Non-adaptive Bellman-Ford: Yen's improvement is optimal
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910807622680576 |
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| author | Hu, Jialu Kozma, László |
| author_facet | Hu, Jialu Kozma, László |
| contents | The Bellman-Ford algorithm for single-source shortest paths repeatedly updates tentative distances in an operation called relaxing an edge. In several important applications a non-adaptive (oblivious) implementation is preferred, which means fixing the entire sequence of relaxations upfront, independently of the edge-weights. Such an implementation performs, in a dense graph on $n$ vertices, $(1 + o(1))n^3$ relaxations. An improvement by Yen from 1970 reduces the number of relaxations by a factor of two. We show that no further constant-factor improvements are possible, and every non-adaptive deterministic algorithm based on relaxations must perform $(\frac{1}{2} - o(1))n^3$ steps. This improves an earlier lower bound of Eppstein of $(\frac{1}{6} - o(1))n^3$. Given that a non-adaptive randomized variant of Bellman-Ford with at most $(\frac{1}{3} + o(1))n^3$ relaxations (with high probability) is known, our result implies a strict separation between deterministic and randomized strategies, answering an open question of Eppstein. On the complexity side, we show that deciding whether a given relaxation sequence is guaranteed to yield correct distances is NP-hard, even with the complete graph as input. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_10343 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-adaptive Bellman-Ford: Yen's improvement is optimal Hu, Jialu Kozma, László Data Structures and Algorithms Combinatorics The Bellman-Ford algorithm for single-source shortest paths repeatedly updates tentative distances in an operation called relaxing an edge. In several important applications a non-adaptive (oblivious) implementation is preferred, which means fixing the entire sequence of relaxations upfront, independently of the edge-weights. Such an implementation performs, in a dense graph on $n$ vertices, $(1 + o(1))n^3$ relaxations. An improvement by Yen from 1970 reduces the number of relaxations by a factor of two. We show that no further constant-factor improvements are possible, and every non-adaptive deterministic algorithm based on relaxations must perform $(\frac{1}{2} - o(1))n^3$ steps. This improves an earlier lower bound of Eppstein of $(\frac{1}{6} - o(1))n^3$. Given that a non-adaptive randomized variant of Bellman-Ford with at most $(\frac{1}{3} + o(1))n^3$ relaxations (with high probability) is known, our result implies a strict separation between deterministic and randomized strategies, answering an open question of Eppstein. On the complexity side, we show that deciding whether a given relaxation sequence is guaranteed to yield correct distances is NP-hard, even with the complete graph as input. |
| title | Non-adaptive Bellman-Ford: Yen's improvement is optimal |
| topic | Data Structures and Algorithms Combinatorics |
| url | https://arxiv.org/abs/2402.10343 |