Computational geometry

: Current systems for graph computation require a distributed computing cluster to handle very large real-world problems, such as analysis on social networks or the web graph. While distributed computational resources have become more accessible developing distributed graph algorithms still remains challenging, especially to non-experts. In this work, we present GraphChi, a disk-based system for computing efficiently on graphs with billions of edges. By using a well-known method to break large graphs into small parts, and a novel Parallel Sliding Windows algorithm, GraphChi is able to execute several advanced data mining, graph mining and machine learning algorithms on very large graphs, using just a single consumer-level computer. We show, through experiments and theoretical analysis, that GraphChi performs well on both SSDs and rotational hard drives. We build on the basis of Parallel Sliding Windows to propose a new data structure Partitioned Adjacency Lists, which we use to design an online graph database GraphChi-DB.We demonstrate that, on a single PC, GraphChi-DB can process over one hundred thousand graph updates per second, while simultaneously performing computation. GraphChi-DB compares favorably to existing graph databases, particularly on data that is much larger than the available memory. We evaluate our work both experimentally and theoretically. Based on the Parallel Sliding Windows algorithm, we propose new I/O efficient algorithms for solving fundamental graph problems. We also propose a novel algorithm for simulating billions of random walks in parallel on a single computer. By repeating experiments reported for existing distributed systems we show that with only fraction of the resources, GraphChi can solve the same problems in a very reasonable time. Our work makes large-scale graph computation available to anyone with a modern PC.

Large-scale Graph Computation on Just a PC

Current systems for graph computation require a distributed computing cluster to handle very large real-world problems, such as analysis on social networks or the web graph. While distributed computational resources have become more accessible, developing distributed graph algorithms still remains challenging, especially to non-experts.In this work, we present GraphChi, a disk-based system for computing efficiently on graphs with billions of edges. By using a well-known method to break large graphs into small parts, and a novel parallel sliding windows method, GraphChi is able to execute several advanced data mining, graph mining, and machine learning algorithms on very large graphs, using just a single consumer-level computer. We further extend GraphChi to support graphs that evolve over time, and demonstrate that, on a single computer, GraphChi can process over one hundred thousand graph updates per second, while simultaneously performing computation. We show, through experiments and theoretical analysis, that GraphChi performs well on both SSDs and rotational hard drives.By repeating experiments reported for existing distributed systems, we show that, with only fraction of the resources, GraphChi can solve the same problems in very reasonable time. Our work makes large-scale graph computation available to anyone with a modern PC.

/pdf/graphchi-large-scale-graph-computation-on-just-a-pc-c5qonwv7c3.pdf

GraphChi: large-scale graph computation on just a PC

We propose shortest path algorithms that use A* search in combination with a new graph-theoretic lower-bounding technique based on landmarks and the triangle inequality. Our algorithms compute optimal shortest paths and work on any directed graph. We give experimental results showing that the most efficient of our new algorithms outperforms previous algorithms, in particular A* search with Euclidean bounds, by a wide margin on road networks and on some synthetic problem families.

/pdf/computing-the-shortest-path-a-search-meets-graph-theory-1bvu8tebrp.pdf

Computing the shortest path: A search meets graph theory

Journal of the ACM

In this paper we determine the computational complexity of the dynamic convex hull problem in the planar case. We present a data structure that maintains a finite set of n points in the plane under insertion and deletion of points in amortized O(log n) time per operation. The space usage of the data structure is O(n). The data structure supports extreme point queries in a given direction, tangent queries through a given point, and queries for the neighboring points on the convex hull in O(log n) time. The extreme point queries can be used to decide whether or not a given line intersects the convex hull, and the tangent queries to determine whether a given point is inside the convex hull. We give a lower bound on the amortized asymptotic time complexity that matches the performance of this data structure.

