Proceedings Article10.1145/1806689.1806772
Towards polynomial lower bounds for dynamic problems
Mihai Patrascu
- 05 Jun 2010
- pp 603-610
TL;DR: This work describes a carefully-chosen dynamic version of set disjointness (the "multiphase problem"), and conjecture that it requires n^Omega(1) time per operation, and forms the first nonalgebraic reduction from 3SUM, which allows3SUM-hardness results for combinatorial problems.
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Abstract: We consider a number of dynamic problems with no known poly-logarithmic upper bounds, and show that they require nΩ(1) time per operation, unless 3SUM has strongly subquadratic algorithms. Our result is modular: (1) We describe a carefully-chosen dynamic version of set disjointness (the "multiphase problem"), and conjecture that it requires n^Omega(1) time per operation. All our lower bounds follow by easy reduction. (2) We reduce 3SUM to the multiphase problem. Ours is the first nonalgebraic reduction from 3SUM, and allows 3SUM-hardness results for combinatorial problems. For instance, it implies hardness of reporting all triangles in a graph. (3) It is plausible that an unconditional lower bound for the multiphase problem can be established via a number-on-forehead communication game.
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