Emit As You Go: Enumerating Edges of a Spanning Tree
Katrin Casel, Stefan Neubert
TL;DR
The paper studies efficient enumeration of edges in a spanning tree to enable interleaved planning and execution, focusing on four MST variants (undirected/unweighted, undirected/weighted, directed/unweighted, directed/weighted). It develops an enumeration framework with lazy initialization and banker's amortization to bound preprocessing time $p$ and per-edge delay $d$, achieving tight upper and lower bounds for three variants and showing no meaningful enumeration for the weighted directed case. Specifically, it proves delays of $O(\overline{\Delta})$ for undirected unweighted graphs, $O(\Delta)$ (with total-time considerations) for undirected weighted graphs, and $O(\Delta^+)$ for directed unweighted graphs with a given root, while the directed weighted case requires $\Omega(m)$ preprocessing or is infeasible. Experiments on random undirected unweighted graphs corroborate the theoretical benefits, demonstrating practical speedups in first-output delay and overall runtime with modest preprocessing overhead.
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.