A note on harmonic subgraphs in labelled geometric graphs

Araujo, G.; Balogh, József; Fabila, Ruy; Salazar, Gelasio; Urrutia, Jorge

doi:10.1016/j.ipl.2007.08.016

Cited by 3 publications

(1 citation statement)

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“…Urrutia, who originally formulated these problems (see discussion in the introduction of Ref. 4), dealt exclusively with the worst-case instances of the labeled set S. Our goal is to study the average (likely) case behavior under the assumption that the point labels are random. More specifically, we assume that each of the n points is labeled with an integer drawn uniformly at random from {0, 1, 2, .…”

Section: Introductionmentioning

confidence: 99%

Large harmonic sets of noncrossing edges for n randomly labeled vertices in convex position

Balogh

Pittel

Salazar

2006

Random Struct Algorithms

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Consider a set of n points in the plane in convex position, where each point has an integer label from {0, 1, . . . , n − 1}. This naturally induces a labeling of the edges: each edge (i, j) is assigned the label i + j, modulo n. We propose algorithms for finding large noncrossing harmonic matchings or paths, i.e., the matchings or paths in which no two edges have the same label. When the point labels are chosen uniformly at random, and independently of each other, our matching algorithm with high probability delivers a nearly-perfect matching, a matching of size n/2 − O(n 1/3 ln n). We show that, in sharp contrast, a near-perfect path is unlikely: with high probability the length of the longest path is below 0.96n. In a series of computer experiments, one of our algorithms invariably found a path of length above 0.76n, and the likely path length for our empirically second best algorithm is proved to be above 0.66n.

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Section: Introductionmentioning

confidence: 99%