TY - GEN

T1 - Near-Linear Algorithms for Visibility Graphs over a 1.5-Dimensional Terrain

AU - Katz, Matthew J.

AU - Saban, Rachel

AU - Sharir, Micha

N1 - Publisher Copyright:
© Matthew J. Katz, Rachel Saban, and Micha Sharir; licensed under Creative Commons License CC-BY 4.0.

PY - 2024/9/1

Y1 - 2024/9/1

N2 - We present several near-linear algorithms for problems involving visibility over a 1.5-dimensional terrain. Concretely, we have a 1.5-dimensional terrain T, i.e., a bounded x-monotone polygonal path in the plane, with n vertices, and a set P of m points that lie on or above T. The visibility graph V G(P, T) is the graph with P as its vertex set and {(p, q) | p and q are visible to each other} as its edge set. We present algorithms that perform BFS and DFS on V G(P, T), which run in O(n log n + m log3(m + n)) time. We also consider three optimization problems, in which P is a set of points on T, and we erect a vertical tower of height h at each p ∈ P. In the first problem, called the reverse shortest path problem, we are given two points s, t ∈ P, and an integer k, and wish to find the smallest height h∗ for which V G(P(h∗), T) contains a path from s to t of at most k edges, where P(h∗) is the set of the tips of the towers of height h∗ erected at the points of P. In the second problem we wish to find the smallest height h∗ for which V G(P(h∗), T) contains a cycle, and in the third problem we wish to find the smallest height h∗ for which V G(P(h∗), T) is nonempty; we refer to that problem as “Seeing the most without being seen”. We present algorithms for the first two problems that run in O∗((m + n)6/5) time, where the O∗(·) notation hides subpolynomial factors. The third problem can be solved by a faster algorithm, which runs in O((n + m) log3(m + n)) time.

AB - We present several near-linear algorithms for problems involving visibility over a 1.5-dimensional terrain. Concretely, we have a 1.5-dimensional terrain T, i.e., a bounded x-monotone polygonal path in the plane, with n vertices, and a set P of m points that lie on or above T. The visibility graph V G(P, T) is the graph with P as its vertex set and {(p, q) | p and q are visible to each other} as its edge set. We present algorithms that perform BFS and DFS on V G(P, T), which run in O(n log n + m log3(m + n)) time. We also consider three optimization problems, in which P is a set of points on T, and we erect a vertical tower of height h at each p ∈ P. In the first problem, called the reverse shortest path problem, we are given two points s, t ∈ P, and an integer k, and wish to find the smallest height h∗ for which V G(P(h∗), T) contains a path from s to t of at most k edges, where P(h∗) is the set of the tips of the towers of height h∗ erected at the points of P. In the second problem we wish to find the smallest height h∗ for which V G(P(h∗), T) contains a cycle, and in the third problem we wish to find the smallest height h∗ for which V G(P(h∗), T) is nonempty; we refer to that problem as “Seeing the most without being seen”. We present algorithms for the first two problems that run in O∗((m + n)6/5) time, where the O∗(·) notation hides subpolynomial factors. The third problem can be solved by a faster algorithm, which runs in O((n + m) log3(m + n)) time.

KW - 1.5-dimensional terrain

KW - parametric search

KW - range searching

KW - reverse shortest path

KW - shrink-and-bifurcate

KW - visibility

KW - visibility graph

UR - http://www.scopus.com/inward/record.url?scp=85205683814&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ESA.2024.77

DO - 10.4230/LIPIcs.ESA.2024.77

M3 - Conference contribution

AN - SCOPUS:85205683814

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 32nd Annual European Symposium on Algorithms, ESA 2024

A2 - Chan, Timothy

A2 - Fischer, Johannes

A2 - Iacono, John

A2 - Herman, Grzegorz

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 32nd Annual European Symposium on Algorithms, ESA 2024

Y2 - 2 September 2024 through 4 September 2024

ER -