Fast algorithms for finding O(congestion+dilation) packet routing schedules

Leighton, Maggs and Rao (1988) showed that for any network and any set of packets whose paths through the network are fixed and edge-simple, there exists a schedule for routing the packets to their destinations in O(c+d) steps using constant-size queues, where c is the congestion of the paths in the network, and d is the length of the longest path (the dilation). The proof, however, used the Lova´sz (1975) local lemma and was not constructive. In this paper, we show how to find such a schedule in O(NE+Elog^εE) time, for any fixed ε>0, where N is the total number of packets, and E is the number of edges in the network. We also show how to parallelize the algorithm so that it runs in NC. The method that we use to construct efficient packet routing schedules is based on the algorithmic form of the Lova´sz local lemma discovered by Beck (1991)