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 G of subgraphs of a given graph G such that, for every set F of at most f edges, there is a subfamily GF ⊆G with the following properties. 1. No subgraph in GF 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 GF. The covering value of the RPC is the total number |G| of subgraphs. The query time is the time needed to compute the subfamily GF given the set F. Weimann and Yuster [TALG'13] devised a randomized RPC with covering value O^e(fL^f) and query time O^e(f²L^f). This was derandomized by Karthik and Parter [TALG'24], who also reduced the query time to Õ(f²L). Their approach uses some heavy algebraic machinery involving error-correcting codes and an increased covering value of O((cfLlog 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 (Formula presented) and decreases the query time to (Formula presented), 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 (Formula presented) subgraphs and show that this is tight up to the L^o(1) term.