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 [16]: Given a set of N nodes (faulty processors or sources) in an mxn rectangular grid where nz, n less than or equal to 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 [5] 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(root mnN) solve the problem in O(d root 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 ii the disjoint paths have to be ended at another set of N nodes (sinks) in the grid instead of the grid boundary.