The two-median problem on Manhattan meshes

We investigate the two-median problem on a mesh with M columns and N rows (M ≥ N ), under the Manhattan (L1) metric. We derive exact algorithms with respect to m, n, and r , the number of columns, rows, and vertices, respectively, that contain requests. Specifically, we give an O(mn2 logm) time, O(r ) space algorithm for general (nonuniform) meshes (assuming m≥n). For uniform meshes, we give two algorithms both using O(MN ) space. One is anO(MN2) time algorithm,while the other is an algorithm running in O(MN logN ) time with high probability and in O(MN2) time in the worst case assuming the weights are independent and identically distributed random variables satisfying certain natural conditions. These improve upon the previously bestknown algorithm that runs in O(mn2r ) time. © 2007 Wiley Periodicals, Inc. NETWORKS, Vol. 49(3), 226–233 2007