https://wwwmayr.in.tum.de/personen/jacob/Publications/dpch.pdf

Dynamic planar convex hull

In this article, we determine the amortized computational complexity of the planar dynamic convex hull problem by querying. 
We present a data structure that maintains a set of n points in the plane under the insertion and deletion of points in amortized O(log n) time per operation. The space usage of the data structure is O(n). The data structure supports extreme point queries in a given direction, tangent queries through a given point, and queries for the neighboring points on the convex hull in O(log n) time. The extreme point queries can be used to decide whether or not a given line intersects the convex hull, and the tangent queries to determine whether a given point is inside the convex hull. 
We give a lower bound on the amortized asymptotic time complexity that matches the performance of this data structure.

Dynamic Planar Convex Hull

Given an alphabet $\Sigma$, a (directed) graph G whose edges are weighted and $\Sigma$-labeled, and a formal language $L\subseteq\Sigma^*$, the formal-language-constrained shortest/simple path problem consists of finding a shortest (simple) path p in G complying with the additional constraint that l(p) \in L$. Here l(p) denotes the unique word obtained by concatenating the $\Sigma$-labels of the edges along the path p. The main contributions of this paper include the following: 
We show that the formal-language-constrained shortest path problem is solvable efficiently in polynomial time when L is restricted to be a context-free language (CFL). When L is specified as a regular language we provide algorithms with improved space and time bounds. In contrast, we show that the problem of finding a simple path between a source and a given destination is NP-hard, even when L is restricted to fixed simple regular languages and to very simple classes of graphs (e.g., complete grids). For the class of treewidth-bounded graphs, we show that (i) the problem of finding a regular-language-constrained simple path between source and destination is solvable in polynomial time and (ii) the extension to finding CFL-constrained simple paths is NP-complete. 
Our results extend the previous results in [SIAM J. Comput., 24 (1995), pp. 1235--1258; Proceedings of the 76th Annual Meeting of the Transportation Research Board, 1997; and Proceedings of the 9th ACM SIGACT-SIGMOD-SIGART Symposium on Database Systems, 1990, pp. 230--242]. Several additional extensions and applications of our results in the context of transportation problems are presented. For instance, as a corollary of our results, we obtain a polynomial-time algorithm for the best k-similar path problem studied in Proceedings of the 76th Annual Meeting of the Transportation Reasearch Board, 1997]. The previous best algorithm was given by [ Proceedings of the 76th Annual Meeting of the Transportation Research Board, 1997] and takes exponential time in the worst case.

/pdf/formal-language-constrained-path-problems-4jryisx38h.pdf

Formal-Language-Constrained Path Problems

We propose a version of cache oblivious search trees which is simpler than the previous proposal of Bender, Demaine and Farach-Colton and has the same complexity bounds. In particular, our data structure avoids the use of weight balanced B-trees, and can be implemented as just a single array of data elements, without the use of pointers. The structure also improves space utilization.For storing n elements, our proposal uses (1 + e)n times the element size of memory, and performs searches in worst case O(logBn) memory transfers, updates in amortized O((log2n)/(eB)) memory transfers, and range queries in worst case O(logBn + k/B) memory transfers, where k is the size of the output.The basic idea of our data structure is to maintain a dynamic binary tree of height log n+O(1) using existing methods, embed this tree in a static binary tree, which in turn is embedded in an array in a cache oblivious fashion, using the van Emde Boas layout of Prokop.We also investigate the practicality of cache obliviousness in the area of search trees, by providing an empirical comparison of different methods for laying out a search tree in memory.

https://www.cs.au.dk/~gerth/papers/soda02.pdf

Cache oblivious search trees via binary trees of small height

http://www.cs.au.dk/~gerth/papers/atmos03.pdf

Riko Jacob

Papers

Dynamic planar convex hull

Dynamic Planar Convex Hull

Formal-Language-Constrained Path Problems

Cache oblivious search trees via binary trees of small height

Time-dependent networks as models to achieve fast exact time-table queries