Algorithms for Enumerating All Spanning Trees ofUndirected and Weighted Graphs
01 Apr 1995-SIAM Journal on Computing (Society for Industrial and Applied Mathematics)-Vol. 24, Iss: 2, pp 247-265
TL;DR: Algorithms for enumeration of spanning trees in undirected graphs, with and without weights, are presented, based on swapping edges in a fundamental cycle to construct a computation tree.
Abstract: In this paper, we present algorithms for enumeration of spanning trees in undirected graphs, with and without weights.
The algorithms use a search tree technique to construct a computation tree. The computation tree can be used to output all spanning trees by outputting only relative changes between spanning trees rather than the entire spanning trees themselves. Both the construction of the computation tree and the listing of the trees is shown to require $O(N+V+E)$ operations for the case of undirected graphs without weights. The basic algorithm is based on swapping edges in a fundamental cycle. For the case of weighted graphs (undirected), we show that the nodes of the computation tree of spanning trees can be sorted in increasing order of weight, in $O(N\log V+VE)$ time. The spanning trees themselves can be listed in $O(NV)$ time. Here $N$, $V$, and $E$ refer respectively to the number of spanning trees, vertices, and edges of the graph.
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07 Aug 2002
TL;DR: DBXplorer, a system that enables keyword-based searches in relational databases using a commercial relational database and Web server and allows users to interact via a browser front-end is discussed.
Abstract: Internet search engines have popularized the keyword-based search paradigm. While traditional database management systems offer powerful query languages, they do not allow keyword-based search. In this paper, we discuss DBXplorer, a system that enables keyword-based searches in relational databases. DBXplorer has been implemented using a commercial relational database and Web server and allows users to interact via a browser front-end. We outline the challenges and discuss the implementation of our system, including results of extensive experimental evaluation.
818 citations
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Abstract: With the tremendous growth of information available to end users through the Web, search engines come to play ever a more critical role. Nevertheless, because of their general-purpose approach, it is always less uncommon that obtained result sets provide a burden of useless pages. The next-generation Web architecture, represented by the Semantic Web, provides the layered architecture possibly allowing overcoming this limitation. Several search engines have been proposed, which allow increasing information retrieval accuracy by exploiting a key content of semantic Web resources, that is, relations. However, in order to rank results, most of the existing solutions need to work on the whole annotated knowledge base. In this paper, we propose a relation-based page rank algorithm to be used in conjunction with semantic Web search engines that simply relies on information that could be extracted from user queries and on annotated resources. Relevance is measured as the probability that a retrieved resource actually contains those relations whose existence was assumed by the user at the time of query definition.
116 citations
Cites background from "Algorithms for Enumerating All Span..."
...Unfortunately, even if many algorithms have been proposed in the literature for addressing the task of finding all the spanning forests (or trees) in a graph [ 14 ], [30], none of them is capable of taking into account forests with a variable number of edges derived from originating spanning forests....
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25 Oct 2008
TL;DR: The fastest known general-purpose algorithm for computing a host of fundamental graph invariants, such as the Jones polynomial of an alternating link in knot theory, and the partition functions of the models of Ising, Potts, and Fortuin-Kasteleyn in statistical physics, runs in time roughly proportional to the number of spanning trees in the input graph.
Abstract: The deletion-contraction algorithm is perhaps the most popular method for computing a host of fundamental graph invariants such as the chromatic, flow, and reliability polynomials in graph theory, the Jones polynomial of an alternating link in knot theory, and the partition functions of the models of Ising, Potts, and Fortuin-Kasteleyn in statistical physics. Prior to this work, deletion-contraction was also the fastest known general-purpose algorithm for these invariants, running in time roughly proportional to the number of spanning trees in the input graph.Here, we give a substantially faster algorithm that computes the Tutte polynomial-and hence, all the aforementioned invariants and more-of an arbitrary graph in time within a polynomial factor of the number of connected vertex sets. The algorithm actually evaluates a multivariate generalization of the Tutte polynomial by making use of an identity due to Fortuin and Kasteleyn. We also provide a polynomial-space variant of the algorithm and give an analogous result for Chung and Graham's cover polynomial.
88 citations
eBay1
TL;DR: A toolkit which implements deterministic learning to support search and text classification tasks and uses syntactic parse tree-based similarity measure instead of bag-of-words and keyword frequency approach is described.
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Cites background or methods from "Algorithms for Enumerating All Span..."
...In terms of operations on trees we could follow along the lines of (Kapoor & Ramesh 1995)....
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...The current work deals with syntactic tree transformation in the graph learning framework (compare with Chakrabarti & Faloutsos 2006, Kapoor & Ramesh 1995), treating various phrasings for the same meaning in a more unified and automated manner....
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TL;DR: The solution to the problem of finding the set of optimal spanning trees of a connected graph is described through an algorithm that builds the family of efficient trees, considering two cost functions defined on theSet of edges.
71 citations
References
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TL;DR: Backtrack algorithms for listing certain kinds of subgraphs of a graph are described and analyzed and their applications are analyzed.
Abstract: Backtrack algorithms for listing certain kinds of subgraphs of a graph are described and analyzed. Included are algorithms for listing all spanning trees, all cycles, all simple cycles, or all of c...
276 citations
TL;DR: An algorithm for finding all spanning trees (arborescences) of a directed graph is presented that uses backtracking and a method for detecting bridges based on depth-first search.
Abstract: An algorithm for finding all spanning trees (arborescences) of a directed graph is presented. It uses backtracking and a method for detecting bridges based on depth-first search. The time required is $O(V + E + EN)$ and the space is $O(V + E)$, where V, E, and N represent the number of vertices, edges, and spanning trees, respectively. If the graph is undirected, the time decreases to $O(V + E + VN)$, which is optimal to within a constant factor. The previously best-known algorithm for undirected graphs requires time $O(V + E + EN)$.
193 citations
01 Jan 1975
TL;DR: Two algorithms for generating spanning trees of a connected graph in order of increasing weight are presented, one of which generates the K smallest weight trees, where K can be specified in advance or during execution of the algorithm.
139 citations
86 citations
TL;DR: While an edge-numbering convention and a criterion for a tree play the key roles in systematic generation of trees, the storage technique makes it possible to obtain all the co-factors and determinants of a node-admittance matrix of any network by merely operating on one single master forest matrix.
Abstract: A new method of listing all possible trees of any given graph without duplication or redundancy, using simple geometrical properties of the graph, is proposed. The procedure given is suitable for both manual and automatic computation, and any modifications to the given graph can be catered to by suitable interpolation and extrapolation. An alternate method for complete graphs, derived from it, gives the trees arranged in an order most suitable for their storage as master forest matrices and for directly obtaining trees and 2-trees of any given graph through simple modifications to them instead of starting from scratch every time. Some properties of master forest matrices are discussed, which., inter alia, lead to a formula for the number of trees in a sub-graph of a complete graph. While an edge-numbering convention and a criterion for a tree play the key roles in systematic generation of trees, the storage technique makes it possible to obtain all the co-factors and determinants of a node-admittance matrix of any network (within the range of storage) by merely operating on one single master forest matrix.
42 citations