New Algorithms for Incremental Minimum Spanning Trees and Temporal Graph Applications

2025-04-06

AI Summary

This work addresses the longstanding gap between poor practical performance of theoretically optimal algorithms and the lack of theoretical guarantees in heuristic solutions for incremental minimum spanning tree (MST) computation on temporal graphs. We propose the first solution that simultaneously achieves optimal asymptotic complexity and high practical efficiency. Our approach introduces AM-tree, a novel dynamic tree data structure that unifies the link-cut tree theoretical framework, principles of incremental graph algorithms, and system-level engineering optimizations. AM-tree maintains O(log n) amortized time complexity for both edge updates and path queries while delivering substantial runtime improvements. Experiments show that AM-tree accelerates edge updates by 7.8–11× and path queries by 7.7–13.7× over standard link-cut trees, and consistently outperforms all existing heuristic methods. To our knowledge, this is the first work to bridge the theory–practice divide in incremental MST maintenance on temporal graphs.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Data Mining & Knowledge Management: Mining of Spatial, Temporal or Spatio-Temporal Data
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications

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. However, to give efficient solutions for incremental MST, we observe a gap between theory and practice. 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 guarantee, and can be significantly faster than the link-cut tree (7.8-11x in update and 7.7-13.7x in query).

Problem

• Develops efficient algorithms for incremental Minimum Spanning Trees (MST)
• Bridges theory-practice gap in temporal graph processing solutions
• Introduces AM-tree for improved performance and theoretical guarantees

Innovation

• New algorithms for Incremental MST
• AM-tree data structure with theoretical guarantees
• Efficient temporal graph processing in practice