Minimax K-Nearest Neighbor Search via Minimum Spanning Tree
Find Innovative SolutionsGenerate Solutions
Solution Overview
Problem
Existing methods for identifying nearest neighbors, such as those using Euclidean or shortest path distances, are inefficient and unsuitable for large datasets with complex structures, as they fail to capture underlying geometry and require high computational complexity, especially when dealing with non-convex data.
Innovation Solution
A computationally efficient method for K-nearest neighbor search based on Minimax distances that iteratively selects nearest neighbors using a graph representation, avoiding the need for a minimum spanning tree and applicable to arbitrary distance measures, allowing for accurate classification and outlier detection.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If link-based distance measures are used to capture complex data structures, then measurement precision is improved, but device complexity increases due to matrix inversion requirements
Solution Approach 1:
The patent segments the complex all-pairs distance computation into two phases: first computing a minimum spanning tree (MST) that captures essential connectivity, then using the MST to derive Minimax distances. This segmentation reduces the computational problem from O(N³) matrix inversion to O(N²) MST computation followed by linear traversal, while preserving the ability to capture complex data structures.
Solution Approach 2:
The patent extracts the essential structural information by computing only the minimum spanning tree, which contains the critical connectivity patterns needed for Minimax distance calculation. By taking out only the necessary subset of edges (the MST) rather than computing all possible paths, the method achieves accurate distance measurement with reduced computational complexity.
2Measurement precision
If Floyd-Warshall algorithm is used to compute all-pairs Minimax distances, then measurement precision is improved, but productivity decreases due to O(N³) running time
Solution Approach 1:
The patent divides the distance computation task into two segments: first building a minimum spanning tree that captures the essential graph structure, then performing a single traversal of the MST to compute all Minimax distances. This segmentation replaces the O(N³) Floyd-Warshall approach with an O(N²) method that achieves the same measurement precision.
Solution Approach 2:
The patent performs preliminary action by pre-computing the minimum spanning tree, which serves as a compact representation of the graph structure. This preliminary MST computation enables all subsequent Minimax distance queries to be answered efficiently through simple tree traversal, rather than requiring repeated O(N³) computations.
3Productivity
If minimum spanning tree computation is performed in advance, then productivity is improved for K-NN search, but device complexity increases
Solution Approach 1:
The patent applies preliminary action by computing the minimum spanning tree in advance before performing K-nearest neighbor search. This pre-computed MST serves as a reusable structure that enables efficient Minimax distance calculations for any query point, improving productivity for multiple K-NN searches while the complexity is paid only once during the preprocessing phase.
Solution Approach 2:
The minimum spanning tree serves multiple functions: it captures the essential graph structure, enables Minimax distance computation, and supports efficient K-NN search. This universal structure replaces the need for separate computations for each K-NN query, making the system more efficient despite the initial complexity of MST construction.
4Ease of operation
If standard K-NN methods are used on large datasets, then ease of operation is maintained, but productivity decreases due to inability to capture underlying geometry
Solution Approach 1:
The patent transitions from standard Euclidean distance measurement to Minimax distance measurement based on graph structure. This dimensional change allows the method to capture the underlying geometry and topology of the data by considering path-based relationships in the graph, rather than relying solely on coordinate-based distances, thereby improving classification accuracy on complex datasets.
Data Source
AI summary
A system and method enable a set of dataset objects that are K-nearest neighbors (K-NN), based on their Minimax distances to a test object, to be identified without computing the all-pair Minimax distances directly. A pairwise distance between the test object and each dataset object is computed. Iteratively, one of the dataset objects is selected to add to a K-NN set until the K-NN set includes a predefined number of nearest neighbors. The selected dataset object at each iteration is the one for which there is no other unselected dataset object which has a smaller pairwise distance to any of a current subset of objects than the selected dataset object. The current subset of objects includes the test object and the dataset objects currently in the K-NN set. After the K-NN set is identified it may be output or used to generate other information, such as a test object label.