🤖 AI Summary
This work addresses real-time network connectivity planning scenarios where tree construction and execution must interleave dynamically, overcoming the high preprocessing overhead and response latency inherent in conventional two-phase MST algorithms. We propose an *edge-enumeration* approach to MST construction, introducing the first unified theoretical framework for efficient edge enumeration across three graph classes: undirected unweighted, directed, and weighted graphs. We derive tight delay bounds for enumeration and prove that meaningful edge enumeration is impossible for weighted directed graphs. Our method employs traversal-based, priority-queue-driven algorithms explicitly optimized for enumeration delay, incorporating adversarial lower-bound analysis and degree-aware delay modeling. On undirected unweighted graphs, it achieves average-case output delay proportional to the graph’s average degree. Experiments demonstrate substantially lower preprocessing cost and significantly reduced first-edge response latency compared to classical MST algorithms, enabling low-latency, real-time connectivity planning in dynamic networks.
📝 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.