Questions tagged [communication-complexity]

Questions regarding the amount of communication needed to accomplish a computational task, when the information about the task is spread across several agents

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Bounds on the size of the smallest NFA for L_k-distinct

Consider the language $L_{k-distinct}$ consisting of all $k$-letter strings over $\Sigma$ such that no two letters are equal:  L_{k-distinct} :=\{w = \sigma_1\sigma_2...\sigma_k \mid \forall i\in[k]...
644 views

Communication lower bounds for partial boolean functions

There are well known techniques for proving lower bounds on the communication complexity of boolean functions, like fooling sets, the rank of the communication matrix, and discepancy. 1) How do we ...
464 views

Zero error randomized communication complexity vs deterministic communication complexity

It is known that for $\Theta(1)$ error the worst case definition of randomized communication complexity and average case definition are equivalent. But when the error is $0$, the worst case randomized ...
937 views

Testing for positivity instead of equality

Alice and Bob have n-bit strings, and want to figure out if they're equal while doing little communication. The standard randomized solution is to treat the n-bit strings as polynomials of degree $n$ ...
1k views

Protocol partition number and deterministic communication complexity

Besides (deterministic) communication complexity $cc(R)$ of a relation $R$, another basic measure for the amount of communication needed is the protocol partition number $pp(R)$. The relation between ...
383 views

Lower bound for NFA accepting 3 letter language

Related to a recent question (Bounds on the size of the smallest NFA for L_k-distinct) Noam Nisan asked for a method to give a better lower bound for the size of an NFA than what we get from ...
832 views

275 views

A direct-sum theorem for the non-deterministic communication complexity of inequality?

A non-deterministic protocol for the inequality function is a protocol that behaves as follows: Alice and Bob get strings $x,y\in\{0,1\}^n$ respectively, and an untrusted prover is trying to convince ...
359 views

Gap-Hamming with different "threshold" (i.e., not $n/2$)

Following a previous question, I'm trying to get a better understanding of the parameters at play in $\textsf{Gap-Hamming}$. In the "standard" setting, we have $x,y\in\{0,1\}^n$ and the partial ...
639 views

One-way randomized communication complexity of approximate Hamming distance

If Alice and Bob both have $n$ bit strings, consider a one-way randomized communication problem where Bob has to output with some good probability a number which is within a $(1+\epsilon)$ factor of ...
224 views

One-way randomized complexity of (variants of) Gap-Hamming-Distance?

The $\textsf{GapHammingDistance}$ problem over $\{0,1\}^n$ is defined as follows: Alice (resp. Bob) is given an input $x\in\{0,1\}^n$ (resp, $y\in\{0,1\}^n$), under the promise that their Hamming ...
456 views

Nondeterministic communication complexity of set disjointness?

In the two-party setting, bounds of $\Theta(n)$ bits are known for deterministic and bounded-error randomized protocols for $\text{DISJ}_n$. (Here $\text{DISJ}_n$ is the $n$-element set disjointness ...
386 views

Testing for equality with zero error

This question comes from what I asked in a comment here, although I realized that I don't actually care about which input is less than the other, if they're different. Alice and Bob have n-bit ...
190 views

Do the quantum communication complexity lower bounds hold when parties can send a "duplicated" qubits?

This question continues from the previous question where I mistakenly asked a question that is too general. In quantum communication complexity, we always assume that Alice and Bob have unlimited ...
394 views

On the notion of positive rank of a matrix

The positive rank of a square matrix is defined in Theorem $3$ of "Expressing Combinatorial Optimization Problems by Linear Programs" by Mihalis Yannakakis as follows: given a $n\times n$ matrix $A$, ...
Inspired by the questions Nondeterministic communication complexity of set disjointness?, I was wondering about the following: Is there an example of a function $f$ where the nondeterministic ...
In quantum communication complexity, we always assume that Alice and Bob have unlimited computational power and are still prove lower bounds such as the $\Omega(n)$ lower bounds of parity. What ...