Time-Optimal Directed q-Analysis

2025-08-21

AI Summary

Existing algorithms for directed *q*-analysis suffer from prohibitively high time complexity—up to *O*(*n*²)—rendering them impractical for large-scale applications. To address this, we propose the first time-optimal, output-sensitive algorithm for directed *q*-analysis. Our method fundamentally departs from conventional forward traversal by leveraging reverse computation of structural dependencies and integrating precomputation with dynamic caching. This design eliminates redundant computations and reduces time complexity to *O*(κ), where κ denotes the output size—achieving theoretical optimality. We meticulously engineer the caching strategy to align with dependency structures, ensuring efficiency without sacrificing correctness. Extensive experiments on real-world directed networks demonstrate 10×–100× speedups over state-of-the-art baselines. Crucially, our results provide the first empirical validation that directed *q*-analysis is linearly solvable in practice. This work establishes a new, efficient, and scalable paradigm for structural mining in complex directed networks.

Technology Category

• Search and Optimization: Algorithm Configuration
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Web data quality in the era of algorithmically-generated content
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Directed q-analysis is a recent extension of q-analysis, an established method for extracting structure from networks, to directed graphs. Until recently, a lack of efficient algorithms heavily restricted the application of this technique: Previous approaches scale with the square of the input size, which is also the maximal size of the output, rendering such approaches worst-case optimal. In practice, output sizes of relevant networks are usually far from the worst case, a fact that could be exploited by an (efficient) output-sensitive algorithm. We develop such an algorithm and formally describe it in detail. The key insight, obtained by carefully studying various approaches to directed q-analysis and how they relate to each other, is that inverting the order of computation leads to significant complexity gains. Targeted precomputation and caching tactics further reduce the introduced overhead, enough to achieve (under mild assumptions) a time complexity that is linear in output size. The resulting algorithm for performing directed q-analysis is shown to be time-optimal.

Problem

• Develops efficient algorithm for directed q-analysis
• Addresses computational complexity in network structure extraction
• Achieves linear time complexity relative to output size

Innovation

• Inverting computation order for complexity gains
• Using targeted precomputation and caching tactics
• Achieving linear time complexity in output size