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Algebraic Methods in the Congested Clique
Keren Censor-Hillel, Petteri Kaski, Janne H. Korhonen, Christoph Lenzen, Ami Paz, Jukka Suomela
TL;DR
The paper asks how algebraic techniques can improve distance computation and subgraph detection in the congested clique. It adapts parallel matrix multiplication to this model and combines it with centralized techniques, obtaining improved bounds across matrix multiplication, cycle problems, girth, and APSP.
Problem
Distance computation and subgraph detection in the congested clique require improved algorithms beyond existing upper bounds.
Method
The paper adapts parallel matrix multiplication implementations to the congested clique and combines them with centralized algorithmic techniques.
Results
The approach yields O(n1−2/ω+ε)-round ring matrix multiplication and improved algorithms including (1 + o(1))-approximate APSP in O(nρ+o(1)) rounds.
Takeaways & Limitations
Algebraic methods provide efficient congested-clique algorithms that outperform previous combinatorial upper bounds across several classical graph problems.
Takeaways & Limitations
The work focuses on the fully connected clique topology, although the authors expect related techniques may extend to CONGEST; corresponding lower bounds remain open.
Abstract
from arXiv · showhide
In this work, we use algebraic methods for studying distance computation and subgraph detection tasks in the congested clique model. Specifically, we adapt parallel matrix multiplication implementations to the congested clique, obtaining an $O(n^{1-2/ω})$ round matrix multiplication algorithm, where $ω< 2.3728639$ is the exponent of matrix multiplication. In conjunction with known techniques from centralised algorithmics, this gives significant improvements over previous best upper bounds in the congested clique model. The highlight results include: -- triangle and 4-cycle counting in $O(n^{0.158})$ rounds, improving upon the $O(n^{1/3})$ triangle detection algorithm of Dolev et al. [DISC 2012], -- a $(1 + o(1))$-approximation of all-pairs shortest paths in $O(n^{0.158})$ rounds, improving upon the $\tilde{O} (n^{1/2})$-round $(2 + o(1))$-approximation algorithm of Nanongkai [STOC 2014], and -- computing the girth in $O(n^{0.158})$ rounds, which is the first non-trivial solution in this model. In addition, we present a novel constant-round combinatorial algorithm for detecting 4-cycles.
1 Introduction
The paper adapts algebraic and fast matrix-multiplication techniques to the congested clique, yielding faster algorithms for matrix multiplication, cycle problems, girth, and shortest paths. Its applications improve several prior bounds, including approximate APSP and triangle detection.
- Matrix multiplication: O(n1−2/ω+ε) rounds suffice for n × n matrix multiplication over rings, while semiring multiplication takes O(n1/3) rounds.Here ω < 2.3728639 is the matrix multiplication exponent; bounds scale with entry encoding length.
- Cycle detection and counting: O(nρ) rounds compute triangle and 4-cycle counts, improving the prior O(n1/3)-round triangle detection bound when ρ ≤ 1−2/ω.The same framework also detects k-cycles in 2O(k)nρ log n rounds for constant k.
- Cycle detection and counting: O(1) rounds detect 4-cycles with a novel algorithm that does not use matrix multiplication.This is separate from the matrix-multiplication-based counting and cycle-detection results.
- Girth: ˜O(nρ) rounds compute the girth of undirected, unweighted graphs, using a trade-off between short-cycle detection and graph sparsity.If short cycles are absent, the resulting edge bound lets all nodes learn the graph.
- Shortest paths: (1 + o(1))-approximate APSP is computable in O(nρ+o(1)) rounds for directed graphs with integer weights in {0, 1, . . . , 2no(1)}.The result improves on the prior (2 + o(1))-approximation in ˜O(n1/2) rounds; exact APSP bounds are also given for several graph classes.
2 Matrix Multiplication Algorithms
The section presents semiring and fast matrix multiplication algorithms for the congested clique by partitioning matrix computations across nodes and adapting bilinear algorithms. These constructions yield O(n^1/3) rounds over semirings and O(n^(1−2/ω+ε)) rounds over rings, with oblivious communication and routing-based implementations.
- 2.1 Semiring matrix multiplication: O(n^1/3) rounds suffice for semiring matrix multiplication using a three-dimensional partition of the multiplication work.Nodes are identified with triples and assigned subcubes of the matrix-multiplication computation; communication is then used to distribute inputs, partial products, and assembled rows.
