Time Complexity and Parallel Speedup of Relational Queries

Time Complexity and Parallel Speedup of Relational Queries to Solve Graph Problems Carlos Ordonez, Predrag T. Tosic

Graph analytics • • Exploration Path Connectivity Structure

Our contributions • • Unify many algorithms into two iterations Relational queries Time complexity based on matrix density Parallel processing

Preliminaries • G definition: G=(V, E), n=|V|, m=|E| • E sparse matrix: m=O(n) • Iterative algorithms – Matrix-Vector multiplication – Matrix-Matrix multiplication

Basic Matrix-Vector Iteration • | S 0 |=1 or |S 0|=n • S= S*E • Semiring switching operators: – */sum() – +/min() – */min

Basic Matrix-Matrix • • R=R*E E+=E+ E 2 +. . + Ek Semiring applies as well Until fixpoint or max path length k

Relational Algebra

Time complexity merge sort • Matrix-vector: O(n log n) • Matrix-matrix: – O(m log m) if m=O(f(n)) not assumed – from O(n) to O(n 3) – clique of size K=O(n)

Fast algorithm termination immediate fixpoint

Matrix-matrix: Time complexity • Tree • Complete graph

Hardness of Matrix-Matrix multiplication

Parallel processing • Matrix-vector – Partition S – replicate S • Matrix-matrix, consider edge (i, j) – Partition by source vertex i: neighbors are local – Partition by destination vertex j: neighbors are local – Partition by edge i, j: neighbors not local

Parallel processing speedup • Speedup – sparse : – dense matrix-vector – dense matrix-matrix • Partitioning – Hashing • Sorting: parallel merge sort • Computation – Local – Distributed

Conclusions • Unified two families of graph algorithms • Relational algebra expressions to evaluate matrix and matrix-vector multiplication • Characterize O() and speedup assuming sparse or dense adjacency matrix