Parametric Shortest Paths in Planar Graphs

Abstract

We construct a family of planar graphs {G n } n≥4 , where G n has n vertices including a source vertex s and a sink vertex t, and edge weights that change linearly with a parameter λ such that, as λ varies in (−∞, +∞), the piece-wise linear cost of the shortest path from s to t has n Ω(log n) pieces. This shows that lower bounds obtained earlier by Carstensen (1983) and Mulmuley & Shah (2001) for general graphs also hold for planar graphs, thereby refuting a conjecture of Nikolova (2009).Gusfield (1980) and Dean (2009) showed that the number of pieces for every n-vertex graph with linear edge weights is n log n+O(1) . We generalize this result in two ways. (i) If the edge weights vary as a polynomial of degree at most d, then the number of pieces is n log n+(α(n)+O(1)) d , where α(n) is the slow growing inverse Ackermann function. (ii) If the edge weights are linear forms of three parameters, then the number of pieces, appropriately defined for R 3 , is n (log n) 2 +O(log n) .