New Algorithms for Incremental Minimum Spanning Trees and Temporal Graph Applications

TL;DR

The paper presents AM-trees, a transformed MST data structure with an anti-monopoly rule that enforces a logarithmic height to enable fast path-max queries and edge insertions. It outlines strict and lazy variants, plus persistence, and demonstrates that AM-trees match the theoretical efficiency of classic structures like the link-cut tree while delivering superior practical performance in temporal-graph workloads. By mapping temporal connectivity tasks (e.g., point-interval, interval-point, bipartiteness, and k-connectivity) to incremental MST problems, the authors provide a unified framework with broad applicability and strong empirical results. The work offers substantial implications for real-time temporal graph processing, enabling scalable online and historical queries with rigorous guarantees and implementable designs.

Abstract

Processing graphs with temporal information (the temporal graphs) has become increasingly important in the real world. In this paper, we study efficient solutions to temporal graph applications using new algorithms for Incremental Minimum Spanning Trees (MST). The first contribution of this work is to formally discuss how a broad set of setting-problem combinations of temporal graph processing can be solved using incremental MST, along with their theoretical guarantees. Despite the importance of the problem, we observe a gap between theory and practice for efficient incremental MST algorithms. While many classic data structures, such as the link-cut tree, provide strong bounds for incremental MST, their performance is limited in practice. Meanwhile, existing practical solutions used in applications do not have any non-trivial theoretical guarantees. Our second and main contribution includes new algorithms for incremental MST that are efficient both in theory and in practice. Our new data structure, the AM-tree, achieves the same theoretical bound as the link-cut tree for temporal graph processing and shows strong performance in practice. In our experiments, the AM-tree has competitive or better performance than existing practical solutions due to theoretical guarantees, and can be significantly faster than the link-cut tree (7.8-11x in updates and 7.7-13.7x in queries).

Figures (7)

• Figure 1: Outline and contributions of this paper.
• Figure 2: Solving point-interval temporal connectivity on graph $G^*$ using incremental MST crouch2013dynamic.
• Figure 3: An example of the transformed MST (T-MST). A T-MST redistribute the edges in an MST, but preserves the answers to path-max queries in the MST.
• Figure 4: An illustration of the Promote algorithm.
• Figure 5: An example of linkBystitch. The figure illusrates linkBystitch$(u,v,8)$. Values on the edges are edge weights. The figure shows all vertices on the path from $u$ and $v$ to the root, and omits all other vertices. An explanation about the process is shown in the figure, and the pseudocode is presented in Alg. \ref{['algo:preemptive-cdt']} Lines \ref{['line:stitch:start']}-\ref{['line:stitch:end']}.

Theorems & Definitions (18)

• Definition 1: Path-Max
• Definition 2: transformed MST (T-MST)
• Definition 3: Anti-Monopoly tree (AM-tree)
• Theorem 4.1: Correctness
• Theorem 4.2: Balance Guarantee
• Lemma 4.1
• Theorem 4.3
• Lemma 4.2
• Lemma 4.3
• Lemma 4.4