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Maximizing Weighted Dominance in the Plane

Waseem Akram, Sanjeev Saxena

Abstract

Let P be a set of n weighted points, Q be a set of m unweighted points in the plane, and k a non-negative integer. We consider the problem of computing a subset $Q'\subseteq Q$ with size at most k such that the sum of the weights of the points of P dominated by at least one point in the set Q' is maximized. A point q in the plane dominates another point p if and only if $x(q)\ge x(p)$ and $y(q)\ge y(p)$, and at least one inequality is strict. We present a solution to the problem that takes O(n + m)-space and $O(k \min\{n+m, \frac{n}{k}+m^2\}\log m)$-time. We (conditionally) improve upon the existing result (the bounds of our solution are interesting when $m= o(\sqrt{n}))$. Moreover, we also present a simple algorithm solving the problem in $O(km^2+n\log m)$-time and $O(n+m)$-space. The bounds of the algorithm are interesting when $m= o(\sqrt{n})$.

Maximizing Weighted Dominance in the Plane

Abstract

Let P be a set of n weighted points, Q be a set of m unweighted points in the plane, and k a non-negative integer. We consider the problem of computing a subset with size at most k such that the sum of the weights of the points of P dominated by at least one point in the set Q' is maximized. A point q in the plane dominates another point p if and only if and , and at least one inequality is strict. We present a solution to the problem that takes O(n + m)-space and -time. We (conditionally) improve upon the existing result (the bounds of our solution are interesting when . Moreover, we also present a simple algorithm solving the problem in -time and -space. The bounds of the algorithm are interesting when .
Paper Structure (8 sections, 8 theorems, 3 equations, 2 figures, 1 algorithm)

This paper contains 8 sections, 8 theorems, 3 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Motivation
  3. Previous and Related Works
  4. Definitions and Notations
  5. Transformation of Input points
  6. An $O(km^2+n\log m)$-Time Solution to the maxDominance Problem
  7. Further Improvements
  8. Conclusion

Key Result

Lemma 1

Each pair of points in $P$ contained in a given cell $C_{ij}$ have the same dominating points from the set $Q$.

Figures (2)

  • Figure 1: Cells between lines through $q_3$ and $q_4$ are shaded with grey. The cell $C_{32}$ has been shown in green shade along with corner points.
  • Figure 2: (a) represents the non-empty cells of the partition, and (b) the cells containing their representatives.

Theorems & Definitions (8)

  • Lemma 1
  • Corollary 1
  • Lemma 2
  • Theorem 1
  • Theorem 2
  • Lemma 3
  • Lemma 4
  • Theorem 3