Cm Chris Gray

Vertical ray shooting and computing depth orders for fat objects

We present new results for three problems dealing with a set P of n convex constant-complexity fat polyhedra in 3-space. (i) We describe a data structure for vertical ray shooting in P that has O(log2 n) query time and uses O(n log2 n) storage. (ii) We give an algorithm to compute in O(n log3 n) time a depth order on P, if it exists. (iii) We give an algorithm to verify in O(n log4 n) time whether a given order on P is a valid depth order. All three results improve on previous results.

Vertical Ray Shooting and Computing Depth Orders for Fat Objects

We present new results for three problems dealing with a set <i>P</i> of <i>n</i> convex constant-complexity fat polyhedra in 3-space. (i) We describe a data structure for vertical ray shooting in <i>P</i> that has <i>O</i>(log² <i>n</i>) query time and uses <i>O</i>(<i>n</i> log² <i>n</i>) storage. (ii) We give an algorithm to compute in <i>O</i>(<i>n</i> log³ <i>n</i>) time a depth order on <i>P</i>, if it exists. (iii) We give an algorithm to verify in <i>O</i>(<i>n</i> log⁴ <i>n</i>) time whether a given order on <i>P</i> is a valid depth order. All three results improve on previous results.

Triangulating and guarding realistic polygons

We propose a new model of realistic input: k-guardable objects. An object is k-guardable if its boundary can be seen by k guards. We show that k-guardable polygons generalize two previously identified classes of realistic input. Following this, we give two simple algorithms for triangulating k-guardable polygons. One algorithm requires the guards as input while the other does not. Both take linear time assuming that k is constant and both are easily implementable.

Ray shooting and intersection searching amidst fat convex polyhedra in 3-space

We present a data structure for ray-shooting queries in a set of convex fat polyhedra of total complexity <i>n</i> in <i>R</i>³. The data structure uses <i>O(n</i>^2+ε) storage and preprocessing time, and queries can be answered in <i>O</i>(log² <i>n</i>) time. A trade-off between storage and query time is also possible: for any <i>m</i> with <i>n < m < n</i>², we can construct a structure that uses <i>O(m</i>^1+ε) storage and preprocessing time such that queries take <i>O((n/√m)</i>log² <i>n</i>) time.We also describe a data structure for simplex intersection queries in a set of <i>n</i> convex fat constant-complexity polyhedra in <i>R</i>³. For any <i>m</i> with <i>n < m < n</i>³, we can construct a structure that uses <i>O(m</i>^1+ε) storage and preprocessing time such that all polyhedra intersecting a query simplex can be reported in <i>O((n/m</i>^1/3)log <i>n+k</i>) time, where <i>k</i> is the number of answers.

Computing the visibility map of fat objects

We give an output-sensitive algorithm for computing the visibility map of a set of n constant-complexity convex fat polyhedra or curved objects in 3-space. Our algorithm runs in O((n + k) polylog n) time, where k is the combinatorial complexity of the visibility map. This is the first algorithm for computing the visibility map of fat objects that does not require a depth order on the objects and is faster than the best known algorithm for general objects. It is also the first output-sensitive algorithm for curved objects that does not require a depth order.

Decompositions and Boundary Coverings of Non-convex Fat Polyhedra

We show that any locally-fat (or (a, s)-covered) polyhedron with convex fat faces can be decomposed into O(n) tetrahedra, where n is the number of vertices of the polyhedron. We also show that the restriction that the faces are fat is necessary: there are locally-fat polyhedra with non-fat faces that require O(n2) pieces in any convex decomposition. Furthermore, we show that if we want the polyhedra in the decomposition to be fat themselves, then the worst-case number of tetrahedra cannot be bounded as a function of n. Finally, we obtain several results on the problem where we want to only cover the boundary of the polyhedron, and not its entire interior.