On the Power of Local Orientations

We consider a network represented by a simple connected undirected graph with N vertices. The nodes of the graph do not have any special unique identifiers, the only requirement is that each incident edge of a vertex v of degree dv has assigned a locally-unique label (portname) between 1 and dv, inclusive. This is called a local orientation. Our main goal is to have an agent visit all vertices of the graph, with the focus on the simplicity of the agent in terms of memory usage. We show that after a suitable preprocessing phase in which we alter the local orientations, the graph can be traversed using the simplest possible agent – an right-hand-rule using agent (RH-agent). Formally, this is an extremely simple finite automaton that always obeys the following rule: “Start by taking the edge with the label 1. Then, whenever you enter a node, continue by taking the successor edge (in local orientation) to the edge you arrived through.”. For the above mentioned pre-processing phase we design an algorithm for an agent that performs the precomputation. The agent will be able to alter the network by modifying the local orientations using a simple operation of exchanging two local labels in one step. The goal of this precomputation is to change the local orientations in such a way that the RH-agent’s algorithm will cause the RH-agent to visit all the nodes. We show a polynomialtime algorithm for this precomputation that needs only one pebble and O(logN) memory in the agent. Furthermore, we also show a modification of this algorithm that needs one pebble and only constant memory for the precomputing agent. In the second case, the termination detection is not solved yet, so one extra bit of information in the initial vertex is needed for terminating the precomputation. As an example of the use of our algorithm, we introduce the problem of building a spanning tree of the graph. More precisely, we present an algorithm for an agent that will construct a rooted spanning tree. This spanning tree will be constructed by altering the local orientations: in any non-root vertex the edge with local label 1 will lead to the parent of this vertex in the spanning tree. Our algorithm only needs to use one pebble and O(log N) memory in the agent that performs the computation.