Distance problems within Helly graphs and k-Helly graphs

The ball hypergraph of a graph G is the family balls all possible centers and radii in G. Balls subfamily are k-wise intersecting if intersection any k always nonempty. Helly number least integer greater than one such that every has nonempty common intersection. A k-Helly (or Helly, k=2) its at most k. We prove central vertex medians an n-vertex m-edge can be computed w.h.p. O˜(mn) time. Both results extend to broader setting where we assign nonnegative cost vertices. For fixed k, also present O˜(mkn)-time randomized algorithm for radius computation within graphs. If relax definition (for what sometimes called “almost Helly-type” property literature), then our approach leads approximation computing with additive one-sided error some constant.