Join-Reachability Problems in Directed Graphs

For a given collection G of directed graphs we define the join-reachability graph of G, denoted by J(G), as the directed graph that, for any pair of vertices a and b, contains a path from a to b if and only if such a path exists in all graphs of G. Our goal is to compute an efficient representation of J(G). In particular, we consider two versions of this problem. In the explicit version we wish to construct the smallest join-reachability graph for G. In the implicit version we wish to build an efficient data structure (in terms of space and query time) such that we can report fast the set of vertices that reach a query vertex in all graphs of G. This problem is related to the well-studied reachability problem and is motivated by emerging applications of graph-structured databases and graph algorithms. We consider the construction of join-reachability structures for two graphs and develop techniques that can be applied to both the explicit and the implicit problem. First we present optimal and near-optimal structures for paths and trees. Then, based on these results, we provide efficient structures for planar graphs and general directed graphs

Fast Parallel Operations on Search Trees

Using (a,b)-trees as an example, we show how to perform a parallel split with logarithmic latency and parallel join, bulk updates, intersection, union (or merge), and (symmetric) set difference with logarithmic latency and with information theoretically optimal work. We present both asymptotically optimal solutions and simplified versions that perform well in practice - they are several times faster than previous implementations

Space-Efficient DFS and Applications: Simpler, Leaner, Faster

The problem of space-efficient depth-first search (DFS) is reconsidered. A particularly simple and fast algorithm is presented that, on a directed or undirected input graph $G=(V,E)$ with $n$ vertices and $m$ edges, carries out a DFS in $O(n+m)$ time with $n+\sum_{v\in V_{\ge 3}}\lceil{\log_2(d_v-1)}\rceil +O(\log n)\le n+m+O(\log n)$ bits of working memory, where $d_v$ is the (total) degree of $v$, for each $v\in V$, and $V_{\ge 3}=\{v\in V\mid d_v\ge 3\}$. A slightly more complicated variant of the algorithm works in the same time with at most $n+({4/5})m+O(\log n)$ bits. It is also shown that a DFS can be carried out in a graph with $n$ vertices and $m$ edges in $O(n+m\log^*\! n)$ time with $O(n)$ bits or in $O(n+m)$ time with either $O(n\log\log(4+{m/n}))$ bits or, for arbitrary integer $k\ge 1$, $O(n\log^{(k)}\! n)$ bits. These results among them subsume or improve most earlier results on space-efficient DFS. Some of the new time and space bounds are shown to extend to applications of DFS such as the computation of cut vertices, bridges, biconnected components and 2-edge-connected components in undirected graphs

Parallel Working-Set Search Structures

In this paper we present two versions of a parallel working-set map on p processors that supports searches, insertions and deletions. In both versions, the total work of all operations when the map has size at least p is bounded by the working-set bound, i.e., the cost of an item depends on how recently it was accessed (for some linearization): accessing an item in the map with recency r takes O(1+log r) work. In the simpler version each map operation has O((log p)^2+log n) span (where n is the maximum size of the map). In the pipelined version each map operation on an item with recency r has O((log p)^2+log r) span. (Operations in parallel may have overlapping span; span is additive only for operations in sequence.) Both data structures are designed to be used by a dynamic multithreading parallel program that at each step executes a unit-time instruction or makes a data structure call. To achieve the stated bounds, the pipelined data structure requires a weak-priority scheduler, which supports a limited form of 2-level prioritization. At the end we explain how the results translate to practical implementations using work-stealing schedulers. To the best of our knowledge, this is the first parallel implementation of a self-adjusting search structure where the cost of an operation adapts to the access sequence. A corollary of the working-set bound is that it achieves work static optimality: the total work is bounded by the access costs in an optimal static search tree.Comment: Authors' version of a paper accepted to SPAA 201