scholarly journals The communication complexity of the Hamming distance problem

2006 ◽  
Vol 99 (4) ◽  
pp. 149-153 ◽  
Author(s):  
Wei Huang ◽  
Yaoyun Shi ◽  
Shengyu Zhang ◽  
Yufan Zhu
Keyword(s):  
Communication Complexity ◽  
Hamming Distance ◽  
Distance Problem

scholarly journals On Ajtai’s Lower Bound Technique for R-way Branching Programs and the Hamming Distance Problem

2000 ◽  
Vol 7 (11) ◽  
Author(s):  
Jakob Pagter
Keyword(s):  
Lower Bound ◽  
Hamming Distance ◽  
Linear Time ◽  
Program Model ◽  
Branching Programs ◽  
Original Proof ◽  
Trade Off ◽  
Time Space ◽  
Distance Problem ◽  
With Memory

In this report we study the proof employed by Miklos Ajtai<br />[Determinism versus Non-Determinism for Linear Time RAMs<br />with Memory Restrictions, 31st Symposium on Theory of <br />Computation (STOC), 1999] when proving a non-trivial lower bound<br />in a general model of computation for the Hamming Distance<br />problem: given n elements: decide whether any two of them have<br />"small" Hamming distance. Specifically, Ajtai was able to show<br />that any R-way branching program deciding this problem using<br />time O(n) must use space Omega(n lg n).<br />We generalize Ajtai's original proof allowing us to prove a<br />time-space trade-off for deciding the Hamming Distance problem<br /> in the R-way branching program model for time between n<br />and alpha n lg n / lg lg n, for some suitable 0 < alpha < 1. In particular we prove<br />that if space is O(n^(1−epsilon)), then time is Omega(n lg n / lg lg n).

scholarly journals Approximate string searching with fast fourier transforms and simplexes

2019 ◽  
Author(s):  
Daniel Liu
Keyword(s):  
Fourier Transform ◽  
Fast Fourier Transform ◽  
Fourier Transforms ◽  
Hamming Distance ◽  
Dimensional Space ◽  
Approximate String Matching ◽  
String Searching ◽  
Novel Approach ◽  
Sigma 1 ◽  
Distance Problem

Previous algorithms for solving the approximate string matching with Hamming distance problem with wildcard ("don't care") characters have been shown to take \(O(|\Sigma| N \log M)\) time, where \(N\) is the length of the text, \(M\) is the length of the pattern, and \(|\Sigma|\) is the size of the alphabet. They make use of the Fast Fourier Transform for efficiently calculating convolutions. We describe a novel approach of the problem, which makes use of special encoding schemes that depend on \((|\Sigma| - 1)\)-simplexes in \((|\Sigma| - 1)\)-dimensional space.

On the orthogonal rank and impossibility of quantum round elimination

2017 ◽  
Vol 17 (1&2) ◽  
pp. 106-116
Author(s):  
Jop Briet ◽  
Jeroen Zuiddam
Keyword(s):  
Lower Bound ◽  
Communication Complexity ◽  
Quantum Communication ◽  
Hamming Distance ◽  
Cayley Graphs ◽  
Affirmative Answer ◽  
Complex Vector ◽  
Total Communication ◽  
Quantum Communication Complexity ◽  
The Cost

After Bob sends Alice a bit, she responds with a lengthy reply. At the cost of a factor of two in the total communication, Alice could just as well have given Bob her two possible replies at once without listening to him at all, and have him select which one applies. Motivated by a conjecture stating that this form of “round elimination” is impossible in exact quantum communication complexity, we study the orthogonal rank and a symmetric variant thereof for a certain family of Cayley graphs. The orthogonal rank of a graph is the smallest number d for which one can label each vertex with a nonzero d-dimensional complex vector such that adjacent vertices receive orthogonal vectors. We show an exp(n) lower bound on the orthogonal rank of the graph on {0, 1} n in which two strings are adjacent if they have Hamming distance at least n/2. In combination with previous work, this implies an affirmative answer to the above conjecture.

scholarly journals Generalizations of the distributed Deutsch–Jozsa promise problem

