Connectivity Oracles for Planar Graphs

We consider dynamic subgraph connectivity problems for planar graphs. In this model there is a fixed underlying planar graph, where each edge and vertex is either "off" (failed) or "on" (recovered). We wish to answer connectivity queries with respect to the "on" subgraph. The model has two natural variants, one in which there are $d$ edge/vertex failures that precede all connectivity queries, and one in which failures/recoveries and queries are intermixed. We present a $d$-failure connectivity oracle for planar graphs that processes any $d$ edge/vertex failures in $sort(d,n)$ time so that connectivity queries can be answered in $pred(d,n)$ time. (Here $sort$ and $pred$ are the time for integer sorting and integer predecessor search over a subset of $[n]$ of size $d$.) Our algorithm has two discrete parts. The first is an algorithm tailored to triconnected planar graphs. It makes use of Barnette's theorem, which states that every triconnected planar graph contains a degree-3 spanning tree. The second part is a generic reduction from general (planar) graphs to triconnected (planar) graphs. Our algorithm is, moreover, provably optimal. An implication of Patrascu and Thorup's lower bound on predecessor search is that no $d$-failure connectivity oracle (even on trees) can beat $pred(d,n)$ query time. We extend our algorithms to the subgraph connectivity model where edge/vertex failures (but no recoveries) are intermixed with connectivity queries. In triconnected planar graphs each failure and query is handled in $O(\log n)$ time (amortized), whereas in general planar graphs both bounds become $O(\log^2 n)$.