Reconstructing Generalized Staircase Polygons with Uniform Step Length

Visibility graph reconstruction, which asks us to construct a polygon that has a given visibility graph, is a fundamental problem with unknown complexity (although visibility graph recognition is known to be in PSPACE). We show that two classes of uniform step length polygons can be reconstructed efficiently by finding and removing rectangles formed between consecutive convex boundary vertices called tabs. In particular, we give an $O(n^2m)$-time reconstruction algorithm for orthogonally convex polygons, where $n$ and $m$ are the number of vertices and edges in the visibility graph, respectively. We further show that reconstructing a monotone chain of staircases (a histogram) is fixed-parameter tractable, when parameterized on the number of tabs, and polynomially solvable in time $O(n^2m)$ under reasonable alignment restrictions.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Guard placement for efficient pointin-polygon proofs

{eppstein, goodrich, nodari} (at) ics.uci.edu We consider the problem of placing a small number of angle guards inside a simple polygon P so as to provide efficient proofs that any given point is inside P. Each angle guard views an infinite wedge of the plane, and a point can prove membership in P if it is inside the wedges for a set of guards whose common intersection contains no points outside the polygon. This model leads to a broad class of new art gallery type problems, which we call “sculpture garden ” problems and for which we provide upper and lower bounds. In particular, we show there is a polygon P such that a “natural” angle-guard vertex placement cannot fully distinguish between points on the inside and outside of P (even if we place a guard at every vertex of P), which implies that Steinerpoint guards are sometimes necessary. More generally, we show that, for any polygon P, there is a set of n + 2(h − 1) angle guards that solve the sculpture garden problem for P, where h is the number of holes in P (so a simple polygon can be defined with n − 2 guards). In addition, we show that, for any orthogonal polygon P, the sculpture garden problem can be solved using n angle guards. We also give an 2 example of a class of simple (non-general-position) polygons that have sculpture garden solutions using O ( √ n) guards, and we show this bound is optimal to within a constant factor. Finally, while optimizing the number of guards solving a sculpture garden problem for a particular P is of unknown complexity, we show how to find in polynomial time a guard placement whose size is within a factor of 2 of the optimal number for any particular polygon

An Efficient Multiway Mergesort for GPU Architectures

Sorting is a primitive operation that is a building block for countless algorithms. As such, it is important to design sorting algorithms that approach peak performance on a range of hardware architectures. Graphics Processing Units (GPUs) are particularly attractive architectures as they provides massive parallelism and computing power. However, the intricacies of their compute and memory hierarchies make designing GPU-efficient algorithms challenging. In this work we present GPU Multiway Mergesort (MMS), a new GPU-efficient multiway mergesort algorithm. MMS employs a new partitioning technique that exposes the parallelism needed by modern GPU architectures. To the best of our knowledge, MMS is the first sorting algorithm for the GPU that is asymptotically optimal in terms of global memory accesses and that is completely free of shared memory bank conflicts. We realize an initial implementation of MMS, evaluate its performance on three modern GPU architectures, and compare it to competitive implementations available in state-of-the-art GPU libraries. Despite these implementations being highly optimized, MMS compares favorably, achieving performance improvements for most random inputs. Furthermore, unlike MMS, state-of-the-art algorithms are susceptible to bank conflicts. We find that for certain inputs that cause these algorithms to incur large numbers of bank conflicts, MMS can achieve up to a 37.6% speedup over its fastest competitor. Overall, even though its current implementation is not fully optimized, due to its efficient use of the memory hierarchy, MMS outperforms the fastest comparison-based sorting implementations available to date

A parallel priority queue with fast updates for GPU architectures

The high computational throughput of modern graphics processing units (GPUs) make them the de-facto architecture for high-performance computing applications. However, to achieve peak performance, GPUs require highly parallel workloads, as well as memory access patterns that exhibit good locality of reference. As a result, many state-of-the-art algorithms and data structures designed for GPUs sacrifice work-optimality to achieve the necessary parallelism. Furthermore, some abstract data types are avoided completely due to there being no corresponding data structure that performs well on the GPU. One such abstract data type is the priority queue. Many well-known algorithms rely on priority queue operations as a building block. While various priority queue structures have been developed that are parallel, cache-aware, or cache-oblivious, none has been shown to be efficient on GPUs. In this paper, we present the parBucketHeap, a parallel, cache-efficient data structure designed for modern GPU architectures that supports standard priority queue operations, as well as bulk update. We analyze the structure in several well-known computational models and show that it provides both optimal parallelism and is cache-efficient. We implement the parBucketHeap and, using it, we solve the single-source shortest path (SSSP) problem. Experimental results indicate that, for sufficiently large, dense graphs with high diameter, we out-perform current state-of-the-art SSSP algorithms on the GPU by up to a factor of 5. Unlike existing GPU SSSP algorithms, our approach is work-optimal and places significantly less load on the GPU, reducing power consumption