A matrix method for finding last common nodes in an origin-based traffic assignment problem

Many algorithms have been presented to solve the traffic assignment problem. Recently, Bar-Gera introduced the concept of “last common node” into an origin-based algorithm to solve the traffic assignment problem. However, how to find the last common nodes has not been investigated in detail. In this paper, we present a matrix method for finding the last common nodes in an origin-based traffic assignment problem. In an acyclic network, the power of binary adjacency matrix (Ak) will record the number of directed simple routes of length k. Taking this feature into consideration, Sp, the total number of the simple routes related to an origin node p in the subnetwork Gp, is counted by Sp=∑kApk=(I−Ap)−1. Then, every common node for OD pair pq is picked out by comparing (Sp)pr×(Sp)rq and (Sp)pq, and the last common node for OD pair pq is filtered out according to the topological order l(r). Our method is implemented to find out all LCNs for all n∗(n−1) OD pairs, then tested on three kinds of model networks and four urban transportation networks. We find that the overall computing time T and the size of network n, has a relation like T∼O(n3), which is better than the theoretical estimation O(n4).