Breadth First Search (BFS) traversal is an archetype for many important graph problems. However, computing a BFS level decomposition for massive graphs was considered nonviable so far, because of the large number of I/Os it incurs. This paper presents the first experimental evaluation of recent external-memory BFS algorithms for general graphs. With our STXXL based implementations exploiting pipelining and diskparallelism, we were able to compute the BFS level decomposition of a web-crawl based graph of around 130 million nodes and 1.4 billion edges in less than 4 hours using single disk and 2.3 hours using 4 disks. We demonstrate that some rather simple external-memory algorithms perform significantly better (minutes as compared to hours) than internal-memory BFS, even if more than half of the input resides internally.

Given a set of points P ⊆ R 2 , a conflict-free coloring of P is an assignment of colors to points of P , such that each non-empty axis-parallel rectangle T in the plane contains a point whose color is distinct from all other points in P ∩ T . This notion has been the subject of recent interest, and is motivated by frequency assignment in wireless cellular networks: one naturally would like to minimize the number of frequencies (colors) assigned to bases stations (points), such that within any range (for instance, rectangle), there is no interference. We show that any set of n points in R 2 can be conflict-free colored withÕ(n .382+ ) colors in expected polynomial time, for any arbitrarily small > 0. This improves upon the previously known bound of O( p n log log n/ log n).

Breadth first search (BFS) traversal on massive graphs in external memory was considered non-viable until recently, because of the large number of I/Os it incurs. Ajwani et al. [3] showed that the randomized variant of the o(n) I/O algorithm of Mehlhorn and Meyer [24] (MM BFS) can compute the BFS level decomposition for large graphs (around a billion edges) in a few hours for small diameter graphs and a few days for large diameter graphs. We improve upon their implementation of this algorithm by reducing the overhead associated with each BFS level, thereby improving the results for large diameter graphs which are more difficult for BFS traversal in external memory. Also, we present the implementation of the deterministic variant of MM BFS and show that in most cases, it outperforms the randomized variant. The running time for BFS traversal is further improved with a heuristic that preserves the worst case guarantees of MM BFS. Together, they reduce the time for BFS on large diameter graphs from days shown in [3] to hours. In particular, on line graphs with random layout on disks, our implementation of the deterministic variant of MM BFS with the proposed heuristic is more than 75 times faster than the previous best result for the randomized variant of MM BFS in [3].

Given a set of points P ⊆ R 2 , a conflict-free coloring of P is an assignment of colors to points of P , such that each non-empty axis-parallel rectangle T in the plane contains a point whose color is distinct from all other points in P ∩ T. This notion has been the subject of recent interest, and is motivated by frequency assignment in wireless cellular networks: one naturally would like to minimize the number of frequencies (colors) assigned to bases stations (points), such that within any range (for instance, rectangle), there is no interference. We show that any set of n points in R 2 can be conflict-free colored with˜Owith˜ with˜O(n .382+) colors in expected polynomial time, for any arbitrarily small > 0. This improves upon the previously known bound of O(p n log log n/ log n).

We study ranked enumeration of join-query results according to very general orders defined by selective dioids. Our main contribution is a framework for ranked enumeration over a class of dynamic programming problems that generalizes seemingly different problems that had been studied in isolation. To this end, we extend classic algorithms that find the k -shortest paths in a weighted graph. For full conjunctive queries, including cyclic ones, our approach is optimal in terms of the time to return the top result and the delay between results. These optimality properties are derived for the widely used notion of data complexity, which treats query size as a constant. By performing a careful cost analysis, we are able to uncover a previously unknown tradeoff between two incomparable enumeration approaches: one has lower complexity when the number of returned results is small, the other when the number is very large. We theoretically and empirically demonstrate the superiority of our techniques over batch algorithms, which produce the full result and then sort it. Our technique is not only faster for returning the first few results, but on some inputs beats the batch algorithm even when all results are produced.

We study techniques for obtaining efficient algorithms for geometric problems on private-cache chip multiprocessors. We show how to obtain optimal algorithms for interval stabbing counting, 1-D range counting, weighted 2-D dominance counting, and for computing 3-D maxima, 2-D lower envelopes, and 2-D convex hulls. These results are obtained by analyzing adaptations of either the PEM merge sort algorithm or PRAM algorithms. For the second group of problems-orthogonal line segment intersection reporting, batched range reporting, and related problems-more effort is required. What distinguishes these problems from the ones in the previous group is the variable output size, which requires I/O-efficient load balancing strategies based on the contribution of the individual input elements to the output size. To obtain nearly optimal algorithms for these problems, we introduce a parallel distribution sweeping technique inspired by its sequential counterpart.

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We present a simple algorithm which maintains the topological order of a directed acyclic graph with n nodes under an online edge insertion sequence in O(n 2.75 ) time, independent of the number of edges m inserted. For dense DAGs, this is an improvement over the previous best result of O(min{m 3 2 log n, m 3 2 + n 2 log n}) by Katriel and Bodlaender. We also provide an empirical comparison of our algorithm with other algorithms for online topological sorting. Our implementation outperforms them on certain hard instances while it is still competitive on random edge insertion sequences leading to complete DAGs.