Advances and challenges in distance geometry: A systematic review in terms of theory, applications, algorithmic efficiency, and solution quality

Abstract

Aim This systematic review investigates advances and challenges in distance geometry (DG), that focusing on algorithmic innovations, theoretical insights, and applications in structural biology and computational chemistry, robotics and motion planning, wireless networking and sensor infrastructure, complex network analysis, artificial intelligence, NLP and machine learning, structural engineering and optimization, theoretical physics and quantum computing, combinatorial optimization, visualization and topological data analysis. Method A systematic literature search of studies published between 2016 and 2025 identified 256 papers (164 via targeted queries, 72 through citation chaining). After relevance ranking, 236 were retained, and finally, 75 papers were classified as highly significant eligible studies that met the predefined relevance and quality criteria. Results The reviewed literature demonstrates continued progress in cycle-based algorithms, Riemannian optimization, combinatorial methods, and other mathematical-programming approaches. Several studies report improvements in reconstruction accuracy, convergence, or computational performance for specific problem classes. Geometric-algebra methods provide alternative mathematical formulations for selected distance-geometry problems, although their applicability and validation vary across studies. Quantum approaches remain comparatively less mature: most identified contributions are theoretical, exploratory, or prototype-based, with limited evidence regarding large-scale implementation, noise robustness, or superiority over established classical methods. Persistent challenges include scalability, incomplete and noisy distance data, inconsistent benchmarking, and limited reproducibility. Conclusion Distance geometry is advancing through the interaction of mathematical theory, algorithmic innovation, and interdisciplinary applications. However, the maturity and empirical support of different approaches vary considerably. Future research should emphasize standardized benchmarks, transparent baseline comparisons, systematic robustness testing, reproducible implementations, and experimentally validated methods for large-scale problems.

The authors' abstract, as published at the source. Array, 2026 · DOI ↗