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A Fast Algorithm for All-Pairs-Shortest-Paths Suitable for Neural Networks.

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  • 1Division of Biology and Biological Engineering, California Institute of Technology, Pasadena, CA 91125, U.S.A. zjing@caltech.edu.

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This study introduces a novel matrix inversion method to efficiently calculate shortest path distances in graphs. The approach offers a computationally faster alternative for dense graphs and extends to weighted networks.

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Area of Science:

  • Graph Theory
  • Computational Mathematics
  • Network Science

Background:

  • Finding shortest paths in directed graphs is a fundamental problem in computer science and network analysis.
  • Existing algorithms can be computationally intensive, especially for large or dense graphs.

Purpose of the Study:

  • To present a new method for calculating all-pairs-shortest-path distances using matrix inversion.
  • To demonstrate the method's efficiency and applicability to various graph types, including weighted graphs.

Main Methods:

  • Utilizing matrix inversion of (I - γA), where A is the adjacency matrix and γ is a small constant.
  • Deriving graph-theoretic bounds for the optimal value of γ.
  • Numerically evaluating the method across different graph structures.

Main Results:

  • Shortest path distances can be computed as Dij = ceil(logγ[(I-γA)-1]ij).
  • The method proves computationally faster than alternatives for many dense graphs.
  • The approach demonstrates local accuracy and extends to graphs with positive edge weights.

Conclusions:

  • Matrix inversion provides an efficient and versatile method for solving the all-pairs-shortest-path problem.
  • This technique offers a foundation for developing neural network solutions for shortest path computations.