[Paper Review] An Improved Algorithm for Incremental DFS Tree in Undirected Graphs

Lijie Chen, Ran Duan+3 more|arXiv (Cornell University)|Jul 17, 2016

Algorithms and Data Compression3 citations

TL;DR

This paper presents an improved algorithm for maintaining a depth-first search (DFS) tree in undirected graphs under incremental updates—adding vertices or edges one at a time. By combining a novel tree-partition lemma with fractional cascading and optimizing data structures, the algorithm achieves an optimal O(n) worst-case update time per operation, matching the theoretical lower bound when explicitly outputting the DFS tree.

ABSTRACT

Depth first search (DFS) tree is one of the most well-known data structures for designing efficient graph algorithms. Given an undirected graph $G=(V,E)$ with $n$ vertices and $m$ edges, the textbook algorithm takes $O(n+m)$ time to construct a DFS tree. In this paper, we study the problem of maintaining a DFS tree when the graph is undergoing incremental updates. Formally, we show: Given an arbitrary online sequence of edge or vertex insertions, there is an algorithm that reports a DFS tree in $O(n)$ worst case time per operation, and requires $O\left(\min\{m \log n, n^2\} ight)$ preprocessing time. Our result improves the previous $O(n \log^3 n)$ worst case update time algorithm by Baswana et al. and the $O(n \log n)$ time by Nakamura and Sadakane, and matches the trivial $Ω(n)$ lower bound when it is required to explicitly output a DFS tree. Our result builds on the framework introduced in the breakthrough work by Baswana et al., together with a novel use of a tree-partition lemma by Duan and Zhan, and the celebrated fractional cascading technique by Chazelle and Guibas.

Motivation & Objective

• To address the challenge of efficiently maintaining a DFS tree in undirected graphs under incremental updates (vertex and edge insertions).
• To close the gap between the best-known previous update time of O(n log³n) and the theoretical Ω(n) lower bound for explicit DFS tree output.
• To design a dynamic data structure that supports fast, worst-case updates while maintaining linear preprocessing time and space.
• To improve upon prior frameworks, particularly Baswana et al.'s approach, by integrating advanced data structures and combinatorial lemmas.

Proposed method

• Leverages a tree-partition lemma by Duan and Zhang to decompose the DFS tree into subtrees with controlled size and structure.
• Employs fractional cascading to accelerate queries over marked vertices along paths, enabling efficient ancestor-descendant relationship lookups.
• Uses a batched incremental algorithm (BatchInsert) to process all updates at once, then de-amortizes it to achieve worst-case O(n) per-update performance.
• Maintains multiple data structures: a 2D-range successor structure for large subtrees and precomputed entries for small subtrees to support fast queries.
• Introduces a marking scheme (M) on vertices to partition paths and reduce query complexity, ensuring logarithmic path lengths in key cases.
• Applies a de-amortization technique from Baswana et al. to convert batched updates into online, worst-case efficient operations.

Experimental results

Research questions

• RQ1 Can we achieve an O(n) worst-case update time for maintaining a DFS tree in undirected graphs under incremental updates?
• RQ2 How can we improve upon the O(n log³n) update time of Baswana et al. and the O(n log n) of Nakamura and Sadakane?
• RQ3 What combinatorial and data structure techniques can be combined to minimize the cost of path traversal and subtree queries during incremental updates?
• RQ4 Is it possible to match the Ω(n) lower bound for explicit DFS tree output while maintaining linear preprocessing time?
• RQ5 How can fractional cascading and tree partitioning be integrated to accelerate dynamic DFS tree queries?

Key findings

• The proposed algorithm maintains a DFS tree in O(n) worst-case time per incremental update, matching the theoretical lower bound for explicit output.
• The algorithm requires O(min{m log n, n²}) preprocessing time and space, which is nearly linear in the size of the graph.
• The use of a tree-partition lemma ensures that only O(n / log n) subtrees require expensive 2D-range successor queries, limiting their number and improving efficiency.
• Fractional cascading enables O(1) or O(log n) time per query for subtree queries, depending on size, with total time bounded by O(n) across all queries.
• The de-amortization technique successfully transforms a batched algorithm into an online dynamic algorithm with worst-case guarantees.
• The algorithm improves upon the previous best-known update time of O(n log³n) by Baswana et al. and O(n log n) by Nakamura and Sadakane, achieving optimal O(n) performance.