Threesomes, Degenerates, and Love Triangles

Grønlund, Allan; Pettie, Seth

doi:10.1109/focs.2014.72

Cited by 27 publications

(4 citation statements)

References 34 publications

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“…In particular, a Õ(n)-time verifier for 3SUM would yield an affirmative answer to our main question in the Boolean setting. Note that an analogous improvement of the O(n 3/2 √ log n) [18] size bound in the decision tree model to a size of O(n log 2 n) has recently been obtained [25].…”

Section: Summentioning

confidence: 72%

On Nondeterministic Derandomization of Freivalds' Algorithm: Consequences, Avenues and Algorithmic Progress

Künnemann

2018

Preprint

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Motivated by studying the power of randomness, certifying algorithms and barriers for fine-grained reductions, we investigate the question whether the multiplication of two n × n matrices can be performed in near-optimal nondeterministic time Õ(n 2 ). Since a classic algorithm due to Freivalds verifies correctness of matrix products probabilistically in time O(n 2 ), our question is a relaxation of the open problem of derandomizing Freivalds' algorithm.We discuss consequences of a positive or negative resolution of this problem and provide potential avenues towards resolving it. Particularly, we show that sufficiently fast deterministic verifiers for 3SUM or univariate polynomial identity testing yield faster deterministic verifiers for matrix multiplication. Furthermore, we present the partial algorithmic progress that distinguishing whether an integer matrix product is correct or contains between 1 and n erroneous entries can be performed in time Õ(n 2 ) -interestingly, the difficult case of deterministic matrix product verification is not a problem of "finding a needle in the haystack", but rather cancellation effects in the presence of many errors.Our main technical contribution is a deterministic algorithm that corrects an integer matrix product containing at most t errors in time Õ( √ tn 2 + t 2 ). To obtain this result, we show how to compute an integer matrix product with at most t nonzeroes in the same running time. This improves upon known deterministic output-sensitive integer matrix multiplication algorithms for t = Ω(n 2/3 ) nonzeroes, which is of independent interest.

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Section: Summentioning

confidence: 72%

On Nondeterministic Derandomization of Freivalds' Algorithm: Consequences, Avenues and Algorithmic Progress

Künnemann

2018

Preprint

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“…The 3-SUM problem can be easily solved in O(n 2 ) time, and with a lot of effort this can be slightly improved, both for integer and real inputs [14,23,42].…”

Section: A Few Hardness Conjecturesmentioning

confidence: 99%

How fast can we play Tetris greedily with rectangular pieces?

Dallant,

Iacono

2024

Theoretical Computer Science

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“…The 3-SUM problem can be easily solved in O(n 2 ) time, and with a lot of effort this can be slightly improved, both for integer and real inputs [15,59].…”

Section: A Few Hardness Conjecturesmentioning

confidence: 99%

How Fast Can We Play Tetris Greedily With Rectangular Pieces?

Dallant¹,

Iacono²

2022

Preprint

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Consider a variant of Tetris played on a board of width w and infinite height, where the pieces are axis-aligned rectangles of arbitrary integer dimensions, the pieces can only be moved before letting them drop, and a row does not disappear once it is full. Suppose we want to follow a greedy strategy: let each rectangle fall where it will end up the lowest given the current state of the board. To do so, we want a data structure which can always suggest a greedy move. In other words, we want a data structure which maintains a set of O(n) rectangles, supports queries which return where to drop the rectangle, and updates which insert a rectangle dropped at a certain position and return the height of the highest point in the updated set of rectangles. We show via a reduction to the Multiphase problem [Pătraşcu, 2010] that on a board of width w = Θ(n), if the OMv conjecture [Henzinger et al., 2015] is true, then both operations cannot be supported in time O(n 1/2− ) simultaneously. The reduction also implies polynomial bounds from the 3-SUM conjecture and the APSP conjecture. On the other hand, we show that there is a data structure supporting both operations in O(n 1/2 log 3/2 n) time on boards of width n O(1) , matching the lower bound up to a n o(1) factor.

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Threesomes, Degenerates, and Love Triangles

Cited by 27 publications

References 34 publications

On Nondeterministic Derandomization of Freivalds' Algorithm: Consequences, Avenues and Algorithmic Progress

On Nondeterministic Derandomization of Freivalds' Algorithm: Consequences, Avenues and Algorithmic Progress

How fast can we play Tetris greedily with rectangular pieces?

How Fast Can We Play Tetris Greedily With Rectangular Pieces?

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