Planar Point Location in Sublogarithmic Time

We give a number of new lower bounds in the cell probe model 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 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 answer to the query, then the update time must be Omega(sqrt(log n/ log log n)). These results allow us to prove lower bounds for a variety of seemingly unrelated dynamic problems. We give a lower bound for the dynamic planar point location in monotone subdivisions of Omega(log n/ log log n) per operation. We give a lower bound for the dynamic transitive closure problem on upward planar graphs with one source and one sink of 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 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.

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Adaptive Planar Point Location

SIAM Journal on Computing ◽

10.1137/18m1218194 ◽

2021 ◽

Vol 50 (4) ◽

pp. 1200-1247

Author(s):

Siu-Wing Cheng ◽

Man-Kit Lau

Keyword(s):

Point Location ◽

Planar Point ◽

Planar Point Location

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Planar point Location Search

Algorithms in Combinatorial Geometry ◽

10.1007/978-3-642-61568-9_11 ◽

1987 ◽

pp. 241-266

Author(s):

Herbert Edelsbrunner

Keyword(s):

Point Location ◽

Planar Point ◽

Planar Point Location

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Towards an Optimal Method for Dynamic Planar Point Location

SIAM Journal on Computing ◽

10.1137/16m1066506 ◽

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

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AN IMPROVED HYPERCUBE BOUND FOR MULTISEARCHING AND ITS APPLICATIONS

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195999000030 ◽

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.

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