- 2.1 Semiring matrix multiplication: Oblivious routing implements the semiring algorithm in O(n^1/3) rounds because each node sends or receives at most O(n^4/3) messages.The communication pattern is independent of the input matrices, enabling the stated routing scheme.
- 2.2 Fast Matrix Multiplication: Fast matrix multiplication first forms linear combinations of matrix entries, computes their products, and combines those products to obtain the matrix product.The input matrices are organized using a two-level grid of submatrices, supporting the distributed simulation of the bilinear algorithm.
- 2.2 Fast Matrix Multiplication: O(n^(1−2/ω+ε)) rounds suffice over rings by translating bilinear matrix multiplication algorithms into the congested clique.A bilinear algorithm using O(n^σ) scalar multiplications leads to O(n^(1−2/σ)) rounds, and known algorithms with O(n^(ω+ε)) multiplications give the theorem's ring bound.
3 Upper Bounds
The paper develops algebraic and combinatorial upper bounds for subgraph detection, cycle counting, girth, and shortest-path problems in the congested clique. Fast matrix multiplication yields broad improvements, while a separate routine detects 4-cycles in constant rounds.
- Subgraph detection and counting: O(n^ρ) rounds suffice to count triangles and 4-cycles in directed and undirected graphs.
- Subgraph detection and counting: O(n^ρ) rounds suffice to compute triangle and 4-cycle counts using adjacency-matrix trace formulas and fast matrix multiplication.The counting algorithms apply fast matrix multiplication to the graph adjacency matrix.
- Subgraph detection and counting: 2^O(k)n^ρ log n rounds suffice to detect directed or undirected k-cycles.The method uses colour-coding and recursively computed colour-subset matrices.
- Subgraph detection and counting: O(1) rounds suffice to detect 4-cycles without directly using matrix multiplication.The algorithm can be interpreted as an efficient sparse matrix multiplication routine.
- Girth: ˜O(n^ρ) rounds suffice to compute undirected and directed girth.The girth algorithm combines cycle detection with a trade-off relating girth and the number of edges.
4 Lower Bounds
The lower bounds show that several congested-clique matrix-multiplication and APSP implementations face communication barriers. These barriers are unconditional for some settings and apply to Strassen-like algorithms under stated structural assumptions.
- Trivial multiplication: Parallel-machine communication lower bounds transfer to congested-clique implementations of trivial Θ(n3) matrix multiplication.The transfer uses simulation of the relevant parallel models by the congested clique.
- Semiring-based bounds: ˜Ω(n1/3) rounds are required for trivial Θ(n3) matrix multiplication and APSP algorithms restricted to summing weights and taking minimums.The APSP statement follows because min-plus matrix multiplication reduces to APSP with constant blowup.
- Fast matrix multiplication: Any Strassen-like implementation using Ω(nσ) element multiplications requires ˜Ω(n1−2/σ) communication rounds.The bound relies on the recursive, connected-decoding, no-repeated-multiplication structure of Strassen-like algorithms.
- Broadcast congested clique: Known CONGEST lower bounds imply that weighted APSP cannot be approximated below factor 2 in fewer than ˜Ω(n) rounds in the broadcast congested clique.This consequence is obtained from a result on weighted-graph APSP approximation in CONGEST.
- Broadcast congested clique: The corresponding broadcast congested-clique lower bound covers Boolean-semiring matrix multiplication and APSP algorithms.The Boolean-semiring qualification allows different matrix-multiplication exponents across semirings; the bound also applies to integers, rationals, and min-plus.
5 Conclusions
The paper demonstrates that algebraic methods, especially fast matrix multiplication, yield congested-clique algorithms outperforming previous combinatorial approaches. It also identifies broader applicability while noting that the current results focus on the fully connected topology.
- Contributions: Algebraic methods, particularly fast matrix multiplication, produce efficient congested-clique algorithms that outperform previous combinatorial algorithms.The paper combines algebraic techniques with distributed computation in the clique model.
- Further applicability: Similar improvements may follow from other known centralized matrix-multiplication-based algorithms not exhausted by this work.The paper presents this as an expectation about further applications, not as an established result.
- Scope: The techniques are expected to apply beyond the fully connected clique, including to the usual CONGEST model.The paper notes sparse regions may make triangle detection easy, while dense regions may provide enough aggregate bandwidth for fast matrix multiplication.