Distributed Algorithms for Finding Central Paths in Tree Networks

Given a graph G = (V,E) and a path P , let d(v, P ) be the distance from vertex v ∈ V to path P . A path-center is a path which minimizes the eccentricity e(P ) = maxv∈V d(v, P ) such that for every path P ′ in G, e(P ) ≤ e(P ′), and for every subpath P ′′ ⊂ P , e(P ) < e(P ′′). Similarly, a core is a path which minimizes the distance d(P ) = ∑v∈V d(v, P ) such that for every path P ′ in G, d(P ) ≤ d(P ′), and for every subpath P ′′ ⊂ P , d(P ) < d(P ′′). We present distributed algorithms for finding path-centers and cores in asynchronous tree networks. We then extend these to compute a subset-centrum path in trees which minimizes the distance with respect to a subset of vertices, dS(v, P ) = ∑ v∈S d(v, P ), where S ⊆ V . The time and communication complexities of these algorithms are O(D) and O(n) respectively, where D is the diameter and n is the number of vertices (edges) of the network. These algorithms are asymptotically optimal.