Holographic Algorithms

Leslie Valiant recently proposed a theory of holographic algorithms. These novel algorithms achieve exponential speed-ups for certain computational problems compared to naive algorithms for the same problems. The methodology uses Pfaffians and (planar) perfect matchings as basic computational primitives, and attempts to create exponential cancellations in computation. In this article we survey this new theory of matchgate computations and holographic algorithms. Key words: Theoretical Computer Science, Computational Complexity Theory, Perfect Matchings, Pfaffians, Matchgates, Matchcircuits, Matchgrids, Signatures, Holographic Algorithms. 1 Some Historical Background There have always been two major strands of mathematical thought since antiquity and across civilizations: Structural Theory and Computation, as exemplified by Euclid's Elements and Diophantus' Arithmetica. Structural Theory prizes the formulation and proof of structural theorems, while Computation seeks efficien...

• Type: Journal
• Year: 2008
• Where: SIAMCOMP
• Authors: Leslie G. Valiant
• Keywords: Algorithms, Holographic Algorithms, SIAMCOMP 2008, Theory