An on-line algorithm for improving performance in navigation
  Recent papers have shown optimally-competitive on-line strategies for a robot traveling from a point $s$ to a point $t$ in certain unknown geometric environments. We consider the question: Having gained some partial information about the scene on its first trip from $s$ to $t$, can the robot improve its performance on subsequent trips it might make? This is a type of on-line problem where a strategy must exploit {\em partial information \/} about the future (e.g., about obstacles that lie ahead). For scenes with axis-parallel rectangular obstacles where the Euclidean distance between $s$ and $t$ is $n$, we present a deterministic algorithm whose {\em average\/} trip length after $k$ trips, $k \leq n$, is $O(\rootnbyk)$ times the length of the shortest $s$-$t$ path in the scene. We also show that this is the best a deterministic strategy can do. This algorithm can be thought of as performing an optimal tradeoff between search effort and the goodness of the path found. We improve this algorithm so that for {\em every\/} $i \leq n$, the robot's $i$th trip length is $O(\rootnbyi)$ times the shortest $s$-$t$ path length. A key idea of the paper is that a {\em tree\/} structure can be defined in the scene, where the nodes are portions of certain obstacles and the edges are ``short'' paths from a node to its children. The core of our algorithms is an on-line strategy for traversing this tree optimally. 
Avrim Blum, Prasad Chalasani