[Paper Review] Advancing Through Terrains

Vincent Froese, Malte Renken|arXiv (Cornell University)|Apr 18, 2019

Data Management and Algorithms32 references4 citations

TL;DR

This paper advances the understanding of terrain visibility graphs—key in time series analysis and computational geometry—by proving they exclude anti-holes of size greater than five and are not unigraphs. It introduces an O(d* log Δ) output-sensitive shortest path algorithm for terrain-like graphs and an O(n⁴)-time polynomial-time algorithm for Dominating Set on funnel visibility graphs, a subclass with practical relevance.

ABSTRACT

We study terrain visibility graphs, a well-known graph class closely related to polygon visibility graphs in computational geometry, for which a precise graph-theoretical characterization is still unknown. Over the last decade, terrain visibility graphs attracted attention in the context of time series analysis with various practical applications in areas such as physics, geography and medical sciences. We make progress in understanding terrain visibility graphs by providing several graph-theoretic results. For example, we show that they cannot contain antiholes of size larger than five. Moreover, we obtain two algorithmic results. We devise a fast output-sensitive shortest path algorithm on terrain-like graphs and a polynomial-time algorithm for extsc{Dominating Set} on special terrain visibility graphs (called funnel visibility graphs).

Motivation & Objective

• To address the long-standing open problem of characterizing terrain visibility graphs graph-theoretically.
• To develop efficient algorithms for fundamental graph problems on terrain-like graphs, especially shortest paths and Dominating Set.
• To clarify structural properties of terrain visibility graphs, such as their relation to unit interval graphs and unigraphs.
• To provide a foundation for practical applications in time series analysis, robotics, and computational geometry.

Proposed method

• Proved that terrain visibility graphs cannot contain anti-holes of size greater than five using geometric visibility constraints.
• Introduced a new class of graphs called 'terrain-like graphs' satisfying a weaker visibility condition, enabling a general shortest path algorithm.
• Designed a dynamic programming approach for Dominating Set on funnel visibility graphs using structural lemmas on neighbor sets and base decomposition.
• Leveraged the X-property and bar-property to ensure recursive decomposition of subproblems in the dynamic programming formulation.
• Used vertex ordering and neighborhood partitioning (left/right chains) to define recursive subproblems for the Dominating Set algorithm.
• Optimized the algorithm to run in O(n⁴) time by memoizing solutions for all valid (ℓ, r) pairs in the base decomposition.

Experimental results

Research questions

• RQ1 Can terrain visibility graphs contain anti-holes of size greater than five?
• RQ2 Are all unit interval graphs realizable as induced subgraphs of terrain visibility graphs?
• RQ3 Is the class of terrain visibility graphs uniquely determined by its degree sequence (i.e., are they unigraphs)?
• RQ4 Can the Dominating Set problem be solved in polynomial time on general terrain visibility graphs?
• RQ5 Can shortest paths be computed efficiently in induced subgraphs of terrain visibility graphs?

Key findings

• Terrain visibility graphs do not contain anti-holes of size greater than five, providing a new structural constraint.
• Terrain visibility graphs are not unigraphs, as shown by a counterexample where multiple non-isomorphic graphs share the same degree sequence.
• The class of terrain visibility graphs does not include all unit interval graphs, despite both being hole-free.
• An output-sensitive shortest path algorithm runs in O(d* log Δ) time on terrain-like graphs, where d* is the shortest path length and Δ is the maximum degree.
• A polynomial-time O(n⁴)-algorithm solves the Dominating Set problem on funnel visibility graphs using dynamic programming on a structured decomposition of the graph.
• The algorithm for Dominating Set relies on a key lemma ensuring that recursive subproblems remain valid bases, enabling correct dynamic programming state transitions.