מושגי ליבה
We present an O(log^2 w)-competitive randomized algorithm for unweighted layered graph traversal, where w is the maximum width of the graph layers.
תקציר
The paper introduces an efficient randomized algorithm for the problem of unweighted layered graph traversal. In this problem, a mobile agent starts at an arbitrary node (the source) of an unknown weighted graph, and the goal is to reach another arbitrary node (the target). The graph is divided into "layers", where the t-th layer refers to the set of nodes at combinatorial depth t from the source. The agent can only see the next layer once it is located at the current layer, but it has a broad view of all nodes and edges going from the current layer to the next.
The key insights of the paper are:
For the unweighted variant of the problem, where all edge lengths are 1, the competitive ratio can be significantly improved compared to the weighted case.
The algorithm leverages a simple entropic regularizer that evolves as the agent progresses in the layered graph. Specifically, the agent moves in a way that maximizes the entropy of the probability distribution over the current layer.
The analysis of the algorithm is split into "dead-end" phases, where the agent is trapped in a leaf node and must redistribute its probability mass, and "growth" phases, where the agent adapts to the distances between nodes in the new layer.
Several novel techniques are used to bound the movement cost during these phases, including separating the analysis based on the magnitude of the probability mass and deriving relationships between the algorithm's probability mass in sibling subtrees and the number of leaves in these subtrees.
The resulting randomized algorithm achieves an O(log^2 w) competitive ratio, which significantly improves over the previous best known result of O(√w) for the unweighted case.