Ordered Tree Encoding via Dyck Word and Leaf Vector
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Solution Overview
Problem
Existing systems using ordered trees for memory representation in computational systems face performance bottlenecks due to high time complexity in operations like subtree retrieval and modification, and inefficiencies in memory usage and cryptographic proof systems, particularly when dealing with incomplete trees and finite field polynomial constraints.
Innovation Solution
The implementation of an encoding system that represents ordered trees using a structure-encoding codeword and a contiguous leaf vector, where the codeword is a Dyck word tracing a depth-first traversal, allowing for O(1) pointer dereferences and amortized O(1) subtree editing, and optimizing zero-knowledge proof systems by avoiding traversal of non-direct subtrees through lookup tables.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Quantity of substance
If standard array implementations are used for ordered trees, then space complexity is improved, but time complexity for subtree retrieval and modification deteriorates to O(log(n)) pointer dereferences causing cache thrashing
Solution Approach 1:
The patent segments the ordered tree into two separate components: a structure-encoding codeword (Dyck word) that represents the tree topology, and a leaf vector that stores the actual data values. This segmentation allows independent optimization of structure access and data access, eliminating the need for O(log(n)) pointer dereferences while maintaining space efficiency.
Solution Approach 2:
The patent introduces a Dyck word as an intermediary representation between the traditional tree structure and the data storage. The Dyck word encodes the tree structure in a linear format that enables O(1) access to subtree positions through polynomial evaluation, acting as a mediator that translates structural queries into efficient computational operations.
2Ease of manufacture
If complete array-based ordered tree implementations are used, then memory allocation is simplified, but memory waste increases when trees are incomplete or skewed
Solution Approach 1:
By separating the structure encoding from the data storage, the patent allows the leaf vector to be dynamically sized to match the actual number of leaves in the tree, regardless of the tree's shape or completeness. This eliminates the need to allocate space for missing nodes in skewed or incomplete trees.
Solution Approach 2:
The patent changes the parameter representation from fixed array indices to polynomial-based position encoding using the Dyck word. This allows the system to adapt to any tree shape (complete, incomplete, skewed) without changing the fundamental data structure, optimizing space usage based on the actual tree configuration.
3Adaptability or versatility
If traditional CPU memory access patterns are used for tree operations, then hardware compatibility is maintained, but performance deteriorates due to cache thrashing from non-contiguous memory access
Solution Approach 1:
The patent transforms the traditional hierarchical tree structure into a linear polynomial representation where subtree positions can be directly calculated using polynomial evaluation. This dimensional change from hierarchical to linear organization enables contiguous memory access patterns that are cache-friendly while maintaining the logical tree structure through the Dyck word encoding.
Data Source
AI summary
A computer-based system, computer-implemented method, and computer program product for state minimization and unlinkable transactions leverage a zero-knowledge (ZK) verifier system implemented on blockchain network(s). The ZK system is configured to, via a ZK virtual machine, verify a block of block(s) of a state-minimized decentralized ledger and recursively verify a corresponding zero-knowledge proof (ZKP) of a predecessor block of the block. The block is associated with an electronic transaction. The ZK system is further configured to generate an unlinkable transaction by obfuscating a correlation between a source node and a destination node of node(s) of the network(s). The source and destination nodes are associated with the transaction. The ZK system is further configured to validate the transaction by verifying a ZKP corresponding to the unlinkable transaction.