A Note on Dynamic Bidirected Dyck-Reachability with Cycles

TL;DR

The paper analyzes dynamic bidirected Dyck-reachability in graphs with cycles and identifies a deficiency in LiSZ22's deletion procedure when the merged graph $G_m$ contains cycles. It proposes a cycle-aware extension: on edge deletions, DFS from the representative node to collect cycle nodes $N_c$, split them, and re-run the optimal static Dyck-reachability algorithm (Opt-Dyck'ChatterjeeCP18LiSZ22) on the updated graph. The authors show that this yields a correct dynamic algorithm for cyclic merged graphs with worst-case $O(n^2)$ time, and discuss trade-offs and potential improvements, such as maintaining a dynamic topological order and leveraging recent Dyck SCC techniques. These results contribute to practical dynamic analysis of Dyck-reachability in bidirected graphs with cycles, with implications for scalable maintenance of Dyck-SCCs and related reachability queries.

Abstract

Recently, Li et al. [2022] presented a dynamic Dyck-reachability algorithm for bidirected graphs. The basic idea is based on updating edge weights in a data structure called the merged graph $G_m$. As noted in Krishna et al. [2023], the edge deletion procedure described in the algorithm of Li et al. [2022] cannot properly update the weights in the presence of cycles in $G_m$. This note discusses the cycle case and the time complexity.

Figures (3)

• Figure 1: Bidirected Dyck-reachability.
• Figure 2: Bidirected Dyck-reachability with cycles.
• Figure 3: Handling bidirected Dyck-reachability with cycles.