Simpler and Improved Replacement Path Coverings

Abstract: An important tool in the design of fault-tolerant graph data structures are $(L,f)$-replacement path coverings (RPCs). An RPC is a family $\mathcal{G}$ of subgraphs of a given graph $G$ such that, for every set $F$ of at most $f$ edges, there is a subfamily $\mathcal{G}_F \,{\subseteq}\, \mathcal{G}$ with the following properties. (1) No subgraph in $\mathcal{G}_F$ contains an edge of $F$. (2) For each pair of vertices $s,t$ that have a shortest path in $G-F$ with at most $L$ edges, one such path also exists in some subgraph in $\mathcal{G}_F$. The covering value of the RPC is the total number $|\mathcal{G}|$ of subgraphs. The query time is the time needed to compute the subfamily $\mathcal{G}_F$ given the set $F$. Weimann and Yuster [TALG'13] devised a randomized RPC with covering value $\widetilde{O}(fL<sup>f)$ and query time $\widetilde{O}(f² L<sup>f)$. This was derandomized by Karthik and Parter [TALG'24], who also reduced the query time to $\widetilde{O}(f² L)$. Their approach uses some heavy algebraic machinery involving error-correcting codes and an increased covering value of $O((cfL \log n)<sup>{f+1})$ for some constant $c &gt; 1$. We instead devise a much simpler derandomization via conditional expectations that lowers the covering value back to $\widetilde{O}(fL<sup>{f+o(1)})$ and decreases the query time to $\widetilde{O}(f<sup>{5/2}L<sup>{o(1)})$, assuming $f = o(\log L)$. We also investigate the optimal covering value of any $(L,f)$-replacement path covering (deterministic or randomized) for different parameter ranges. We provide a new randomized construction as well as improving a known lower bound, also by Karthik and Parter. For example, for $f = o(\log L)$, we give an RPC with $\widetilde{O}( (L/f)^f L<sup>{o(1)})$ subgraphs and show that this is tight up to the $L<sup>{o(1)}$ term.

• The paper introduces a conditional expectation-based derandomization for replacement path coverings that achieves a covering value of ~O(f L^(f+o(1))) and significantly faster queries.
• It employs hierarchical sampling trees to balance edge-failure sensitivity and maintain efficient query performance in dynamic, fault-prone networks.
• The work narrows the gap between upper and lower bounds, offering a simpler alternative to algebraic methods for designing robust fault-tolerant graph data structures.

Improvements in Replacement Path Coverings: Simpler Derandomization and Tight Bounds

Background and Motivation

Fault-tolerant graph data structures are essential in dynamic network environments, where edge failures are both unpredictable and bounded in number. $(L, f)$-replacement path coverings (RPCs) serve as core primitives for constructing $f$-edge fault-tolerant distance sensitivity oracles ($f$-DSOs) and related structures, supporting rapid response to queries involving sets of failed edges. The goal is to balance the covering value (number of subgraphs), query efficiency, and determinism while ensuring correctness for all queries concerning failures of up to $f$ edges and shortest paths limited by $L$ hops.

Previous work established randomized constructions with efficient covering values (e.g., $\widetilde{O}(f L^f)$ in [WY13]) but relatively high query times. Later, deterministic approaches (notably Karthik and Parter [KarthikParter24DeterministicRPC_TALG]) derandomized the construction using algebraic error-correcting codes, sacrificing the covering value for faster queries.

Results: Simpler Derandomization and Covering Value Optimization

This paper introduces a substantially simpler derandomization method based on conditional expectations, bypassing the algebraic complexities of code-based approaches. The new deterministic construction achieves a covering value of $\widetilde{O}(f L^{f + o(1)})$ and a query time of $\widetilde{O}(f^{5/2} L^{o(1)})$, matching or improving prior randomized bounds in the regime $f = o(\log L)$.

Construction Overview

• Hierarchical Sampling Trees: The RPCs are formed by multi-level sampling trees, leveraging branching and depth parameters to control the probability distributions over edge presence. Leaves represent subgraphs; internal structure ensures efficient query navigation.
• Conditional Expectation Derandomization: Each decision in the tree construction maximizes expected coverage for well-separated failure-path pairs, maintaining locality and query structure. The method ensures balanced coverage without introducing cross-tree correlations that would negate probabilistic guarantees.
• Parameter Regime: The construction is particularly effective for $f = o(\log L)$, a common setting for practical fault tolerance in large-scale networks.

Strong Numerical Results and Tight Bounds

• Covering Value: The deterministic approach yields covering value $f$0 for $f$1, reducing to $f$2 in high-sensitivity regimes ($f$3).
• Query Time: Achieves sub-polynomial time in $f$4, specifically $f$5.
• Tighter Randomized Bounds: For the same parameter range, a refined analysis lowers the covering value of sampling trees to $f$6, closing gaps to lower bounds.
• Lower Bounds: The paper improves known lower bounds, showing that for $f$7, no $f$8-RPC can have covering value less than $f$9, and for larger $f$0, at least $f$1 is required.

Implications and Theoretical Significance

The results have several noteworthy implications:

• Derandomization without Algebraic Machinery: The introduction of a conditional expectation-based derandomization demonstrates that heavy algebraic techniques are not necessary for constructing deterministic RPCs, provided small sensitivity is assumed.
• Parameter Matching and Gap Reduction: The improved constructions close the asymptotic gap between upper and lower bounds for the covering value, approaching $f$2 for $f$3 and ensuring near-optimality in relevant regimes.
• Query Efficiency: Significantly faster query times reduce practical deployment costs for large networks.
• Generalizability: The simplicity of the new derandomization framework makes it amenable to extension for other sensitivity data structures beyond shortest-path problems.

Open Questions and Future Directions

Although the deterministic construction matches the randomized approach in key parameters for $f$4, achieving deterministic covering value $f$5 without excess remains open. Improvements in both constructions (upper and lower bounds) are possible by further optimizing the balance in hierarchical sampling and exploring alternative combinatorial arrangements.

The method could be further generalized to sensitivity oracles for $f$6-hard graph problems, connectivity variants, and distributed fault-tolerant network primitives. The practical impact hinges on extending efficient fault-tolerance beyond shortest path and distance queries, potentially yielding sublinear and near-optimal data structures in broader algorithmic contexts.

Conclusion

This paper advances the fundamental theory of replacement path coverings, significantly simplifying deterministic constructions, tightening parameter bounds, and clarifying lower-bound barriers. The conditional expectation derandomization achieves both practical query-time improvements and optimality in covering value for low-sensitivity scenarios. These contributions enhance the toolkit for designing robust, scalable fault-tolerant data structures in graph algorithms, with implications for both distributed and centralized network analysis and sensitivity oracle applications (2604.27966).