Abstract
We establish strong connections between the non-emptiness of intersection problem for two and three DFA’s over a binary alphabet and the triangle finding and 3sum problems. In particular, we introduce efficient reductions from triangle finding to non-emptiness of intersection for two DFA’s over a binary alphabet and from 3sum to non-emptiness of intersection for three DFA’s over a binary alphabet. Additionally, in our main result, we show that for every \(\alpha \ge 2\), non-emptiness of intersection for three DFA’s over a unary alphabet can be solved in \(O(n^{\frac{\alpha }{2}})\) time if and only if triangle finding can be solved in \(O(n^{\alpha })\) time.
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Notes
- 1.
- 2.
Additional hardness results were shown in [3].
- 3.
Each path corresponds with an ordered pair of vertices where the underlying vertices are adjacent in G.
- 4.
This follows because \(m = O(n^2)\) for all graphs with n vertices and m edges.
- 5.
This intuition was communicated to the second author from Michael Blondin.
- 6.
The segment may be empty.
- 7.
Zero is permitted to have either sign.
- 8.
The authors would especially like the thank Joseph Swernofsky for advice and feedback that helped in obtaining this result as well as Ronald Fagin for some early feedback and encouragement.
- 9.
Such integers exist because for every n, there exist powers of 2, 3, and 5 in [n, 5n].
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de Oliveira Oliveira, M., Wehar, M. (2018). Intersection Non-emptiness and Hardness Within Polynomial Time. In: Hoshi, M., Seki, S. (eds) Developments in Language Theory. DLT 2018. Lecture Notes in Computer Science(), vol 11088. Springer, Cham. https://doi.org/10.1007/978-3-319-98654-8_23
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