EXPLORING PLANAR GRAPHS USING UNORIENTED MAPS

Abstract

A mobile agent (robot) has to explore an unknown terrain modeled as a planar embedding of an undirected planar connected graph. Exploration consists in visiting all nodes and traversing all edges of the graph, and should be completed using as few edge traversals as possible. The agent has an unlabeled map of the terrain which is another planar embedding of the same graph, preserving the clockwise order of neighbors at each node. The starting node of the agent is marked in the map but the map is unoriented: the agent does not know which direction in the map corresponds to which direction in the terrain. The quality of an exploration algorithm [Formula: see text] is measured by comparing its cost (number of edge traversals) to that of the optimal algorithm having full knowledge of the graph. The ratio between these costs, for a given input consisting of a graph and a starting node, is called the overhead of algorithm [Formula: see text] for this input. We seek exploration algorithms with small overhead. We show an exploration algorithm with overhead of at most 7/5 for all trees, which is the best possible overhead for some trees. We also show an exploration algorithm with the best possible overhead, for any tree with starting node of degree 2. For a large class of planar graphs, called stars of graphs, we show an exploration algorithm with overhead of at most 3/2. Finally, we show a lower bound 5/3 on the overhead of exploration algorithms for some planar graphs.