Optimal dynamic vertical ray shooting in rectilinear planar subdivisions
ACM Transactions on Algorithms (TALG)
Optimal in-place and cache-oblivious algorithms for 3-d convex hulls and 2-d segment intersection
Computational Geometry: Theory and Applications
Testing 2-vertex connectivity and computing pairs of vertex-disjoint s-t paths in digraphs
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
Delaunay triangulations in O(sort(n)) time and more
Journal of the ACM (JACM)
Orthogonal range searching on the RAM, revisited
Proceedings of the twenty-seventh annual symposium on Computational geometry
SIAM Journal on Computing
Persistent predecessor search and orthogonal point location on the word RAM
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Succinct indices for range queries with applications to orthogonal range maxima
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
A faster algorithm for computing motorcycle graphs
Proceedings of the twenty-ninth annual symposium on Computational geometry
Persistent Predecessor Search and Orthogonal Point Location on the Word RAM
ACM Transactions on Algorithms (TALG) - Special Issue on SODA'11
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Given a planar subdivision whose coordinates are integers bounded by $U\leq2^w$, we present a linear-space data structure that can answer point-location queries in $O(\min\{\lg n/\lg\lg n,$ $\sqrt{\lg U/\lg\lg U}\})$ time on the unit-cost random access machine (RAM) with word size $w$. This is the first result to beat the standard $\Theta(\lg n)$ bound for infinite precision models. As a consequence, we obtain the first $o(n\lg n)$ (randomized) algorithms for many fundamental problems in computational geometry for arbitrary integer input on the word RAM, including: constructing the convex hull of a three-dimensional (3D) point set, computing the Voronoi diagram or the Euclidean minimum spanning tree of a planar point set, triangulating a polygon with holes, and finding intersections among a set of line segments. Higher-dimensional extensions and applications are also discussed. Though computational geometry with bounded precision input has been investigated for a long time, improvements have been limited largely to problems of an orthogonal flavor. Our results surpass this long-standing limitation, answering, for example, a question of Willard (SODA'92).