2015 ◽  
Vol 27 (3) ◽  
pp. 311-331 ◽  
Author(s):  
JOZEF GRUSKA ◽  
DAOWEN QIU ◽  
SHENGGEN ZHENG
Keyword(s):  
Communication Complexity ◽  
Quantum Communication ◽  
Hamming Distance ◽  
Finite Automata ◽  
Classical Communication ◽  
Deterministic Finite Automata ◽  
Exact Quantum ◽  
Quantum Communication Complexity

In the distributed Deutsch–Jozsa promise problem, two parties are to determine whether their respective strings x, y ∈ {0,1}n are at the Hamming distanceH(x, y) = 0 or H(x, y) = $\frac{n}{2}$. Buhrman et al. (STOC' 98) proved that the exact quantum communication complexity of this problem is O(log n) while the deterministic communication complexity is Ω(n). This was the first impressive (exponential) gap between quantum and classical communication complexity. In this paper, we generalize the above distributed Deutsch–Jozsa promise problem to determine, for any fixed $\frac{n}{2}$ ⩽ k ⩽ n, whether H(x, y) = 0 or H(x, y) = k, and show that an exponential gap between exact quantum and deterministic communication complexity still holds if k is an even such that $\frac{1}{2}$n ⩽ k < (1 − λ)n, where 0 < λ < $\frac{1}{2}$ is given. We also deal with a promise version of the well-known disjointness problem and show also that for this promise problem there exists an exponential gap between quantum (and also probabilistic) communication complexity and deterministic communication complexity of the promise version of such a disjointness problem. Finally, some applications to quantum, probabilistic and deterministic finite automata of the results obtained are demonstrated.

scholarly journals Communication Complexity of Computing the Hamming Distance

1986 ◽  
Vol 15 (4) ◽  
pp. 932-947 ◽  
Author(s):  
King F. Pang ◽  
Abbas El Gamal
Keyword(s):  
Communication Complexity ◽  
Hamming Distance

scholarly journals Approximate string searching with fast fourier transforms and simplexes

2019 ◽  
Author(s):  
Daniel Liu
Keyword(s):  
Fourier Transform ◽  
Fast Fourier Transform ◽  
Fourier Transforms ◽  
Hamming Distance ◽  
Dimensional Space ◽  
Approximate String Matching ◽  
String Searching ◽  
Novel Approach ◽  
Sigma 1 ◽  
Distance Problem

Previous algorithms for solving the approximate string matching with Hamming distance problem with wildcard ("don't care") characters have been shown to take \(O(|\Sigma| N \log M)\) time, where \(N\) is the length of the text, \(M\) is the length of the pattern, and \(|\Sigma|\) is the size of the alphabet. They make use of the Fast Fourier Transform for efficiently calculating convolutions. We describe a novel approach of the problem, which makes use of special encoding schemes that depend on \((|\Sigma| - 1)\)-simplexes in \((|\Sigma| - 1)\)-dimensional space.

scholarly journals Compressed Communication Complexity of Hamming Distance

Algorithms ◽  
2021 ◽  
Vol 14 (4) ◽  
pp. 116
Author(s):  
Shiori Mitsuya ◽  
Yuto Nakashima ◽  
Shunsuke Inenaga ◽  
Hideo Bannai ◽  
Masayuki Takeda
Keyword(s):  
Communication Complexity ◽  
Complexity Analysis ◽  
Hamming Distance ◽  
Common Prefix

We consider the communication complexity of the Hamming distance of two strings. Bille et al. [SPIRE 2018] considered the communication complexity of the longest common prefix (LCP) problem in the setting where the two parties have their strings in a compressed form, i.e., represented by the Lempel-Ziv 77 factorization (LZ77) with/without self-references. We present a randomized public-coin protocol for a joint computation of the Hamming distance of two strings represented by LZ77 without self-references. Although our scheme is heavily based on Bille et al.’s LCP protocol, our complexity analysis is original which uses Crochemore’s C-factorization and Rytter’s AVL-grammar. As a byproduct, we also show that LZ77 with/without self-references are not monotonic in the sense that their sizes can increase by a factor of 4/3 when a prefix of the string is removed.

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