Core Concepts
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.
Abstract
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.