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Finding the k Shortest Paths
, 1997
"... We give algorithms for finding the k shortest paths (not required to be simple) connecting a pair of vertices in a digraph. Our algorithms output an implicit representation of these paths in a digraph with n vertices and m edges, in time O(m + n log n + k). We can also find the k shortest pat ..."
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Cited by 290 (1 self)
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We give algorithms for finding the k shortest paths (not required to be simple) connecting a pair of vertices in a digraph. Our algorithms output an implicit representation of these paths in a digraph with n vertices and m edges, in time O(m + n log n + k). We can also find the k shortest paths from a given source s to each vertex in the graph, in total time O(m + n log n +kn). We describe applications to dynamic programming problems including the knapsack problem, sequence alignment, maximum inscribed polygons, and genealogical relationship discovery. 1 Introduction We consider a longstudied generalization of the shortest path problem, in which not one but several short paths must be produced. The k shortest paths problem is to list the k paths connecting a given sourcedestination pair in the digraph with minimum total length. Our techniques also apply to the problem of listing all paths shorter than some given threshhold length. In the version of these problems studi...
The Practical Use of the A* Algorithm for Exact Multiple Sequence Alignment
 Journal of Computational Biology
, 1997
"... Multiple alignment is an important problem in computational biology. It is well known that it can be solved exactly by a dynamic programming algorithm which in turn can be interpreted as a shortest path computation in a directed acyclic graph. The A algorithm (or goal directed unidirectional search ..."
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Cited by 18 (3 self)
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Multiple alignment is an important problem in computational biology. It is well known that it can be solved exactly by a dynamic programming algorithm which in turn can be interpreted as a shortest path computation in a directed acyclic graph. The A algorithm (or goal directed unidirectional search) is a technique that speeds up the computation of a shortest path by transforming the edge lengths without losing the optimality of the shortest path. We implemented the A algorithm in a computer program similar to MSA [GKS95] and FMA [SI97b]. We incorporated in this program new bounding strategies for both, lower and upper bounds and show that the A algorithm, together with our improvements, can speed up computations considerably. Additionally we show that the A algorithm together with a standard bounding technique is superior to the well known CarilloLipman bounding since it excludes more nodes from consideration. 1 Introduction One of the most prominent problems in computational mo...
New Approaches To Flexible Alignment Of Multiple Biological Sequences
, 1997
"... The multiple sequence alignment problem is applicable and important in various fields in molecular biology such as the prediction of three dimensional structures of proteins and the inference of phylogenetic tree. However, the optimal alignment based on the scoring criterion is not always the biolog ..."
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Cited by 2 (0 self)
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The multiple sequence alignment problem is applicable and important in various fields in molecular biology such as the prediction of three dimensional structures of proteins and the inference of phylogenetic tree. However, the optimal alignment based on the scoring criterion is not always the biologically most significant alignment. We here propose two flexible and efficient approaches to solve this problem. One approach is to provide many suboptimal alignments as alternatives for the optimal one. Although this suboptimal problem is wellstudied for the alignment of two sequences, it has been considered impossible to investigate such suboptimal alignments of more than two sequences because of the enormous size of the problem. We first introduce techniques for computation of E1 , or a set of all aligned groups of residues in optimal and suboptimal alignments, and then propose algorithms for enumeration of suboptimal alignments. To reduce the amount of suboptimal solutions, we discuss wh...