Adaptive Point Location in Planar Convex Subdivisions

2017 ◽  
Vol 27 (01n02) ◽  
pp. 3-12
Author(s):  
Siu-Wing Cheng ◽  
Man-Kit Lau
Keyword(s):  
Query Sequence ◽  
Point Location ◽  
Worst Case ◽  
Planar Point ◽  
Running Time ◽  
Planar Convex ◽  
Planar Point Location ◽  
Convex Subdivision ◽  
Text Query ◽  
Time Required

We present a planar point location structure for a convex subdivision [Formula: see text]. Given a query sequence of length [Formula: see text], the total running time is [Formula: see text], where [Formula: see text] is the number of vertices in [Formula: see text] and [Formula: see text] is the minimum time required by any linear decision tree for answering planar point location queries in [Formula: see text] to process the same query sequence. The running time includes the preprocessing time. Therefore, for [Formula: see text], our running time is only worse than the best possible bound by [Formula: see text] per query, which is much smaller than the [Formula: see text] query time offered by a worst-case optimal planar point location structure.

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

1996 ◽  
Vol 3 (9) ◽  
Author(s):  
Thore Husfeldt ◽  
Theis Rauhe ◽  
Søren Skyum
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Transitive Closure ◽  
Unit Cost ◽  
Random Access ◽  
Point Location ◽  
Worst Case ◽  
Planar Point ◽  
Planar Point Location ◽  
Prefix Sums

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.

Adaptive Planar Point Location

2021 ◽  
Vol 50 (4) ◽  
pp. 1200-1247
Author(s):  
Siu-Wing Cheng ◽  
Man-Kit Lau
Keyword(s):  
Point Location ◽  
Planar Point ◽  
Planar Point Location

scholarly journals A STATIC OPTIMALITY TRANSFORMATION WITH APPLICATIONS TO PLANAR POINT LOCATION

2012 ◽  
Vol 22 (04) ◽  
pp. 327-340 ◽  
Author(s):  
JOHN IACONO ◽  
WOLFGANG MULZER
Keyword(s):  
Data Structures ◽  
Prior Information ◽  
Query Sequence ◽  
Binary Search ◽  
Asymptotic Performance ◽  
Search Trees ◽  
Point Location ◽  
Static Structure ◽  
Additional Information ◽  
Planar Point Location

Over the last decade, there have been several data structures that, given a planar subdivision and a probability distribution over the plane, provide a way for answering point location queries that is fine-tuned for the distribution. All these methods suffer from the requirement that the query distribution must be known in advance. We present a new data structure for point location queries in planar triangulations. Our structure is asymptotically as fast as the optimal structures, but it requires no prior information about the queries. This is a 2-D analogue of the jump from Knuth's optimum binary search trees (discovered in 1971) to the splay trees of Sleator and Tarjan in 1985. While the former need to know the query distribution, the latter are statically optimal. This means that we can adapt to the query sequence and achieve the same asymptotic performance as an optimum static structure, without needing any additional information.

scholarly journals Towards an Optimal Method for Dynamic Planar Point Location

2018 ◽  
Vol 47 (6) ◽  
pp. 2337-2361
Author(s):  
Timothy M. Chan ◽  
Yakov Nekrich
Keyword(s):  
Point Location ◽  
Optimal Method ◽  
Planar Point ◽  
Planar Point Location

scholarly journals AN IMPROVED HYPERCUBE BOUND FOR MULTISEARCHING AND ITS APPLICATIONS

1999 ◽  
Vol 09 (01) ◽  
pp. 29-38
Author(s):  
MIKHAIL J. ATALLAH
Keyword(s):  
Data Structure ◽  
Constant Time ◽  
Search Problem ◽  
Parallel Search ◽  
Point Location ◽  
Planar Point ◽  
Geometric Problems ◽  
Trapezoidal Decomposition ◽  
Planar Point Location ◽  
Hypercube Model

We give a result that implies an improvement by a factor of log log  n in the hypercube bounds for the geometric problems of batched planar point location, trapezoidal decomposition, and polygon triangulation. The improvements are achieved through a better solution to the multisearch problem on a hypercube, a parallel search problem where the elements in the data structure S to be searched are totally ordered, but where it is not possible to compare in constant time any two given queries q and q′. Whereas the previous best solution to this problem took O( log  n( log log  n)3) time on an n-processor hypercube, the solution given here takes O( log  n( log log  n)2) time on an n-processor hypercube. The hypercube model for which we claim our bounds is the standard one, SIMD, with O(1) memory registers per processor, and with one-port communication. Each register can store O( log  n) bits, so that a processor knows its ID.

scholarly journals Planar Point Location Using Persistent Search Trees

1989 ◽  
Vol 555 (1 Combinatorial) ◽  
pp. 352-362
Author(s):  
NEIL SARNAK ◽  
ROBERT E. TARJAN
Keyword(s):  
Search Trees ◽  
Point Location ◽  
Planar Point ◽  
Planar Point Location
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