Polynomially Improved Efficiency for Fast Parallel Single-Source Lexicographic Depth-First Search, Breadth-First Search, and Topological-First Search

Abstract.

Although lexicographic (lex) variants of greedy algorithms are often P -complete, NC -algorithms are known for the following lex-search problems: lexicographic depth-first search (lex-dfs) for dags [12], [17], lexicographic breadth-first search (lex-bfs) for digraphs [12], [17], and lexicographic topological-first search (lex-tfs) for dags [12]. For the all-sources version of the problem for dense digraphs, the lex-dfs (lex-bfs, lex-tfs) in [12] is (within a log factor of) work-optimal with respect to the all-sources sequential solution that performs a dfs (bfs, tfs) from every vertex. By contrast, to solve the single-source lexicographic version on inputs of size n , all known NC -algorithms perform work that is at least an n factor away from the work performed by their sequential counterparts.

We present parallel algorithms that solve the single-source version of these lex-search problems in O(log ² n) time using M(n) processors on the EREW PRAM. (M(n) denotes the number of processors required to multiply two n\times n integer matrices in O(log  n) time and has O(n ^2.376 ) as tightest currently known bound.) They all offer a polynomial improvement in work-efficiency over that of their corresponding best previously known and close the gap between the requirements of the best known parallel algorithms for the lex and the nonlex versions of the problems.

Key to the efficiency of these algorithms is the novel idea of a lex-splitting tree and lex-conquer subgraphs of a dag G from source s . These structures provide a divide-and-conquer skeleton from which NC -algorithms for several lexicographic search problems emerge, in particular, an algorithm that places in the class NC the lex-dfs for reducible flow graphs—an interesting class of graphs which arise naturally in connection with code optimization and data flow analysis [4], [19]. A notable aspect of these algorithms is that they solve the lex-search problem instance at hand by efficiently transforming solutions of appropriate instances of (nonlex) path problems. This renders them potentially capable of transferring significant algorithmic advances—such as Driscoll et al.'s [14] single-source shortest paths algorithm and Ullman and Yannakakis' [34] transitive closure algorithm—from fundamental (nonlex) path problems to lex-search problems.