Simpler and Improved Replacement Path Coverings

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 $\widetilde{O}(fL^f)$ and query time $\widetilde{O}(f^2 L^f)$. This was derandomized by Karthik and Parter [TALG'24], who also reduced the query time to $\widetilde{O}(f^2 L)$. Their approach uses some heavy algebraic machinery involving error-correcting codes and an increased covering value of $O((cfL \log n)^{f+1})$ for some constant $c > 1$. We instead devise a much simpler derandomization via conditional expectations that lowers the covering value back to $\widetilde{O}(fL^{f+o(1)})$ and decreases the query time to $\widetilde{O}(f^{5/2}L^{o(1)})$, 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 $\widetilde{O}( (L/f)^f L^{o(1)})$ subgraphs and show that this is tight up to the $L^{o(1)}$ term.