arXiv subjects
Publications and source records attributed to Anil Maheshwari.
This survey gives a brief overview of theoretically and practically relevant algorithms to compute geodesic paths and distances on three-dimensional surfaces. The survey focuses on polyhedral three-dimensional surfaces.
We propose to design data structures called succinct geometric indexes of negligible space (more precisely, o(n) bits) that, by taking advantage of the n points in the data set permuted and stored elsewhere as a sequence, to support geometric queries in optimal time. Our first and main result is a succinct geometric index that can answer point location queries, a fundamental problem in computational geometry, on planar triangulations in O(lg n) time. We also design three variants of this index. The first supports point location using $\lg n + 2\sqrt{\lg n} + O(\lg^{1/4} n)$ point-line comparisons. The second supports point location in o(lg n) time when the coordinates are integers bounded by U. The last variant can answer point location in O(H+1) expected time, where H is the entropy of the query distribution. These results match the query efficiency of previous point location structures that use O(n) words or O(n lg n) bits, while saving drastic amounts of space. We then generalize our succinct geometric index to planar subdivisions, and design indexes for other types of queries. Finally, we apply our techniques to design the first implicit data structures that support point location in $O(\lg^2 n)$ time.
A path from s to t on a polyhedral terrain is descending if the height of a point p never increases while we move p along the path from s to t. No efficient algorithm is known to find a shortest descending path (SDP) from s to t in a polyhedral terrain. We give two approximation algorithms (more precisely, FPTASs) that solve the SDP problem on general terrains. Both algorithms are simple, robust and easy to implement.
We address the following problem: Given a complete $k$-partite geometric graph $K$ whose vertex set is a set of $n$ points in $\mathbb{R}^d$, compute a spanner of $K$ that has a ``small'' stretch factor and ``few'' edges. We present two algorithms for this problem. The first algorithm computes a $(5+ε)$-spanner of $K$ with O(n) edges in $O(n \log n)$ time. The second algorithm computes a $(3+ε)$-spanner of $K$ with $O(n \log n)$ edges in $O(n \log n)$ time. The latter result is optimal: We show that for any $2 \leq k \leq n - Θ(\sqrt{n \log n})$, spanners with $O(n \log n)$ edges and stretch factor less than 3 do not exist for all complete $k$-partite geometric graphs.
Given an integer $k \geq 2$, we consider the problem of computing the smallest real number $t(k)$ such that for each set $P$ of points in the plane, there exists a $t(k)$-spanner for $P$ that has chromatic number at most $k$. We prove that $t(2) = 3$, $t(3) = 2$, $t(4) = \sqrt{2}$, and give upper and lower bounds on $t(k)$ for $k>4$. We also show that for any $ε>0$, there exists a $(1+ε)t(k)$-spanner for $P$ that has $O(|P|)$ edges and chromatic number at most $k$. Finally, we consider an on-line variant of the problem where the points of $P$ are given one after another, and the color of a point must be assigned at the moment the point is given. In this setting, we prove that $t(2) = 3$, $t(3) = 1+ \sqrt{3}$, $t(4) = 1+ \sqrt{2}$, and give upper and lower bounds on $t(k)$ for $k>4$.