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Questions tagged [communication-complexity]

Questions about the amount of communication required to compute a function whose input is distributed between two or more parties

Communication complexity is a model of computation invented by Yao. In this model, two or more parties are cooperating in computing a function whose input is distributed among the parties. Their goal is to minimize the number of bits communicated.

Interesting on its own, communication complexity also has many applications to other parts of complexity theory. Most such applications consist of applying communication complexity lower bounds.

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A Myhill-Nerode type characterization of the regular languages using fooling sets?

Ultimately, my question is whether it's possible to exactly characterize the regular languages in terms of fooling sets. To help explain my motivation for asking this, here's a quick exposition. Let $L$ be a language. A set of strings $S$ is called…
templatetypedef
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Problems with Θ(n³) complexity on TMs with lower bounds by communication complexity arguments

One of the most used simple examples of application of Communication Complexity is the $\Omega(n^2)$ lower bound for recognizing palindromes of length $2n$ on a single tape Turing machine. Is there a similarly simple example that proves a lower…
Vor
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Worst known case for log rank conjecture

The log rank conjecture states that there is some universal constant $c > 0$ so that $$CC(f) = O(\log^c \text{rk}\,(M_f))$$ where $f : X \times Y \to \{0, 1\}$ is a boolean function, $CC$ denotes the deterministic communication complexity, $M_f$ is…
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Do correlated inputs imply existence of efficient communication protocols?

Suppose that I have 2 parties Alice and Bob. Alice gets an input $X$ and Bob gets input $Y$ where $X, Y$ are $n$-bit strings. In the classic communication complexity world, computing a function such as disjointness of $X$ and $Y$ requires…
EagerToLearn
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The Clique vs. Independent Set Problem

Suppose you have an undirected graph $G = (V, E)$, known to both Alice and Bob, Alice gets an independent set of $G$. Bob gets a Clique $B ⊆ V$. Is there any algorithm in $O(\log^2 n)$ bits that finds whether $ A ∩ B = Ø $? This is a well known…
Jay
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Compute the union of two sets between two endpoints minimizing communication complexity

I have two endpoints, $a$ and $b$, that can communicate through a channel. $a$ is storing a set of fixed-length strings $A = \{a_1, \ldots, a_{N_A}\}$, and $b$ is storing another set of fixed-length strings $B = \{b_1, \ldots, b_{N_B}\}$, with in…
Matteo Monti
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The communication complexity of Hamming distance mod $4$

If Alice and Bob each have a bit string of length $n$, what is the randomized communication complexity (either one or two-way) of computing the Hamming distance mod $4$? It seems this is hard to search for online but I am sure it must be well…
Simd
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Direct sum non-deterministic protocol for the non-equality function

Brief introduction: Let $n \in \mathbb N$. The non-equality function, denoted $ NE : \{0,1\}^n \times \{0,1\}^n \to \{0,1\} $ is defined as follows: \begin{align} \forall x,y\in \{0,1\}^n \;\;\; NE(x,y) = \begin{cases}1 & x\neq y \\ 0 & x =…
Itay Hazan
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The communication complexity of non-equality

I'm familiar with the fooling set technique to obtain lower bounds for communication complexity protocols. The most basic example is the equality function for which the diagonal matrix gives the fooling set of size $2^n$, because each 1-output needs…
somebody
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Communication complexity of comparing sets, for Bitcoin

In Bitcoin, when one node wants to tell another node about a block, it sends the block header, then all the transactions it contains. This is inefficient, because the receiving node might already have some or all of those transactions. The sending…
Nick ODell
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Does O(1) communication complexity imply that a language is regular?

Let's say that we have a function $g(i,j)$, which is an arbitrary boolean-valued function over $i,j \in \{a,b\}^*$, such that $|i| = |j| = m.$ Moreover, we can also say that $g$ has communication complexity $c(m),$ and we let $L = \{ij \mid g(i,j) =…
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Randomized and Deterministic Communication Complexity of a function

I have a problem I'm trying to answer for my homework. The question is: Let $p$ be a prime number and let $GF(p)$ denote the finite field of size $p$. Suppose that A has input $x∈GF(p)$ encoded with $⌈log2p⌉$ bits and B has input $y∈GF(p)$ encoded…
user45232
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Understanding the flaw in a proof attempt of the Communication Complexity of Equality

I'm new to communication theory and I've been wondering where the following simple argument fails: Equality Problem We have two players, player 1 Alice who gets an $n$-bit vector $X$ and player 2 Bob who gets an $n$-bit vector $Y$. We want one of…
new_here
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Communication complexity of testing balancedness of a Boolean polynomial

The problem I consider is the following: given the $2^n$ coefficients of a Boolean polynomial $f : \{0, 1\}^n \rightarrow \{0, 1\}$, determine if $f$ is balanced namely if the truth table of $f$ contains exactly $2^{n-1}$ zeroes and $2^{n-1}$ ones.…
Charles Bouillaguet
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Upper bound for set disjointness under product distributions

For the set disjointness problem in the 2-party model of communication complexity, Alice is given an input $X$ and Bob is given input $Y$, $X$ and $Y$ are $n$-length bitstrings (sampled from some distribution), which are interpreted as subsets of…
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