[+]Approximate distance oracles for planar graphs. 
We discuss data structures that answer approximate nodetonode shortestpath and distance queries in planar graphs. Such data structures are also known as approximate distance oracles. Analogously to exact distance oracles, approximate distance oracles consist of two algorithms: (1) given a graph G and an approximation parameter ε, a preprocessing algorithm constructs a data structure and (2) given the data structure and a pair of nodes (v,w), a query algorithm computes and outputs an estimate for the distance d(v,w) with respect to G. The estimate is supposed to be at most (1+ε) times larger than the actual shortestpath distance (and no smaller). We shall see that, for any planar graph and for any ε>0, there is a (1+ε)approximate distance oracle with preprocessing time O((n/ε) log^{2} n) and space O((n/ε) log n) and query time O((1/ε) log n). To obtain these data structures, we recursively separate the graph by shortest paths. On these path separators, we carefully select a set of portals, through which we approximate shortest paths that intersect with the separator path. 
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