Journal of the Korea Institute of Information and Communication Engineering (한국정보통신학회논문지)

The Korea Institute of Information and Commucation Engineering (한국정보통신학회)
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Minimizing the Diameter by Augmenting an Edge to a Path in a Metric Space

거리공간속 경로 그래프에 간선추가를 통한 지름의 최소화

  • Kim, Jae-Hoon (Department of Computer Engineering, Busan University of Foreign Studies)
  • Received : 2021.10.14
  • Accepted : 2021.12.08
  • Published : 2022.01.31
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Abstract

This paper deals with the graph in which the weights of edges are given the distances between two end vertices on a metric space. In particular, we will study about a path P with n vertices for these graphs. We obtain a new graph $\bar{P}$ by augmenting an edge to P. Then the length of the shortest path between two vertices on $\bar{P}$ is considered and we focus on the maximum of these lengths. This maximum is called the diameter of the graph $\bar{P}$. We wish to find the augmented edge to minimize the diameter of $\bar{P}$. Especially, for an arbitrary real number λ > 0, we should determine whether the diameter of $\bar{P}$ is less than or equal to λ and we propose an O(n)-time algorithm for this problem, which improves on the time complexity O(nlogn) previously known. Using this decision algorithm, for the length D of P, we provide an O(nlogD)-time algorithm to find the minimum of the diameter of $\bar{P}$.

본 논문은 거리 공간(metric space) 속에 포함된 그래프에서 각 간선의 가중치가 거리 공간 상의 두 끝 정점간의 거리로 주어지는 그래프를 다룬다. 특별히 우리는 이러한 그래프 중 n개 정점을 가진 경로 P에 관해서 연구한다. 우리는 경로 P에 하나의 간선을 추가해서 새로운 그래프 $\bar{P}$ 얻을 수 있다. 그러면 그래프 $\bar{P}$의 두 정점 사이의 최단 경로의 길이를 생각하고 이 길이들 중 최댓값에 주목한다. 이 최댓값을 그래프 $\bar{P}$의 지름(diameter)라고 부른다. 우리는 그래프 $\bar{P}$의 지름이 최소가 되도록 추가하는 간선을 찾고 싶다. 특별히 임의의 실수 λ > 0에 대해서, $\bar{P}$의 지름이 λ 이하가 되는 추가 간선이 존재하는지 여부를 결정하는 문제에 대해 O(n)시간 알고리즘을 제안한다. 이것은 이전 알려진 시간복잡도 O(nlogn)을 개선한다. 이 결정 알고리즘을 이용해서 주어진 경로 P의 길이 D에 대해서, $\bar{P}$의 지름의 최솟값을 찾는 O(nlogD) 시간 알고리즘을 제안한다

Keywords

Acknowledgement

This work was supported by the research grant of the Busan University of Foreign Studies in 2021

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