Laura Toma

Efficient Flow Computation on Massive Grid Terrain Datasets

As detailed terrain data becomes available, GIS terrain applications target larger geographic areas at finer resolutions. Processing the massive datasets involved in such applications presents significant challenges to GIS systems and demands algorithms that are optimized for both data movement and computation. In this paper we present efficient algorithms for flow routing on massive grid terrain datasets, extending our previous work on flow accumulation. Our algorithms are developed in the framework of external memory algorithms and use I/O-techniques to achieve efficiency. We have implemented the algorithms in the Terraflow system, which is the first comprehensive terrain flow software system designed and optimized for massive data. We compare the performance of Terraflow with that of state-of-the-art commercial and open-source GIS systems. On large terrains, Terraflow outperforms existing systems by a factor of 2 to 1,000, and is capable of solving problems no system was previously able to solve.

On external-memory MST, SSSP and multi-way planar graph separation

Recently external memory graph problems have received considerable attention because massive graphs arise naturally in many applications involving massive data sets. Even though a large number of I/O-efficient graph algorithms have been developed, a number of fundamental problems still remain open.The results in this paper fall in two main classes. First we develop an improved algorithm for the problem of computing a minimum spanning tree (MST) of a general undirected graph. Second we show that on planar undirected graphs the problems of computing a multi-way graph separation and single source shortest paths (SSSP) can be reduced I/O-efficiently to planar breadth-first search (BFS). Since BFS can be trivially reduced to SSSP by assigning all edges weight one, it follows that in external memory planar BFS, SSSP, and multi-way separation are equivalent. That is, if any of these problems can be solved I/O-efficiently, then all of them can be solved I/O-efficiently in the same bound. Our planar graph results have subsequently been used to obtain I/O-efficient algorithms for all fundamental problems on planar undirected graphs.

Computing visibility on terrains in external memory

Given an arbitrary viewpoint v and a terrain, the visibility map or viewshed of v is the set of points in the terrain that are visible from v. In this article we consider the problem of computing the viewshed of a point on a very large grid terrain in external memory. We describe algorithms for this problem in the cache-aware and cache-oblivious models, together with an implementation and an experimental evaluation. Our algorithms are a novel application of the distribution sweeping technique and use O(sort(n)) I/Os, where sort(n) is the complexity of sorting n items of data in the I/O-model. The experimental results demonstrate that our algorithm scales up and performs significantly better than the traditional internal-memory plane sweep algorithm and can compute visibility for terrains of 1.1 billion points in less than 4 hours on a low-cost machine compared to more than 32 hours with the internal-memory algorithm.

On External-Memory MST, SSSP, and Multi-way Planar Graph Separation

Recently external memory graph algorithms have received considerable attention because massive graphs arise naturally in many applications involving massive data sets. Even though a large number of I/O-efficient graph algorithms have been developed, a number of fundamental problems still remain open. In this paper we develop an improved algorithm for the problem of computing a minimum spanning tree of a general graph, as well as new algorithms for the single source shortest paths and the multi-way graph separation problems on planar graphs.

I/O-efficient topological sorting of planar DAGs

We present algorithms that solve a number of fundamental problems on planar directed graphs (planar digraphs) in O((N)) I/Os, where (N) is the number of I/Os needed to sort N elements. The problems we consider are breadth-first search, the single-source shortest path problem, computing a directed ear decomposition of a strongly connected planar digraph, computing an open directed ear decomposition of a strongly connected biconnected planar digraph, and topologically sorting a planar directed acyclic graph.

Flow computation on massive grids

As detailed terrain becomes available, GIS applications target larger geographic areas at finer resolutions. Processing the massive data presents significant challenges to GIS systems and demands algorithms that are optimized for both data movement and computation.In this paper we develop effcient algorithms for flow routing on massive terrains, extending our previous work on flow accumulation. Our implementations of these algorithms constitute the first comprehensive terrain flow software system designed and optimized for massive data. We compare the performance of our system, called TERRAFLOW, with that of state of the art commercial and open-source GIS systems. On large terrains, TERRAFLOW outpreforms existing systems by a factor of 2 to 1000, and is capable of solving problems of a scope and scale that are impractical with previous algorithms.

The complexity of flow on fat terrains and its i/o-efficient computation

We study the complexity and the i/o-efficient computation of flow on triangulated terrains. We present an acyclic graph, the descent graph, that enables us to trace flow paths in triangulations i/o-efficiently. We use the descent graph to obtain i/o-efficient algorithms for computing river networks and watershed-area maps in O(Sort(d+r))i/os, where r is the complexity of the river network and d of the descent graph. Furthermore we describe a data structure based on the subdivision of the terrain induced by the edges of the triangulation and paths of steepest ascent and descent from its vertices. This data structure can be used to report the boundary of the watershed of a query point q or the flow path from q in O(l(s)+Scan(k))i/os, where s is the complexity of the subdivision underlying the data structure, l(s) is the number of i/os used for planar point location in this subdivision, and k is the size of the reported output. On @a-fat terrains, that is, triangulated terrains where the minimum angle of any triangle is bounded from below by @a, we show that the worst-case complexity of the descent graph and of any path of steepest descent is O(n/@a^2), where n is the number of triangles in the terrain. The worst-case complexity of the river network and the above-mentioned data structure on such terrains is O(n^2/@a^2). When @a is a positive constant this improves the corresponding bounds for arbitrary terrains by a linear factor. We prove that similar bounds cannot be proven for Delaunay triangulations: these can have river networks of complexity @Q(n^3).

Terracost: Computing least-cost-path surfaces for massive grid terrains

This paper addresses the problem of computing least-cost-path surfaces for massive grid terrains. Consider a grid terrain T and let C be a cost grid for T such that every point in C stores a value that represents the cost of traversing the corresponding point in T. Given C and a set of sources S ∈ T, a least-cost-path grid Δ for T is a grid such that every point in Δ represents the distance to the source in S that can be reached with minimal cost. We present a scalable approach to computing least-cost-path grids. Our algorithm, terracost, is derived from our previous work on I/O-efficient shortest paths on grids and uses O(sort(n)) I/Os, where sort(n) is the complexity of sorting n items of data in the I/O-model of Aggarwal and Vitter. We present the design, the analysis, and an experimental study of terracost. An added benefit of the algorithm underlying terracost is that it naturally lends itself to parallelization. We have implemented terracost in a distributed environment using our cluster management tool and report on experiments that show that it obtains speedup near-linear with the size of the cluster. To the best of our knowledge, this is the first experimental evaluation of a multiple-source least-cost-path algorithm in the external memory setting.

External data structures for shortest path queries on planar digraphs

In this paper we present space-query trade-offs for external memory data structures that answer shortest path queries on planar directed graphs. For any

Simplified External Memory Algorithms for Planar DAGs

S = {\it \Omega}(N^{1 + \epsilon}