Combinatorial Hypermap Encoding for Fast Graph Isomorphism
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Solution Overview
Problem
Current graph databases face challenges in efficiently retrieving and matching graph structures due to the NP-complete nature of structure matching, especially in large databases with dynamically changing topological information, leading to high computational costs and memory requirements.
Innovation Solution
The method employs combinatorial hypermaps (CHYMAPS) with a reversible numeric encoding and decoding system, using tree-based numbering to represent graphs and sub-graphs, enabling fast unification and arithmetic operations, and optimizing storage and retrieval through a three-bit self-delimiting binary code.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If graph structures are stored and retrieved using conventional data structures, then data can be stored and accessed, but computational time and memory requirements become prohibitively high due to the NP-complete nature of structure matching
Solution Approach 1:
The patent replaces conventional graph data structures with a numerical encoding system based on combinatorial hypermaps. Graph structures are transformed into integer codes through a bijective mapping, allowing structural operations to be performed using arithmetic operations on integers rather than complex graph traversals. This substitution of mechanical graph manipulation with numerical computation resolves the contradiction by enabling fast pattern recognition and isomorphism testing through simple arithmetic operations.
Solution Approach 2:
The patent fundamentally changes the parameter representation of graph structures by mapping complex graph topologies to integer parameters through combinatorial hypermap encoding. This parameter transformation allows graph isomorphism and pattern matching problems to be solved through integer arithmetic operations, dramatically reducing computational complexity and memory requirements while maintaining the ability to represent and manipulate graph structures.
2Reliability
If conventional graph matching algorithms are used, then structure matching can be performed, but the NP-complete complexity leads to exponential time consumption for large databases
Solution Approach 1:
The patent replaces conventional graph matching algorithms with numerical operations on combinatorial hypermap encodings. By transforming graph structures into integer codes, the system can perform pattern recognition and isomorphism testing through arithmetic operations that are computationally efficient, maintaining matching accuracy while reducing time complexity from exponential to polynomial or linear time.
Solution Approach 2:
The patent creates a numerical copy of graph structures through bijective encoding into combinatorial hypermaps. This numerical representation preserves all structural information necessary for accurate matching while enabling faster computational operations. The encoding acts as a lossless copy that can be manipulated arithmetically to perform graph matching tasks.
3Measurement precision
If detailed graph topological information is maintained for accurate pattern recognition, then matching precision is improved, but memory consumption increases significantly
Solution Approach 1:
The patent transforms detailed graph topological information into compact integer parameters through combinatorialhypermap encoding. This parameter compression maintains all necessary structural information for accurate pattern recognition while representing complex graphs as concise numerical values, thereby reducing memory consumption without sacrificing matching precision.
Solution Approach 2:
The patent creates a compressed numerical copy of graph topological structures using bijective encoding. This encoding preserves the essential structural properties needed for pattern recognition while using significantly less memory than conventional graph representations, as each graph is represented by an integer code rather than explicit node and edge data structures.
Data Source
AI summary
A method and apparatus is provided for implementing combinatorial hypermaps (CHYMAPS) and/or generalized combinatorial maps (G-Maps) based data representations and operations, comprising: mapping term-algebras to tree-based numbers using a fast algorithm and representing a graph of the mapping structure as a CHYMAPS using reversible numeric encoding and decoding; generating a representation of CHYMAPS in a form optimized for sub-map (sub-graph) to map (graph) isomorphism and partial matching with a general matching process; performing operations on the CHYMAPS as operations on respective numerical representations; performing compression and decompression using a three bit self-delimiting binary code; and storing and retrieving codes.