[Paper Review] On the streaming complexity of fundamental geometric problems

Arijit Bishnu, Arijit Ghosh+2 more|arXiv (Cornell University)|Mar 19, 2018

Complexity and Algorithms in Graphs41 references3 citations

TL;DR

This paper establishes tight space complexity lower bounds and matching approximation algorithms for fundamental geometric problems in the one-pass streaming model. It analyzes Klee's measure, convex body approximation, and discrepancy problems, proving that randomized streaming algorithms require Ω(1/ε) space for ε-additive approximations, with algorithms achieving optimal space usage, and shows that sorted input enables exact O(1)-space computation for color discrepancy.

ABSTRACT

In this paper, we focus on lower bounds and algorithms for some basic geometric problems in the one-pass (insertion only) streaming model. The problems considered are grouped into three categories: (i) Klee's measure (ii) Convex body approximation, geometric query, and (iii) Discrepancy Klee's measure is the problem of finding the area of the union of hyperrectangles. Under convex body approximation, we consider the problems of convex hull, convex body approximation, linear programming in fixed dimensions. The results for convex body approximation implies a property testing type result to find if a query point lies inside a convex polyhedron. Under discrepancy, we consider both the geometric and combinatorial discrepancy. For all the problems considered, we present (randomized) lower bounds on space. Most of our lower bounds are in terms of approximating the solution with respect to an error parameter $ε$. We provide approximation algorithms that closely match the lower bound on space for most of the problems.

Motivation & Objective

• To establish strong space complexity lower bounds for fundamental geometric problems in the one-pass streaming model.
• To design optimal approximation algorithms that match the derived lower bounds for problems like Klee's measure and convex body approximation.
• To investigate the impact of input order on streaming complexity, particularly for discrepancy problems.
• To explore connections between streaming complexity and property testing, especially for convex hull membership.
• To demonstrate that sorted input reduces the space complexity of color discrepancy from Ω(n) to O(1) bits.

Proposed method

• Uses randomized communication complexity techniques to derive ϵ-additive space lower bounds for geometric problems.
• Reduces geometric problems to communication problems like Index and Disjointness to prove lower bounds.
• Designs one-pass streaming algorithms that achieve space complexity matching the lower bounds, using sampling and sketching techniques.
• Applies a multi-pass transformation to convert ϵ-additive algorithms into ϵ-multiplicative ones using logarithmic passes.
• Leverages the structure of sorted streams to compute color discrepancy exactly in O(1) space via prefix difference tracking.
• Employs the Hausdorff distance to define ε-approximation of convex bodies and relates it to geometric and linear programming problems.

Experimental results

Research questions

• RQ1 What is the minimum space required for a one-pass streaming algorithm to compute an ϵ-additive approximation of Klee’s measure in d-dimensional space?
• RQ2 Can convex body approximation and linear programming in fixed dimensions be solved with space complexity matching the information-theoretic lower bound?
• RQ3 How does the input order affect the streaming complexity of geometric discrepancy problems?
• RQ4 What is the streaming complexity of property testing for convex hull membership, and how is it related to convex body approximation?
• RQ5 Can exact computation of color discrepancy be achieved in constant space when the stream is sorted by coordinate?

Key findings

• The paper proves that Ω(1/ε) space is required for any one-pass randomized algorithm to compute an ϵ-additive approximation of Klee’s measure in the streaming model.
• For convex body approximation and related problems, the paper presents algorithms that achieve space complexity matching the derived Ω(1/ε) lower bounds, establishing tight bounds.
• The color discrepancy problem requires Ω(n) space in general streams, but can be solved exactly in O(1) space when the input is sorted by coordinate.
• A multi-pass algorithm is constructed that converts an ϵ-additive one-pass algorithm into an ϵ-multiplicative one using O(log(1/O)) passes, where O is the optimal solution value.
• For sorted streams, the color discrepancy D_c(P) can be computed exactly using only O(1) space by tracking the maximum and minimum of the running difference between red and blue point counts.
• The paper establishes that the optimal solution for color discrepancy is D_c(P) = max_p(R([0,p]) - B([0,p])) - min_q(R([0,q]) - B([0,q])) + 1, enabling constant-space computation.