I/O-Efficient Planar Separators

We present I/O-efficient algorithms for computing optimal separator partitions of planar graphs. Our main result shows that, given a planar graph G with N vertices and an integer r > 0, a vertex separator of size O(N/ √ r) that partitions G into O(N/r) subgraphs of size at most r and boundary size O( √ r) can be computed in O(sort(N)) I/Os. This bound holds provided that M ≥ 56r log B. Together with an I/O-efficient planar embedding algorithm presented in [33], this result is the basis for I/O-efficient solutions to many other fundamental problems on planar graphs, including breadth-first search and shortest paths [5,8], depth-first search [6,9], strong connectivity [9], and topological sorting [7, 8]. Our second result shows that, given I/O-efficient solutions to these problems, a general separator algorithm for graphs with costs and weights on their vertices [3] can be made I/O-efficient. Many classical separator theorems are special cases of this result. In particular, our I/O-efficient version allows the computation of a separator as produced by our first separator algorithm, but without placing any constraints on r in relation to the memory size.