Navigating Planar Topologies in Near-Optimal Space and Time
José Fuentes-Sepúlveda, Gonzalo Navarro, and Diego Seco
We show that any embedding of a planar graph can be encoded succinctly
while efficiently answering a number of topological queries near-optimally.
More precisely, we build on a succinct representation that encodes an embedding
of m edges within 4m bits, which is close to the
information-theoretic lower bound of about 3.58m. With 4m +
o(m) bits of space, we show how to answer a number of topological
queries relating nodes, edges, and faces, most of them in any time in
ω(1). Indeed, 3.58m + o(m) bits suffice if the graph has
no self-loops and no nodes of degree one. Further, we show that with
O(m) bits of space we can solve all those operations in O(1) time.