Enumeration of Spanning Trees Using Edge Exchange with Minimal Partitioning

TL;DR

This thesis introduces the Minimal Partitioning (MP) algorithm for efficiently enumerating all spanning trees in a graph, maintaining minimal partition size and utilizing edge promotion to improve performance and applicability.

Contribution

The paper presents a novel MP algorithm that preserves spanning tree structure during enumeration, with unique properties like minimal partition size and edge promotion, outperforming previous methods.

Findings

MP algorithm requires $O(log V+E/V)$ expected time per spanning tree.

It maintains a total expected space of $O(E log V)$.

Outperforms existing algorithms by $O(V)$ time complexity.

Abstract

In this thesis, Minimal Partitioning (MP) algorithm, an innovative algorithm for enumerating all the spanning trees in an undirected graph is presented. While MP algorithm uses a computational tree graph to traverse all possible spanning trees by the edge exchange technique, it has two unique properties compared to previous algorithms. In the first place, the algorithm maintains a state of minimal partition size in the spanning tree due to edge deletion. This is realized by swapping peripheral edges, more precisely leaf edges, in most of edge exchange operations. Consequently, the main structure of the spanning trees is preserved during the steps of the enumeration process. This extra constraint proves to be advantageous in many applications where the partition size is a factor in the solution cost. Secondly, we introduce, and utilize, the new concept of edge promotion: the exchanged edges always share one end. Practically, and as a result of this property, the interface between the two partitions of the spanning tree during edge exchange has to be maintained from one side only. For a graph $G(V,E)$, MP algorithm requires $O(log V+E/V)$ expected time and $OV log V)$ worst case time for generating each spanning tree. MP algorithm requires a total expected space limit of $O(E log V)$ with worst case limit of $O(EV)$. Like all edge exchange algorithms, MP algorithm retains the advantage of compacted output of $O(1)$ per spanning tree by listing the relative differences only. Three sample real-world applications of spanning trees enumeration are explored and the effects of using MP algorithm are studied. Namely: construction of nets of polyhedra, multi-robots spanning tree routing, and computing the electric current in edges of a network. We report that MP algorithm outperforms other algorithm by $O(V)$ time complexity.