| Summary: | The median path problem (min-sum criterion) is a common problem in graph theory and tree networks. This problem is open to study because its applications are growing and extending in different fields, such as providing insight for decision-makers when selecting the optimal location for non-emergency services, including railroad lines, highways, pipelines, and transit routes. Also, the min-sum criterion can deal with several networks in different applications. The location problem has traditionally been concerned with the optimal location of a single-point facility at either a vertex or along an edge in a network. Recently, numerous investigators have investigated this classic problem and have studied the location of many facilities, such as paths, trees, and cycles. The concept of the median, which measures the centrality of a vertex in a graph, is extended to the paths in a graph. In this paper, we consider the problem of locating path-shaped facilities on a tree network. A new modified and improved algorithm for finding a median single path facility of a specified length in a tree network is proposed. The median criterion for optimality considers the sum of the distances from all vertices of the tree to the path facility. This problem under the median criterion is called the <i>ℓ</i>-core problem. The distance between any two vertices in the tree is equal to the length of the unique path connecting them. This location problem usually has applications in distributed database systems, pipelines, the design of public transportation routes, and communication networks.
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