Communication Complexity and Algorithmic Lower Bounds
Summary
Communication complexity examines the minimum volume of information exchange required among distributed agents to compute a function of their combined inputs. Established nearly four decades ago, this paradigm has matured into a core tool not only for analysing protocols but also for deriving algorithmic lower bounds across models such as circuits, streaming algorithms, property testing and data-structure design. Central themes include the separation of deterministic, randomized and quantum communication costs, the study of combinatorial measures such as matrix rank, discrepancy and information complexity, and their interrelations. By interpreting a communication matrix as a representation of input–output pairs, one can translate lower bounds on message complexity into hardness results for computational resources like time, space or circuit depth. This approach has yielded profound insights into longstanding open problems such as the log-rank conjecture, the gap-Hamming problem and set disjointness, while informing practical applications ranging from secure multiparty computation to distributed optimisation.
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Communication Complexity and Algorithmic Lower Bounds publication trend
The graph below shows the total number of articles in communication complexity and algorithmic lower bounds across all publications each year (not limited to Nature Index journals).
Technical terms
Communication complexity: A measure of the minimum number of bits exchanged between parties to jointly compute a function on distributed inputs.
Algorithmic lower bound: A proof that any algorithm or protocol for a problem requires at least a specified amount of a computational resource, such as time, space or communication.
Log-rank: The logarithm of the rank of the communication matrix of a function; conjectured to characterise deterministic communication complexity up to a constant power.
Discrepancy: A measure of imbalance in a matrix with respect to all rectangles, used to bound communication complexity for deterministic and randomized protocols.
VC dimension: A combinatorial parameter that quantifies the complexity of set families, influencing learning theory and communication costs in restricted disjointness problems.
Blocky rank: The smallest number of “blocky” matrices (blowups of permutation matrices) needed to linearly span a given matrix, linking combinatorial structure to communication lower bounds.
References
- On Blocky Ranks Of Matrices. computational complexity (2024).
- Matrix discrepancy and the log-rank conjecture. Mathematical Programming (2024).
- Disjointness through the Lens of Vapnik–Chervonenkis Dimension: Sparsity and Beyond. computational complexity (2022).
- On the (in)efficiency of non-interactive secure multiparty computation. Designs, Codes and Cryptography (2018).
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