Optimal Robust Algorithms for Finding Triangles and Computing the Girth in Unit Disk and Transmission Graphs

Conceitos Básicos

Optimal robust algorithms for finding a triangle and computing the unweighted girth in unit disk graphs, as well as finding a triangle in transmission graphs.

Resumo

The paper presents robust algorithms for solving fundamental graph problems in the domains of unit disk graphs and transmission graphs.
For unit disk graphs:

The algorithm for finding a triangle runs in O(n) time and is optimal. It exploits the fact that if the maximum degree of a vertex is greater than 5, then the graph must contain a triangle.
The algorithm for computing the girth (the length of the shortest cycle) also runs in O(n) time. It first checks if the graph contains a triangle, and if not, tests if the graph is planar using a linear-time planarity testing algorithm. If the graph is planar, it then computes the girth using an existing linear-time algorithm for planar graphs.
For transmission graphs:

The algorithm for finding a directed triangle runs in O(n+m) time, where n is the number of vertices and m is the number of edges. It first preprocesses the graph to identify the set of vertices that have both incoming and outgoing edges. If any vertex has more than 6 such vertices, then a triangle must exist. Otherwise, the algorithm explicitly checks for triangles.
The key insight is that the robust setting, where the input may or may not be realizable as a unit disk or transmission graph, allows for sublinear algorithms that are faster than the best known algorithms for the general graph case. This is because the algorithms can exploit the underlying geometric structure of these graph classes, even without being given the geometric representation.

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Principais Insights Extraídos De

Robust Algorithms for Finding Triangles and Computing the Girth in Unit Disk and Transmission Graphs

by Katharina Kl... às arxiv.org 05-03-2024

https://arxiv.org/pdf/2405.01180.pdf

Perguntas Mais Profundas

How can the ideas behind these robust algorithms be extended to other geometric graph classes, such as general disk graphs or other proximity-based graphs

The ideas behind the robust algorithms for finding triangles and computing the girth in unit disk and transmission graphs can potentially be extended to other geometric graph classes, such as general disk graphs or other proximity-based graphs. One approach could involve analyzing the structural properties of these graph classes to identify key characteristics that allow for efficient algorithm design. For general disk graphs, similar to unit disk graphs, exploring properties related to vertex degrees and geometric constraints could lead to the development of sublinear robust algorithms. By adapting the concepts of triangle detection and girth computation to suit the specific features of these graph classes, it may be possible to devise algorithms that guarantee correct results within a sublinear time complexity.

Can similar sublinear robust algorithms be developed for other fundamental graph problems, such as finding the maximum clique or computing the diameter

Developing similar sublinear robust algorithms for other fundamental graph problems, such as finding the maximum clique or computing the diameter, is a challenging yet promising direction. To achieve this, it would be essential to identify specific properties or structural constraints that can be leveraged to efficiently solve these problems in the robust setting. For instance, in the case of finding the maximum clique, exploring the connectivity patterns and neighborhood relationships within different graph classes could lead to the design of algorithms that operate within sublinear time bounds. Similarly, for computing the diameter, analyzing the distance metrics and graph connectivity could provide insights into developing sublinear algorithms that ensure accurate results even in the presence of non-standard input graphs.

What are the implications of these robust algorithms for the complexity of recognizing unit disk graphs and transmission graphs as abstract graphs, compared to the known hardness results

The implications of the robust algorithms for recognizing unit disk graphs and transmission graphs as abstract graphs, compared to the known hardness results, are significant in terms of algorithmic efficiency and practical applicability. While the hardness results indicate the computational challenges associated with determining whether an abstract graph corresponds to a specific geometric graph class, the existence of sublinear robust algorithms offers a ray of hope for efficient graph recognition in practical scenarios. By providing algorithms that can correctly identify unit disk graphs and transmission graphs within sublinear time complexities, these robust approaches bridge the gap between theoretical complexity and practical feasibility. This not only enhances the understanding of the structural properties of geometric graphs but also opens up possibilities for developing similar efficient recognition algorithms for other graph classes with specific geometric or proximity-based characteristics.