Dynamic Planar Point Location Research Articles

Towards an Optimal Method for Dynamic Planar Point Location

We describe a fully dynamic linear-space data structure for point location in connected planar subdivisions, or more generally vertical ray shooting among nonintersecting line segments, that suppor...

Three-Dimensional Layers of Maxima

We present an O(n log n)-time algorithm to solve the three-dimensional layers-of-maxima problem. This is an improvement over the prior O(n log n log log n)-time solution. A previous claimed O(n log n)-time solution due to Atallah et al. has technical flaws. Our algorithm is based on a common framework underlying previous work, but to implement it we devise a new data structure to solve a special case of dynamic planar point location in a staircase subdivision. Our data structure itself relies on a new extension to dynamic fractional cascading that allows vertices of high degree in the control graph.

Lower Bounds for Dynamic Transitive Closure, Planar Point Location, and Parentheses Matching

We give a number of new lower bounds in the cell probe model<br />with logarithmic cell size, which entails the same bounds on the random access computer with logarithmic word size and unit cost operations. We study the signed prefix sum problem: given a string of length n of zeroes and signed ones, compute the sum of its ith prefix during updates. We show a<br />lower bound of Omega(log n/log log n) time per operations, even if the prefix sums are bounded by log n/log log n during all updates. We also show that if the update time is bounded by the product of the worst-case update time and the<br />answer to the query, then the update time must be Omega(sqrt(log n/ log log n)).<br /> These results allow us to prove lower bounds for a variety of seemingly unrelated<br />dynamic problems. We give a lower bound for the dynamic planar point location in monotone subdivisions of <br />Omega(log n/ log log n) per operation. We give<br />a lower bound for the dynamic transitive closure problem on upward planar graphs with one source and one sink of <br />Omega(log n/(log logn)^2) per operation. We give a lower bound of Omega(sqrt(log n/log log n)) for the dynamic membership problem of any Dyck language with two or more letters. This implies the same<br />lower bound for the dynamic word problem for the free group with k generators. We also give lower bounds for the dynamic prefix majority and prefix equality problems.

Dynamic Point Location in General Subdivisions

The dynamic planar point location problem is the task of maintaining a dynamic set S of n nonintersecting (except possibly at endpoints) line segments in the plane under the following operations: • Locate (: point): Report the segment immediately above , i.e., the first segment intersected by an upward vertical ray starting at ; • Insert (: segment): Add segment to the collection of segments; • Delete (: segment): Remove segment from the collection of segments. We present a solution which requires space O( n) and has query and insertion time O(log n log log n) and deletion time O(log 2 n). The bounds for insertions and deletions are amortized. A query time below O(log 2 n) was previously only known for monotone subdivisions and subdivisions consisting of horizontal segments and required nonlinear space.

New Results on Dynamic Planar Point Location

A point location scheme is presented for a dynamic planar subdivision whose underlying graph is only required to be connected. The operations supported include: reporting the name of the region containing a query point, inserting/deleting an edge, and inserting/deleting/moving a degree-2 vertex. The scheme uses $O(n)$ space, has a worst-case query time of $O(\log ^2 n)$, and a worst-case update time of $O(\log n)$, where n is the number of vertices currently in the subdivision. Insertion (respectively, deletion) of an arbitrary k-edge chain inside a region can be performed in $O(k\log (n + k))$ (respectively, $O(k\log n)$) time in worst-case. The scheme outperforms the solutions given in works by Fries, Mehlhorn, and Naeher and by Overmars and also handles more general subdivisions than the schemes given in works by Preparata and Tamassia. The result is based on a new solution to a dynamic visibility problem for a set of line segments in the plane that are nonintersecting, except possibly at endpoints. Th...

Dynamic planar point location with optimal query time

We present a new dynamic technique for locating a point in a convex planar subdivision whose n vertices lie on a fixed set of N horizontal lines. The supported update operations are insertion/deletion of vertices and edges, and (horizontal) translation of vertices. Our method achieves query time O n + log IV, space O( N+ n log N). and insertion/deletion time O( log n log N). Hence, for N=O( n), the query time is O( log n), which is optimal. The proposed technique, based on the trapezoid method, provides an efficient solution to many significant applications where the most frequent operation is the point location query, while updates are more rarely executed.