A Frequency-first Heuristic for Shortest Linear Programs

The shortest linear program has been proved to be a NP-hard problem. In order to obtain the better approximate solution, a frequency-first heuristic method is proposed, which can optimize the number of XOR gates required by linear components while ensuring the stability of the algorithm. Firstly, we combine the pre-emptive gate strategy with frequency-first (FQCY) in the selection stage to reduce the increase of time complexity caused by exhaustive search, so that the high-density matrix can obtain the optimal result within a reasonable time. Secondly, minimization of vector and appropriate randomization are added to deal with the tie, so as to give full play to the advantages of cancellation-allowed circuit and increase the possibility of obtaining the optimal solution. Finally, compared with Paar, BP, RNBP, RSDF algorithms on random matrices of various sizes and densities, it is proved that the probability of obtaining the optimal solution of the proposed algorithm in circuit depth is more than 30% higher than RNBP and RSDF. In terms of the number of XOR gates, especially for larger matrix, the probability of obtaining the optimal solution increases by more than 10%. The stability of the optimal circuit generated by this algorithm is about 90%.