Discrete Integral and Discrete Derivative on Graphs and Switch Problem of Trees

Abstract

For a vertex and edge weighted (VEW) graph G with a vertex weight function fG let Wα,β(G)=∑{u,v}⊆V(G)[αfG(u)×fG(v)+β(fG(u)+fG(v))]dG(u,v) where, α,β∈ℝ and dG(u,v) denotes the distance, the minimum sum of edge weights across all the paths connecting u,v∈V(G). Assume T is a VEW tree, and e∈ E(T) fails. If we reconnect the two components of T−e with new edge ϵ≠e such that, Wα,β(Tϵ\e=T−e+ϵ) is minimum, then ϵ is called a best switch (BS) of e w.r.t. Wα,β. We define three notions: convexity, discrete derivative, and discrete integral for the VEW graphs. As an application of the notions, we solve some BS problems for positively VEW trees. For example, assume T is an n-vertex VEW tree. Then, for the inputs e∈ E(T) and w,α,β ∈ℝ+, we return ϵ, Tϵ\e, and Wα,β(Tϵ\e) with the worst average time of O(logn) and the best time of O(1) where ϵ is a BS of e w.r.t. Wα,β and the weight of ϵ is w.