An undirected graph \(G=(V,A)\) by a set V of n nodes, a set A of m edges, and a set \(D\subseteq V\) consists of h demand nodes are given. Peeters (Eur J Oper Res 104:299–309, 1998) presented the absolute 1-center problem, which finds a point x placed on nodes or edges of the graph G with the property that the cost distance from the most expensive demand node to x is as cheap as possible. In the absolute 1-center problem, the distance between two nodes is computed through a shortest path between them. This paper expands the idea of Peeters (1998) and presents a new version of the absolute 1-center problem, which is called the absolute quickest 1-center problem. A value \(\sigma \) is given, and the problem finds a point \(x^*\) placed on nodes or edges of the graph G with the property that the transmission time of the quickest path to send \(\sigma \) units of data from the farthest demand node to \(x^*\) is the minimum value. We presented an \(O(r|D|(m+n\mathrm{{log}}\ n))\) time algorithm to solve the absolute quickest 1-center problem, where r is the number of distinct capacity values.

给出了一个无向图\（G =（V，A）\），该图由n个节点的集合V，m个边的集合A和由h个需求节点组成的集合\（D \ subseteq V \）组成。Peeters（Eur J Oper Res 104：299–309，1998）提出了绝对的1中心问题，该问题找到位于图G的节点或边缘上的点x，其属性为从最昂贵的需求节点到成本点的距离X尽可能便宜。在绝对1中心问题中，通过两个节点之间的最短路径来计算两个节点之间的距离。本文扩展了Peeters（1998）的思想，并提出了绝对一中心问题的新版本，称为绝对最快一中心问题。给定值\（\ sigma \），问题发现位于图G的节点或边缘上的点\（x ^ * \）具有以下特性：最快路径的发送时间为\（\ sigma \）从最远需求节点到\（x ^ * \）的数据单位为最小值。我们提出了一个\（O（r | D |（m + n \ mathrm {{log}} \ n））\）时间算法来解决绝对最快的1中心问题，其中r是不同容量值的数量。