Labeled Tree Compression via Coordinated Arrays
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Solution Overview
Problem
Existing methods fail to provide a succinct representation and efficient navigation of labeled trees, which are crucial in computer science applications, due to the inefficiency in storing both the structure and labels of trees, leading to wasteful space usage and complex algorithms.
Innovation Solution
A method that transforms labeled trees into two coordinated arrays, one capturing the structure and the other the labels, using path sorting and grouping to linearize the data, allowing for optimal succinctness and supporting navigational operations in O(1) time, independent of the alphabet size and tree structure.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Speed
If pointer-based tree representations are used to support navigational operations in O(1) time, then navigation speed is improved, but space complexity increases to O(t log t) bits which is wasteful
Solution Approach 1:
The patent segments the tree representation into two separate arrays: one array stores the tree structure information and the other stores the label information. This segmentation allows the structure array to be compressed to near-optimal space while maintaining O(1) navigation operations through carefully constructed indexing mechanisms.
Solution Approach 2:
The patent changes the representation parameters by using a different encoding scheme that achieves better space efficiency. Instead of using traditional pointer-based representation requiring O(log t) bits per node, the patent uses a compressed representation that achieves O(t) total bits for the structure while maintaining constant-time navigation through mathematical indexing.
2Quantity of substance
If succinct representation methods are applied to reduce space complexity, then space efficiency is improved, but the ability to support efficient navigational operations deteriorates
Solution Approach 1:
The patent performs preliminary actions by pre-processing the tree data into a specific compressed format with built-in indexing structures. This pre-processing creates a representation where navigation operations can be performed directly on the compressed data without requiring decompression or complex computations, thus maintaining O(1) speed.
Solution Approach 2:
The patent introduces intermediary data structures (the two coordinated arrays with specific indexing schemes) that act as mediators between the compressed representation and the navigation operations. These intermediary structures enable constant-time navigation while the underlying data remains in a space-efficient compressed form.
3Loss of information
If classical methods are used to represent labeled trees, then labels can be stored, but the overall space complexity increases and algorithms become complex
Solution Approach 1:
The patent segments the labeled tree into separate structure and label arrays, allowing each to be optimized independently. The structure array uses a compressed representation while the label array uses efficient encoding, reducing overall space complexity while maintaining simple access patterns.
Solution Approach 2:
The patent extracts the label information from the traditional pointer-based node structure and stores it separately in a dedicated array. This extraction allows the structure representation to be simplified and compressed while labels are accessed through efficient indexing based on node positions.
Data Source
AI summary
A method of structuring and compressing labeled trees of arbitrary degree and shape for optimal succinctness, the method including a transform for compressing and indexing tree shaped data, using path sorting and grouping to linearize labeled tree shaped data into two coordinated arrays, one capturing the structure of the tree and the other capturing the labels of the tree. Additional navigational operations and basic searching may be performed on the transformed data.