Communication Complexity of Collision

@inproceedings{Gs2022CommunicationCO,
  title={Communication Complexity of Collision},
  author={Mika G{\"o}{\"o}s and Siddhartha Jain},
  booktitle={Electron. Colloquium Comput. Complex.},
  year={2022}
}
The Collision problem is to decide whether a given list of numbers ( x 1 , . . . , x n ) ∈ [ n ] n is 1-to-1 or 2-to-1 when promised one of them is the case. We show an n Ω(1) randomised communication lower bound for the natural two-party version of Collision where Alice holds the first half of the bits of each x i and Bob holds the second half. As an application, we also show a similar lower bound for a weak bit-pigeonhole search problem, which answers a question of Itsykson and Riazanov ( CCC… 

Figures from this paper

References

SHOWING 1-10 OF 26 REFERENCES

Quantum lower bound for the collision problem

A lower bound of Ω(n1/5) is shown on the number of queries needed by a quantum computer to solve the problem of deciding whether two sets are equal or disjoint on a constant fraction of elements.

Quantum Lower Bound for the Collision Problem with Small Range

  • S. Kutin
  • Mathematics, Computer Science
    Theory Comput.
  • 2005
A modified version of Aaronson and Shi's quantum lower bound for the r-to-one collision problem that removes a restriction that applies only when the range has size at least 3n/2.

Quantum lower bounds for the collision and the element distinctness problems

  • Yaoyun Shi
  • Computer Science
    The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings.
  • 2002
It is proved that any quantum algorithm for finding a collision in an r-to-one function must evaluate the function /spl Omega/ ((n/r)/sup 1/3/) times, where n is the size of the domain and r|n.

Impossibility of succinct quantum proofs for collision-freeness

We show that any quantum algorithm to decide whether a function f : [n] → [n] is a permutation or far from a permutation must make Ω (n1/3/w) queries to f, even if the algorithm is given a w-qubit

Quantum cryptanalysis of hash and claw-free functions

A quantum algorithm that finds collisions in arbitrary functions after only O(3√N/τ) expected evaluations of the function, more efficient than the best possible classical algorithm, even allowing probabilism.

A Polynomial Quantum Query Lower Bound for the Set Equality Problem

It is shown that any error-bounded quantum query algorithm that solves the set equality problem must evaluate oracles \(\Omega(\sqrt[5]{\frac{n}{\ln n}})\) times, where n=|A|=|B|.

Proof complexity of natural formulas via communication arguments

The result implies that the bit pigeonhole requires exponential tree-like Th( k) proofs, where Th(k) is the semantic proof system operating with polynomial inequalities of degree at most k and k = O(log1--ϵ n) for some ϵ > 0.

How significant are the known collision and element distinctness quantum algorithms?

The criterion that an algorithm width requires O(S) hardware to be considered significant if it produces a speedup of better than O(√S) over asimple quantum search algorithm is proposed.

Quantum lower bounds for approximate counting via laurent polynomials

  • S. Aaronson
  • Computer Science, Mathematics
    Electron. Colloquium Comput. Complex.
  • 2018
The complexity of approximate counting, the problem of multiplicatively estimating the size of a nonempty set S ⊆ [N], is resolved in two natural generalizations of quantum query complexity.

Quantum Algorithms for Testing Properties of Distributions

It is shown that the L-distance ∥p-q∥<sub>1</sub> can be estimated with a constant precision using only O(N-1/2) queries in the quantum settings, whereas classical computers need Ω(N<sup>1-o</sup>(1)) queries.