TRULY SUBCUBIC ALGORITHMS FOR LANGUAGE EDIT DISTANCE AND RNA FOLDING VIA FAST BOUNDED-DIFFERENCE MIN-PLUS PRODUCT
TRULY SUBCUBIC ALGORITHMS FOR LANGUAGE EDIT DISTANCE AND RNA FOLDING VIA FAST BOUNDED-DIFFERENCE MIN-PLUS PRODUCT
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DOI：
10.1137/17m112720x
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发表时间：
2019-01-01
影响因子：
1.6
通讯作者：
Williams, Virginia Vassilevska
中科院分区：
文献类型：
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作者：
Bringmann, Karl;Grandoni, Fabrizio;Williams, Virginia Vassilevska
无论是（最小值， +） - 两个N X矩阵的乘积（即Epsilon> 0）的（即n（3- epsilon））的（即o（n（3- epsilon）））的（即o（n（3- epsilon）））是一个主要的开放问题；特别是，由于它等效于N-Vertex图中著名的全对最短的路径问题（APSP）。已知（最小值， +）的某些限制是特殊类型的矩阵的产品，可以接收真正的亚地带算法，每种算法都会产生一种特殊的APSP案例，可以更快地解决。在本文中，我们考虑了一个新的，不同和有力的限制，其中所有矩阵条目都是整数，一个矩阵可以是任意的，只要另一个矩阵在其列或行中具有“有界差异”，即任何两个连续的两个连续的差异条目只有少量不同。我们获得了该有限差异（min， +） - 产品（回答Chan和Lewenstein的开放问题）的第一个真正的亚采算法。我们的新算法，加上加强英勇的方法来解决无上下文的语法解析，以矩阵乘法来解决以下问题的第一个真正的亚基算法：语言编辑距离（解析社区中的主要问题），RNA折叠（生物信息学中的一个主要问题）和最佳的堆栈生成（回答了Tarjan的开放问题）。
It is a major open problem whether the (min, +)-product of two n x n matrices has a truly subcubic (i.e., O(n(3-epsilon)) for epsilon > 0) time algorithm; in particular, since it is equivalent to the famous all-pairs-shortest-paths problem (APSP) in n-vertex graphs. Some restrictions of the (min, +)-product to special types of matrices are known to admit truly subcubic algorithms, each giving rise to a special case of APSP that can be solved faster. In this paper we consider a new, different, and powerful restriction in which all matrix entries are integers and one matrix can be arbitrary, as long as the other matrix has "bounded differences" in either its columns or rows, i.e., any two consecutive entries differ by only a small amount. We obtain the first truly subcubic algorithm for this bounded-difference (min, +)-product (answering an open problem of Chan and Lewenstein). Our new algorithm, combined with a strengthening of an approach of Valiant for solving context-free grammar parsing with matrix multiplication, yields the first truly subcubic algorithms for the following problems: language edit distance (a major problem in the parsing community), RNA folding (a major problem in bioinformatics), and optimum stack generation (answering an open problem of Tarjan).