Paths and Intersections: Characterization of Quasi-metrics in Directed Okamura-Seymour Instances
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916454052397056 |
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| author | Chen, Yu Tan, Zihan |
| author_facet | Chen, Yu Tan, Zihan |
| contents | We study the following distance realization problem. Given a quasi-metric $D$ on a set $T$ of terminals, does there exist a directed Okamura-Seymour graph that realizes $D$ as the (directed) shortest-path distance metric on $T$? We show that, if we are further given the circular ordering of terminals lying on the boundary, then Monge property is a sufficient and necessary condition. This generalizes previous results for undirected Okamura-Seymour instances.
With the circular ordering, we give a greedy algorithm for constructing a directed Okamura-Seymour instance that realizes the input quasi-metric. The algorithm takes the dual perspective concerning flows and routings, and is based on a new way of analyzing graph structures, by viewing graphs as \emph{paths and their intersections}. We believe this new understanding is of independent interest and will prove useful in other problems in graph theory and graph algorithms.
We also design an efficient algorithm for finding such a circular ordering that makes $D$ satisfy Monge property, if one exists. Combined with our result above, this gives an efficient algorithm for the distance realization problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_19246 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Paths and Intersections: Characterization of Quasi-metrics in Directed Okamura-Seymour Instances Chen, Yu Tan, Zihan Data Structures and Algorithms Discrete Mathematics Combinatorics We study the following distance realization problem. Given a quasi-metric $D$ on a set $T$ of terminals, does there exist a directed Okamura-Seymour graph that realizes $D$ as the (directed) shortest-path distance metric on $T$? We show that, if we are further given the circular ordering of terminals lying on the boundary, then Monge property is a sufficient and necessary condition. This generalizes previous results for undirected Okamura-Seymour instances. With the circular ordering, we give a greedy algorithm for constructing a directed Okamura-Seymour instance that realizes the input quasi-metric. The algorithm takes the dual perspective concerning flows and routings, and is based on a new way of analyzing graph structures, by viewing graphs as \emph{paths and their intersections}. We believe this new understanding is of independent interest and will prove useful in other problems in graph theory and graph algorithms. We also design an efficient algorithm for finding such a circular ordering that makes $D$ satisfy Monge property, if one exists. Combined with our result above, this gives an efficient algorithm for the distance realization problem. |
| title | Paths and Intersections: Characterization of Quasi-metrics in Directed Okamura-Seymour Instances |
| topic | Data Structures and Algorithms Discrete Mathematics Combinatorics |
| url | https://arxiv.org/abs/2410.19246 |