Sparse Matrix Hierarchical Tree Structure Parallel Processing
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Solution Overview
Problem
Conventional compressed formats for sparse matrices are not optimized for highly parallel processors, leading to increased latencies and reduced processing efficiency due to inefficient access patterns and storage of metadata.
Innovation Solution
A tree structure is used to represent a sparse matrix, allowing for efficient access and distribution of non-empty regions across multiple processing elements, with regions divided recursively to ensure balanced workloads and compact storage.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Quantity of substance
If conventional compressed formats (CSR/CSC) are used to store sparse matrices, then storage space is reduced by omitting zero elements, but processing efficiency and access latency increase due to unoptimized access patterns for parallel processors
Solution Approach 1:
The sparse matrix is divided into multiple rectangular tiles that partition the non-zero elements into balanced groups. Each tile contains a subset of non-zero elements with roughly equal counts, enabling parallel processing elements to work on disjoint sets of data simultaneously. This segmentation resolves the contradiction by organizing compressed data into access-friendly units while maintaining space efficiency.
Solution Approach 2:
The invention transitions from conventional one-dimensional compressed storage (row-major or column-major) to a two-dimensional tile-based organization. Each tile is represented by metadata including row indices, column indices, and non-zero element counts, creating a hierarchical structure that optimizes both storage density and parallel access patterns without increasing overall storage requirements.
2Ease of operation
If non-zero elements are organized along entire rows (CSR format), then row-based access is simplified, but column access becomes inefficient and data fetch requirements increase during matrix multiplication
Solution Approach 1:
The matrix is segmented into rectangular tiles that can be efficiently accessed along both row and column dimensions. Each tile stores non-zero elements with metadata that enables direct access to elements needed for both row-based and column-based operations, eliminating the inefficiency of traversing entire rows to access column elements.
3Productivity
If tiles are created to group non-zero elements, then parallel processing can be enabled, but the tiles must satisfy multiple constraints (balanced non-zero counts, compact structure, arbitrary accessibility) that are difficult to meet simultaneously with conventional formats
Solution Approach 1:
The sparse matrix is systematically divided into rectangular tiles that naturally satisfy the parallel processing requirements. Each tile is defined by row and column boundaries and contains a balanced number of non-zero elements, allowing multiple processing elements to operate independently on different tiles while maintaining load balance.
Solution Approach 2:
The invention changes the organizational parameters from conventional row-major or column-major ordering to a tile-based structure defined by spatial boundaries and non-zero element distributions. This parameter change enables simultaneous satisfaction of balanced workload, compact storage, and efficient access patterns required for high-performance parallel processing.
Data Source
AI summary
One embodiment sets forth a technique for performing matrix operations. The technique includes traversing a tree structure to access one or more non-empty regions within a matrix. The tree structure includes a first plurality of nodes and a second plurality of nodes corresponding to non-empty regions in the matrix. The first plurality of nodes includes a first node representing a first region and one or more second nodes that are children of the first node and represent second region(s) with an equal size formed within the first region. The second plurality of nodes include a third node representing a third region and one or more fourth nodes that are children of the third node and represent fourth region(s) with substantially equal numbers of non-zero matrix values formed within the third region. The technique also includes performing matrix operation(s) based on the non-empty region(s) to generate a matrix operation result.