Emit As You Go: Enumerating Edges of a Spanning Tree

TL;DR

This paper introduces an efficient enumeration algorithm for parts of a minimum spanning tree in various graph types, enabling interleaved planning and execution with minimal delay, supported by theoretical bounds and practical experiments.

Contribution

It presents the first efficient enumeration algorithm for MST edges in undirected and directed graphs with weights, with proven bounds and practical validation.

Findings

Efficient enumeration possible for undirected unweighted graphs.

Enumeration delay depends on graph degree properties.

No meaningful enumeration for weighted directed graphs.

Abstract

Classically, planning tasks are studied as a two-step process: plan creation and plan execution. In situations where plan creation is slow (for example, due to expensive information access or complex constraints), a natural speed-up tactic is interleaving planning and execution. We implement such an approach with an enumeration algorithm that, after little preprocessing time, outputs parts of a plan one by one with little delay in-between consecutive outputs. As concrete planning task, we consider efficient connectivity in a network formalized as the minimum spanning tree problem in all four standard variants: (un)weighted (un)directed graphs. Solution parts to be emitted one by one for this concrete task are the individual edges that form the final tree. We show with algorithmic upper bounds and matching unconditional adversary lower bounds that efficient enumeration is possible for three of four problem variants; specifically for undirected unweighted graphs (delay in the order of the average degree), as well as graphs with either weights (delay in the order of the maximum degree and the average runtime per emitted edge of a total-time algorithm) or directions (delay in the order of the maximum degree). For graphs with both weighted and directed edges, we show that no meaningful enumeration is possible. Finally, with experiments on random undirected unweighted graphs, we show that the theoretical advantage of little preprocessing and delay carries over to practice.