Election in a Complete Network with a Sense of Direction

Consider a complete network of N asynchronous processors. Every pair of processors is joined by a bidirectional link. Each processor has a unique integer identifier ( id) . The Election Problem is to identify the processor with the largest id. Fix a Hamiltonian cycle that includes all the processors. The network has a sense of direction [6] if at each processor the label on each link gives the distance along this Hamil tonian cycle to the processor at the other end of the link. In particular, if processor x is at distance d from processor y, then y is at distance N d from x. We present an algorithm that uses O(N) messages to solve the Election Problem in a complete network with a sense of direction. In contrast, if at each processor the links are unlabeled, then the network has no sense of d i rect ion. In this latter case, f~(N log N) messages are required to solve the Election Problem [2], and O(N log N) messages suffice [1,2,4].