Lightning graph matching

Binrui Shen, Qiang Niu, Shengxin Zhu

Abstract: The spectral matching algorithm is a classic method for finding correspondences between two graphs, a fundamental task in pattern recognition. It has a time complexity of O(n4)<math><mrow is="true"><mi mathvariant="script" is="true">O</mi><mrow is="true"><mo is="true">(</mo><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mn is="true">4</mn></mrow></msup><mo is="true">)</mo></mrow></mrow></math> and a space complexity of O(n4)<math><mrow is="true"><mi mathvariant="script" is="true">O</mi><mrow is="true"><mo is="true">(</mo><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mn is="true">4</mn></mrow></msup><mo is="true">)</mo></mrow></mrow></math>, where n<math><mi is="true">n</mi></math> is the number of nodes. However, such complexity limits its applicability to large-scale graph matching tasks. This paper proposes a redesign of the algorithm by transforming the graph matching problem into a one-dimensional linear assignment problem. This transformation enables efficient solving by sorting two n×1<math><mrow is="true"><mi is="true">n</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">×</mo><mn is="true">1</mn></mrow></math> vectors. The resulting algorithm is named the Lightning Spectral Assignment Method (LiSA), which enjoys a complexity of O(n2)<math><mrow is="true"><mi mathvariant="script" is="true">O</mi><mrow is="true"><mo is="true">(</mo><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msup><mo is="true">)</mo></mrow></mrow></math>. Numerical experiments demonstrate the efficiency of LiSA, supported by theoretical analysis.