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| Format: | Preprint |
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2024
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| Accès en ligne: | https://arxiv.org/abs/2409.18469 |
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| _version_ | 1866917985454653440 |
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| author | Bhadra, Ronak Tewari, Raghunath |
| author_facet | Bhadra, Ronak Tewari, Raghunath |
| contents | Kallampally and Tewari showed in 2016 that there can be a trade-off between determinism and time in space-bounded computations. This they did by describing an unambiguous non-deterministic algorithm to solve Directed Graph Reachability that requires O(log^2 n) space and simultaneously runs in polynomial time. Savitch's 1970 algorithm that solves the same problem deterministically also requires O(log^2 n) space but doesn't guarantee polynomial running time and hence the trade off. We describe a new problem for which we can show a similar trade off between determinism and time.
We consider a collection P of f directed paths. We show that the problem of finding reachability from one vertex to another in the union G of these path graphs via a path that switches amongst the paths in P at most k times can be solved in O(klog f+log n) space but the algorithm doesn't guarantee polynomial runtime. On the other hand, we also show that the same problem can be solved by an unambiguous non-deterministic algorithm that simultaneously runs in O(klog f+log n) space and polynomial time. Since these two algorithms are not dependent on Savitch, therefore this example sheds new light on how such a trade off between determinism and time happens in space-bounded computations and makes the phenomenon less elusive. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_18469 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Trading Determinism for Time: The k-Reach Problem Bhadra, Ronak Tewari, Raghunath Computational Complexity Kallampally and Tewari showed in 2016 that there can be a trade-off between determinism and time in space-bounded computations. This they did by describing an unambiguous non-deterministic algorithm to solve Directed Graph Reachability that requires O(log^2 n) space and simultaneously runs in polynomial time. Savitch's 1970 algorithm that solves the same problem deterministically also requires O(log^2 n) space but doesn't guarantee polynomial running time and hence the trade off. We describe a new problem for which we can show a similar trade off between determinism and time. We consider a collection P of f directed paths. We show that the problem of finding reachability from one vertex to another in the union G of these path graphs via a path that switches amongst the paths in P at most k times can be solved in O(klog f+log n) space but the algorithm doesn't guarantee polynomial runtime. On the other hand, we also show that the same problem can be solved by an unambiguous non-deterministic algorithm that simultaneously runs in O(klog f+log n) space and polynomial time. Since these two algorithms are not dependent on Savitch, therefore this example sheds new light on how such a trade off between determinism and time happens in space-bounded computations and makes the phenomenon less elusive. |
| title | Trading Determinism for Time: The k-Reach Problem |
| topic | Computational Complexity |
| url | https://arxiv.org/abs/2409.18469 |