We present a parallel algorithm for permanent mod 2^k of a matrix of univariate integer polynomials. It places the problem in ParityL subset of NC^2. This extends the techniques of [Valiant], [Braverman, Kulkarni, Roy] and [Bj\"orklund, Husfeldt], and yields a (randomized) parallel algorithm for shortest 2-disjoint paths improving upon the recent result from (randomized) polynomial time. We also recognize the disjoint paths problem as a special case of finding disjoint cycles, and present (randomized) parallel algorithms for finding a shortest cycle and shortest 2-disjoint cycles passing through any given fixed number of vertices or edges.
翻译:我们为单体整形多元分子矩阵的永久模式 2Qk 提出了一个平行算法。 它将问题放在 NC2 的对等L子集中。 它扩大了[Valit]、[Braverman、Kulkarni、Roy]和[Bj\'orklund、Husfeldt]的技术,并产生了一种(随机化的)平行算法,用于在(随机化的)多元时间最近的结果基础上改善的两条最短的平行路径。 我们还认识到,脱节路径问题是找到脱节周期的特殊案例,目前(随机化)平行算法用于寻找一个最短周期和最短的两条断交周期,通过任何固定数量的脊椎或边缘。