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Exact solutions to the Weighted Region Problem

Sarita de Berg, Guillermo Esteban, Rodrigo I. Silveira, Frank Staals

TL;DR

The paper investigates the Weighted Region Problem with two weights $\{1,\alpha\}$, focusing on a single axis-aligned rectangle $R$ and a source point $s$ relative to $R$. It proves ACMQ-unsolvability when $s$ lies outside $R$ while deriving exact, ACMQ-expressible formulas for the lengths of all 13 shortest-path types when $s$ is on the boundary of $R$, with similar results for $s$ inside $R$; several path-length expressions involve solving quartic auxiliary equations. The authors also study the Shortest Path Map (SPM) by deriving bisectors between path types and discuss which bisectors can be computed exactly in $ACMQ$; they show that many $b_{i,j}$ involving the complex type $\pi_{10}$ are not ACMQ-computable, and that even a single quadrant with a positive weight is not ACMQ-solvable in general. Overall, the work delineates the boundary between exact solvability and hardness for the WRP under two weights, and lays groundwork for exact formulas and SPM construction in restricted geometric configurations.

Abstract

In this paper, we consider the Weighted Region Problem. In the Weighted Region Problem, the length of a path is defined as the sum of the weights of the subpaths within each region, where the weight of a subpath is its Euclidean length multiplied by a weight $ α\geq 0 $ depending on the region. We study a restricted version of the problem of determining shortest paths through a single weighted rectangular region. We prove that even this very restricted version of the problem is unsolvable within the Algebraic Computation Model over the Rational Numbers (ACMQ). On the positive side, we provide the equations for the shortest paths that are computable within the ACMQ. Additionally, we provide equations for the bisectors between regions of the Shortest Path Map for a source point on the boundary of (or inside) the rectangular region.

Exact solutions to the Weighted Region Problem

TL;DR

The paper investigates the Weighted Region Problem with two weights , focusing on a single axis-aligned rectangle and a source point relative to . It proves ACMQ-unsolvability when lies outside while deriving exact, ACMQ-expressible formulas for the lengths of all 13 shortest-path types when is on the boundary of , with similar results for inside ; several path-length expressions involve solving quartic auxiliary equations. The authors also study the Shortest Path Map (SPM) by deriving bisectors between path types and discuss which bisectors can be computed exactly in ; they show that many involving the complex type are not ACMQ-computable, and that even a single quadrant with a positive weight is not ACMQ-solvable in general. Overall, the work delineates the boundary between exact solvability and hardness for the WRP under two weights, and lays groundwork for exact formulas and SPM construction in restricted geometric configurations.

Abstract

In this paper, we consider the Weighted Region Problem. In the Weighted Region Problem, the length of a path is defined as the sum of the weights of the subpaths within each region, where the weight of a subpath is its Euclidean length multiplied by a weight depending on the region. We study a restricted version of the problem of determining shortest paths through a single weighted rectangular region. We prove that even this very restricted version of the problem is unsolvable within the Algebraic Computation Model over the Rational Numbers (ACMQ). On the positive side, we provide the equations for the shortest paths that are computable within the ACMQ. Additionally, we provide equations for the bisectors between regions of the Shortest Path Map for a source point on the boundary of (or inside) the rectangular region.
Paper Structure (13 sections, 22 theorems, 8 equations, 2 figures)

This paper contains 13 sections, 22 theorems, 8 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Our results.
  3. Shortest paths and their properties
  4. Computing a shortest path
  5. The source point $s$ lies on the boundary of $R$
  6. Shortest path types.
  7. Length of $\pi_i(s,t)$.
  8. The source point $s$ lies inside $R$
  9. The source point $s$ lies outside of $R$
  10. Computing a Shortest Path Map
  11. The source point $s$ lies on the boundary of $R$
  12. The source point $s$ lies inside $R$
  13. Conclusion

Key Result

Lemma 1

Let $\mathcal{S}\xspace$ be a polygonal subdivision for which each region has a weight in the set $\{1, \alpha\}$, with $\alpha \geq 0$. A shortest path $\pi(s,t)$ visits any edge of the subdivision at most once.

Figures (2)

  • Figure 1: Examples of shortest paths between two points---shown in orange---for two weighted regions. The unbounded region has weight 1, the squares have varying weight $\alpha$.
  • Figure 2: Path types for $s$ on the boundary of $R$ of weight $\alpha < 1$ (blue) and $1 \leq \alpha < \sqrt{2}$ (orange).

Theorems & Definitions (23)

  • Lemma 1
  • Corollary 2
  • Theorem 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • Lemma 10
  • Lemma 11
  • ...and 13 more