Parametric Shortest Paths in Planar Graphs
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We construct a family of planar graphs (G_n: n≥ 1), 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 increases, the cost of the shortest path from s to t has n^Ω( n) break points. This shows that lower bounds obtained earlier by Carstensen (1983) and Mulmuley & Shah (2000) for general graphs also hold for planar graphs. A conjecture of Nikolova (2009) states that the number of break points in n-vertex planar graphs is bounded by a polynomial in n; our result refutes this conjecture. Gusfield (1980) and Dean (2009) showed that the number of break points for an n-vertex graph is n^ n + O(1) assuming linear edge weights; we show that if the edge weights are allowed to vary as a polynomial of degree at most d, then the number of break points is n^ n + ψ(n)^d, where ψ(n) ≪^* n is a slowly growing function arising from Davenport-Schinzel sequences.
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