A Note on <i>K</i>-Best Solutions to the Chinese Postman Problem

Saruwatari, Yasufumi; Matsui, Tomomi

doi:10.1137/0803037

Cited by 1 publication

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“…In natural language processing applications, an important generalization of the k shortest paths problem involves finding the k best parse trees of a context-free grammar [35,103,107,148,149,191]. Other problems whose k-best solutions have been studied include the Chinese postman problem [160], the traveling salesman problem [155], the k best spanning arborescences in a directed network [28], the matroid intersection problem [29,190], binary search trees and Huffman coding [5], chess strategies [3], the k best integer flows [92,93,164], the k smallest cuts in a network [91,94,95], and, in probabilistic reasoning, the k best solutions to a graphical model [55,69,78,145].…”

Section: Other Problemsmentioning

confidence: 99%

k-Best Enumeration

Eppstein¹

2016

Encyclopedia of Algorithms

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We survey k-best enumeration problems and the algorithms for solving them, including in particular the problems of finding the k shortest paths, k smallest spanning trees, and k best matchings in weighted graphs.in both the input size and k. There are two general techniques that can be used to achieve this, based on optimal substructures and solution space partitioning. In turn, these methods rely on fast algorithms for the selection problem in certain sets of structured values. SelectionUnderlying many k-best algorithms is the problem of selection, finding the k smallest values from a larger set of values [21]. If there are n values in the set, then selection may be solved in time O(n + k), for instance by quickselect, an algorithm that chooses a random pivot value from the set, partitions the rest of the set into subsets that are less than, equal to, or greater than the pivot, and then recurses into one of these subsets [131].However, in k-best problems, we do not wish to generate all solutions before finding the best of them: we wish to avoid the O(n) part of this O(n + k) time bound. Often, the solution values among which we are selecting the k best have some additional structure that allows finding the k best without examining all solutions. For instance, suppose that the space of solutions can be represented as the vertex set of an (implicitly defined) edge-weighted graph, that the degrees of the vertices in this graph are bounded, and that the quality of any particular solution equals the length of the shortest path in this graph from some designated starting vertex to the solution vertex. In such a case, Dijkstra's algorithm can be used to recover the k smallest of these values in time O(k log k), independent of the size of the graph, without requiring that the whole graph be explored. Other structures that allow the k best solutions to be found more efficiently than enumeration of the whole solution set include representations of the solution space as the set of sums of pairs of values drawn from two sorted sets [76,111], or as the elements of an (implicitly defined) matrix whose rows and columns are sorted [77]. Again, for such structures, the k smallest values can be found efficiently without examining the whole matrix or the whole set of pair sums.A general framework that can be used to describe many of these structured selection problems is the problem of selection in a d-ary heap. A min-heap is a rooted tree, with values associated with its nodes, such that each non-root node has a value that is at least as large as its parent. In order to avoid having to construct all solutions before performing a selection algorithm, and in order to handle situations in which this tree has infinitely many nodes, we will assume that our selection algorithm is given as input an implicit representation of such a heap. That is, its input consists of a binary representation of the root node of the heap together with pointers to two subroutines that take a single node as input: one subroutine to list the children of any...

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Section: Other Problemsmentioning

confidence: 99%

k-Best Enumeration

Eppstein¹

2016

Encyclopedia of Algorithms

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A Note on K-Best Solutions to the Chinese Postman Problem

Cited by 1 publication

References 9 publications

k-Best Enumeration

k-Best Enumeration

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