DeepAI AI Chat
Log In Sign Up

Preprocessing Ambiguous Imprecise Points

by   Ivor van der Hoog, et al.
TU Eindhoven

Let R = {R_1, R_2, ..., R_n} be a set of regions and let X = {x_1, x_2, ..., x_n} be an (unknown) point set with x_i ∈ R_i. Region R_i represents the uncertainty region of x_i. We consider the following question: how fast can we establish order if we are allowed to preprocess the regions in R? The preprocessing model of uncertainty uses two consecutive phases: a preprocessing phase which has access only to R followed by a reconstruction phase during which a desired structure on X is computed. Recent results in this model parametrize the reconstruction time by the ply of R, which is the maximum overlap between the regions in R. We introduce the ambiguity A(R) as a more fine-grained measure of the degree of overlap in R. We show how to preprocess a set of d-dimensional disks in O(n n) time such that we can sort X (if d=1) and reconstruct a quadtree on X (if d≥ 1 but constant) in O(A(R)) time. If A(R) is sub-linear, then reporting the result dominates the running time of the reconstruction phase. However, we can still return a suitable data structure representing the result in O(A(R)) time. In one dimension, R is a set of intervals and the ambiguity is linked to interval entropy, which in turn relates to the well-studied problem of sorting under partial information. The number of comparisons necessary to find the linear order underlying a poset P is lower-bounded by the graph entropy of P. We show that if P is an interval order, then the ambiguity provides a constant-factor approximation of the graph entropy. This gives a lower bound of Ω(A(R)) in all dimensions for the reconstruction phase (sorting or any proximity structure), independent of any preprocessing; hence our result is tight.


page 1

page 2

page 3

page 4


Preprocessing Imprecise Points for the Pareto Front

In the preprocessing model for uncertain data we are given a set of regi...

Lower Bounds on Retroactive Data Structures

We prove essentially optimal fine-grained lower bounds on the gap betwee...

On the Shannon entropy of the number of vertices with zero in-degree in randomly oriented hypergraphs

Suppose that you have n colours and m mutually independent dice, each of...

Worst-Case Efficient Dynamic Geometric Independent Set

We consider the problem of maintaining an approximate maximum independen...

Query-Competitive Sorting with Uncertainty

We study the problem of sorting under incomplete information, when queri...

Online algorithms for finding distinct substrings with length and multiple prefix and suffix conditions

Let two static sequences of strings P and S, representing prefix and suf...