Given a set $\mathcal{P}$ of $h$ pairwise disjoint simple polygonal obstacles in $\mathbb{R}^2$ defined with $n$ vertices, we compute a sketch $Ω$ of $\mathcal{P}$ whose size is independent of $n$, depending only on $h$ and the input parameter $ε$. We utilize $Ω$ to compute a $(1+ε)$-approximate geodesic shortest path between the two given points in $O(n + h((\lg{n}) + (\lg{h})^{1+δ} + (\frac{1}ε\lg{\frac{h}ε})))$ time. Here, $ε$ is a user parameter, and $δ$ is a small positive constant (resulting from the time for triangulating the free space of $\cal P$ using the algorithm in \cite{journals/ijcga/Bar-YehudaC94}). Moreover, we devise a $(2+ε)$-approximation algorithm to answer two-point Euclidean distance queries for the case of convex polygonal obstacles.

Let P be a set of h pairwise-disjoint polygonal obstacles with a total of n vertices in the plane. In this paper, we consider the problem of building a data structure that can quickly compute an L1 shortest obstacle-avoiding path between any two query points s and t. We build a data structure of size O(n + h2 · log h · 4√log h) in O(n + h2 · log2 h · 4√log h) time that answers each query in O(log n + k) time, where k is the number of edges of the output path. Note that n + h2 · log2 h · 4√log h = O(n+h2+ϵ) for any constant ϵ > 0. We also extend our techniques to the weighted rectilinear version in which the "obstacles" of P are rectilinear regions with "weights" and allow L1 paths to travel through them with weighted costs. Our algorithm answers each query in O(log n + k) time with a data structure of size O(n2 · log n · 4√log n) that is built in O(n2 · log2 n · 4√log n) time.

Irregular programs are programs organized around pointer-based data structures such as trees and graphs. Recent investigations by the Galois project have shown that many irregular programs have a generalized form of data-parallelism called amorphous data-parallelism. However, in many programs, amorphous data-parallelism cannot be uncovered using static techniques, and its exploitation requires runtime strategies such as optimistic parallel execution. This raises a natural question: how much amorphous data-parallelism actually exists in irregular programs?