniform random walks, by considering the hypothesis that the walk moves from a vertex v to one of its neighbours with the same probability 1 d(v) where d(v) is the degree of v. It seems, however, that this direct assumption does not simplify the computation of interesting ....

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A.Z. Broder. Generating random spanning trees. In Proc. 30th Ann. IEEE Symp. on Foundations of Computer Science, pages 442--453, October 1989.

....resulting tree, rooted at I. It is easy to check that the stationary distribution is uniform. This chain may be regarded as the specialization (to the complete graph) of the now well known random walk algorithm for constructing a uniform random spanning tree of a general finite undirected graph [2, 8], which is part of the circle of ideas surrounding the Markov chain tree theorem [18, 15] Now define F n i to be the subtree of Q n i rooted at vertex 1 obtained by cutting the edge at vertex 1 which leads toward root(Q n i ) if root(Q n i ) 1 let F n i = Q n i ) Regard (F n i ; i 0) ....

A. Broder. Generating random spanning trees. In Proc. 30'th IEEE Symp. Found. Comp. Sci., pages 442--447, 1989.

....in the context of highly symmetric Markov chains (e.g. those associated with card shuffling [Ald81, Dia88] but seems difficult to apply to the kind of irregular Markov chains that arise in the analysis of Monte Carlo algorithms. Two exceptions are the analyses of Aldous [Ald90] and Broder [Bro89] for a Markov chain on spanning trees of a graph, and of Matthews [Mat91] for a Markov chain related to linear extensions of a partial order. A glance at the latter paper will give an impression of the technical complexities that can arise. 3 We should point out that the coupling method has very ....

A.Z. Broder. Generating random spanning trees, Proceedings of the 30th Annual IEEE Symposium on Foundations of Computer Science, 442--447, 1989.

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A. Broder, Generating random spanning trees, Proceedings of 30th Annual Symposium on Foundations of Computer Science (Research Triangle Park, NC,

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A. Broder, Generating random spanning trees, Proceedings of 30th Annual Symposium on Foundations of Computer Science (Research Triangle Park, NC,

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A. Broder, Generating random spanning trees, Proceedings of 30th Annual Symposium on Foundations of Computer Science (Research Triangle Park, NC, 1989), 442-447, IEEE, 1989.

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A. Broder. Generating random spanning trees. In Proc. 30'th IEEE Symp. Found. Comp. Sci., pages 442--447, 1989.

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A. Broder. Generating random spanning trees. In IEEE 30th Annual Symposium on Foundations of Computer Science, pages 442--447. IEEE, 1989.

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A. Broder. Generating random spanning trees. In Proc. 30'th IEEE Syrup. Found. Comp. Sci., pages 442-447, 1989.

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A. Broder. Generating random spanning trees. In Proc. 30'th IEEE Symp. Found. Comp. Sci., pages 442--447, 1989.

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Broder, A. (1988). Generating random spanning trees. Unpublished.

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