11.3 Communication complexity and additive combinatorics

Communication complexity studies the amount of information exchange needed between parties to compute functions on their inputs. It's crucial in understanding distributed computing limitations and designing efficient protocols. Additive combinatorics provides tools to analyze set structures and prove lower bounds on communication complexity.

The connection between these fields has led to significant results in theoretical computer science. Techniques like the discrepancy method and Fourier analysis have been used to prove lower bounds for various communication problems, shedding light on the inherent complexity of distributed computation.

Communication Complexity Theory

Basic Concepts and Models

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• Studies the amount of communication required for two or more parties to compute a function on their private inputs with the goal of minimizing the total number of bits exchanged between the parties
• Involves two players, Alice and Bob, who each have private inputs and wish to compute a function on their joint inputs by exchanging messages according to a predetermined protocol
• Deterministic communication complexity represents the minimum number of bits that need to be exchanged by the best deterministic protocol for computing the function correctly on all inputs
• Randomized communication complexity allows the protocol to use random bits and requires the correct output to be computed with a certain probability (2/3) on all inputs
• Nondeterministic communication complexity is the minimum number of bits exchanged by the best nondeterministic protocol, where the protocol accepts an input if there exists a certificate that can be verified by exchanging a certain number of bits

Additional Communication Models

• One-way communication model involves only one party sending messages to the other
• Simultaneous message passing model requires both parties to send messages to a third party who computes the output
• Examples of functions studied in communication complexity include:
  • Equality function determines whether Alice and Bob's input strings are equal
  • Set disjointness function checks if Alice and Bob's input sets have any common elements
  • Inner product function computes the inner product of Alice and Bob's input vectors over a finite field

Additive Combinatorics and Communication Complexity

Connections between Additive Combinatorics and Communication Complexity

• Additive combinatorics studies the behavior of subsets of additive groups and their combinatorial properties, providing tools for analyzing the structure of sets and their sumsets (sets of pairwise sums)
• Combinatorial properties of sets can be used to prove lower bounds on the communication complexity of specific functions
• Example: The discrepancy method relates the communication complexity of a function to the discrepancy of its communication matrix, which measures the maximum imbalance between the number of 1s and -1s in any rectangle of the matrix
• Lower bounds on the discrepancy of the communication matrix of a function imply lower bounds on its communication complexity

Techniques for Proving Lower Bounds

• The partition bound method relates the communication complexity of a function to the number of monochromatic rectangles needed to partition its communication matrix
• Fourier analysis on the Boolean cube has been used to prove lower bounds on the randomized communication complexity of certain functions by analyzing their Fourier spectra
• Example: The inner product function, which computes the inner product of Alice and Bob's input vectors over a finite field, has a randomized communication complexity of Ω(n) proved using the discrepancy method and Fourier analysis

Lower Bounds for Communication Complexity

Specific Problems and Their Lower Bounds

• Disjointness problem, where Alice and Bob need to determine whether their input sets are disjoint, has a deterministic communication complexity of Ω(n) proved using the discrepancy method and additive combinatorics
• Gap Hamming distance problem, where Alice and Bob need to distinguish between input strings with Hamming distance less than n/2-√n and greater than n/2+√n, has a randomized communication complexity of Ω(n) proved using additive combinatorics
• Equality function, which determines if Alice and Bob's input strings are equal, has a deterministic communication complexity of Ω(log n) proved using the partition bound and additive combinatorics
• Set disjointness problem for k parties, where each party has a subset of {1,...,n} and they need to determine if there is an element common to all sets, has a randomized communication complexity of Ω(n/k) proved using additive combinatorics

Applications of Lower Bounds

• Lower bounds on communication complexity help understand the inherent limitations of distributed computing and the amount of communication required to solve specific problems
• They provide insights into the design of efficient communication protocols and the trade-offs between communication and computation
• Lower bounds can be used to prove separations between different communication complexity classes (deterministic, randomized, nondeterministic) and to establish the power of randomness in communication
• Example: The Ω(n) lower bound on the deterministic communication complexity of the set disjointness problem implies that any deterministic protocol for this problem must exchange at least a linear number of bits, demonstrating the hardness of the problem

Additive Combinatorics in Multiparty Communication Complexity

Multiparty Communication Models

• Multiparty communication complexity extends the two-party model to k parties, where each party has a private input and they communicate to compute a function on their joint inputs
• Number-on-forehead (NOF) model is a popular multiparty communication model where each party sees the inputs of all other parties but not their own
  • In the k-party NOF model, the input is partitioned into k parts, and each party has access to all parts except the one corresponding to their forehead
• Multiparty simultaneous message passing (SMP) model requires each party to send a message to a referee who computes the output

Lower Bounds in Multiparty Communication Complexity

• Additive combinatorics has been instrumental in proving lower bounds for various functions in the multiparty NOF model
  • Generalized inner product function, where k parties need to compute the product of their input numbers modulo a prime p, has a deterministic k-party NOF communication complexity of Ω(n/2^k) proved using additive combinatorics
  • Set disjointness problem in the k-party NOF model has a randomized communication complexity of Ω(n/(k log k)) proved using additive combinatorics and the discrepancy method
• Additive combinatorics techniques have also been used to prove lower bounds in the multiparty SMP model
• Example: The Ω(n/k) lower bound on the randomized communication complexity of the set disjointness problem in the k-party NOF model shows that the problem remains hard even when each party has access to almost all the input, highlighting the power of the NOF model

Insights from Additive Combinatorics

• Additive combinatorics has provided insights into the role of randomness in multiparty communication complexity
• It has helped establish separations between deterministic, randomized, and nondeterministic multiparty communication complexity classes
• Techniques from additive combinatorics have been used to analyze the structure of communication matrices and to develop new lower bound methods for multiparty communication complexity
• Example: The use of Fourier analysis and the discrepancy method from additive combinatorics has led to tight lower bounds for several multiparty communication problems, showcasing the effectiveness of these techniques in the multiparty setting