Linear complexity algorithms for hierarchical routing

A hierarchical procedure for net routing on l×m ×n grid-graphs, where l is the number of layers, m is the number of rows, and n is the number of columns, is presented. The hierarchy reduces the overall problem to a sequence of subproblems, where each subproblem works on an l×m´×n portion of the overall grid. For each subproblem, an initial constructive placement (CP) of as many nets as possible is used; then a generalized dynamic programming (DP) algorithm to solve the Steiner problem of an l×2×n grid-graph is used for any remaining nets. The CP procedure uses simplistic routing assumptions to route quickly as many nets as possible. If all nets are routed, the subproblem solution is (locally) optimal and the corresponding branch of the binary recursion tree generated by the hierarchy is pruned. For any unrouted nets, the generalized DP procedure is called to route each net, one at a time, with a run time complexity of O(K×n ) where n is the number of columns and K is a function of the grid cross section and layer/wiring restrictions. There are no a priori layer restrictions or limitations on the number of layers used; three-layer cases and even some four-layer cases are feasible for the PYRAMID 90-X