Efficient Distributed Algorithm for the Location Design and Routing Problem

Abstract

The Location Design and Routing problem asks to find a subset of “depot” nodes and a spanning forest of a graph such that every connected component in the forest contains at least one depot. This problem arises in a number of both logistical and computer networking problems, for example, in selecting the number and location of distribution centers in vehicle routing networks. This problem is functionally equivalent to that of supernode selection in peer-to-peer networks. A distributed algorithm approximates a solution to this problem that runs in a logarithmic number of communication rounds with respect to the number of nodes (independent of the topology of the network), and, under assumptions on the embedding of the edge weights, whose solutions are within a factor of 2 of optimal.

1 . A computer-implemented method of identifying nodes or distribution centers for distributing information or materials throughout a network or throughout a geographic area, comprising:
 (a) starting with an empty forest of potential nodes or distribution centers of said network or said geographic area, the computer assigning each potential node or distribution center of the forest to be a member of its own connected component;   (b) for each potential node or distribution center that has not yet been assigned to be a node or distribution center, the computer adding a cut edge of a connected component to the forest and merging the cut edge with a potential node or distribution center on the other end of the cut edge;   (c) when a potential node or distribution center is merged with another potential node or distribution center that has been assigned to a node or distribution center, the computer assigning the same node or distribution center to the potential node or distribution center and stopping a resulting merged node from actively growing; and   (d) the computer repeating steps (a)-(c) until all potential nodes or distribution centers have been assigned to a node or distribution center.   
     
     
         2 . The method of  claim 1 , wherein the computer proceeds asynchronously between respective connected components of the forest of potential nodes or distribution centers. 
     
     
         3 . The method of  claim 1 , wherein the cut edges of the connected components have different weightings. 
     
     
         4 . A computer system including a memory and a processor programmed by executable instructions to implement a method of identifying nodes or distribution centers for distributing information or materials throughout a network or throughout a geographic area, the executable instructions when processed by the processor implementing the steps of:
 (a) starting with an empty forest of potential nodes or distribution centers of said network or said geographic area, assigning each potential node or distribution center of the forest to be a member of its own connected component;   (b) for each potential node or distribution center that has not yet been assigned to be a node or distribution center, adding a cut edge of a connected component to the forest and merging the cut edge with a potential node or distribution center on the other end of the cut edge;   (c) when a potential node or distribution center is merged with another potential node or distribution center that has been assigned to a node or distribution center, assigning the same node or distribution center to the potential node or distribution center and stopping a resulting merged node from actively growing; and   (d) repeating steps (a)-(c) until all potential nodes or distribution centers have been assigned to a node or distribution center.   
     
     
         5 . The computer system of  claim 4 , wherein the processor is further programmed by executable instructions to proceed asynchronously between respective connected components of the forest of potential nodes or distribution centers. 
     
     
         6 . The computer system of  claim 4 , wherein the cut edges of the connected components have different weightings. 
     
     
         7 . A computer readable storage medium having executable instructions stored thereon that when executed by a processor cause said processor to implement a method of identifying nodes or distribution centers for distributing information or materials throughout a network or throughout a geographic area, the method comprising the steps of:
 (a) starting with an empty forest of potential nodes or distribution centers of said network or said geographic area, assigning each potential node or distribution center of the forest to be a member of its own connected component;   (b) for each potential node or distribution center that has not yet been assigned to be a node or distribution center, adding a cut edge of a connected component to the forest and merging the cut edge with a potential node or distribution center on the other end of the cut edge;   (c) when a potential node or distribution center is merged with another potential node or distribution center that has been assigned to a node or distribution center, assigning the same node or distribution center to the potential node or distribution center and stopping a resulting merged node from actively growing; and   (d) repeating steps (a)-(c) until all potential nodes or distribution centers have been assigned to a node or distribution center.   
     
     
         8 . The storage medium of  claim 7 , wherein the processor proceeds asynchronously between respective connected components of the forest of potential nodes or distribution centers. 
     
     
         9 . The storage medium of  claim 7 , wherein the cut edges of the connected components have different weightings.