## Undiscretized Dynamic Programming: Faster Algorithms for Facility Location and Related Problems on Trees

### BibTeX

@MISC{_undiscretizeddynamic,

author = {},

title = {Undiscretized Dynamic Programming: Faster Algorithms for Facility Location and Related Problems on Trees},

year = {}

}

### OpenURL

### Abstract

In the Uncapacitated Facility Location (UFL) problem, there is a xed cost for opening a facility, and some distance matrix d that determines the cost of distributing commodities from any facility i to any consumer j. The problem is NP-hard in general and when d consists of a distance metric in a graph [7, 12]. However, for the case where the commodity transportation costs are given by path lengths in a tree, an O(n 2) dynamic programming algorithm was given by [4,7]. We improve this dynamic programming algorithm by using the geometry of piecewise linear functions and fast merging of the binary search trees used to store these functions. We achieve the complexity bound of O(n log n) fortheTree Location Problem and some related problems. Our approach gives a general method for solving tree dynamic programming problems. 1

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Citation Context ...ome distance matrix d that determines the cost of distributing commodities from any facility i to any consumer j. The problem is NP-hard in general and when d consists of a distance metric in a graph =-=[7, 12]-=-. However, for the case where the commodity transportation costs are given by path lengths in a tree, an O(n 2 ) dynamic programming algorithm was given by [4,7]. We improve this dynamic programming a... |

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Citation Context ...complexity bound of O(n log n) fortheTree Location Problem and some related problems. Our approach gives a general method for solving tree dynamic programming problems. 1 Introduction The UFL problem =-=[3, 4, 7]-=- has been studied extensively in location theory. The essence of the model is a trade-o between the facility placement cost and the transportation cost. The problem is to open a subset of facilities i... |

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Citation Context ...ists of a distance metric in a graph [7, 12]. However, for the case where the commodity transportation costs are given by path lengths in a tree, an O(n 2 ) dynamic programming algorithm was given by =-=[4,7]-=-. We improve this dynamic programming algorithm by using the geometry of piecewise linear functions and fast merging of the binary search trees used to store these functions. We achieve the complexity... |

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