Computing the Center of Uncertain Points on Cactus Graphs

In this paper, we consider the (weighted) one-center problem of uncertain points on a cactus graph. Given are a cactus graph $G$ and a set of $n$ uncertain points. Each uncertain point has $m$ possible locations on $G$ with probabilities and a non-negative weight. The (weighted) one-center problem aims to compute a point (the center) $x^{\ast }$ on $G$ to minimize the maximum (weighted) expected distance from $x^{\ast }$ to all uncertain points. No previous algorithm is known for this problem. In this paper, we propose an $O(\mathrm{\mid }G\mathrm{\mid }+mn\log mn)$-time algorithm for solving it. Since the input is $O(\mathrm{\mid }G\mathrm{\mid }+mn)$, our algorithm is almost optimal.