Efficient algorithms for finding disjoint paths in grids

The reconfiguration problem on VLSI/WSI processor arrays in the presence of faulty processors can be stated as the following integral multi-source routing problem: Given a set of N nodes (faulty processors or sources) in an m×n rectangular grid where m, n≤N, the problem to be solved is to connect the N nodes to distinct nodes at the grid boundary using a set of `disjoint' paths. This problem can be referred to as an escape problem which can be solved trivially in O(mnN) time. 

By exploiting all the properties of the network, planarity and regularity of a grid, integral flow, and unit capacity source/sink/flow, we can optimally compress the size of the grid from O(mn) to O(√mnN) and solve the problem in O(d√mnN), where d is the maximum number of disjoint paths found, for both the edge-disjoint and vertex-disjoint cases. In the worst case, d, m, n are O(N) and the result is O(N^2.5). Note that this routing problem can also be solved with the same time complexity even if the disjoint paths have to be ended at another set of N nodes (sinks) in the grid instead of the grid boundary.