4 research outputs found
Evaluating Stability in Massive Social Networks: Efficient Streaming Algorithms for Structural Balance
Structural balance theory studies stability in networks. Given a -vertex
complete graph whose edges are labeled positive or negative, the
graph is considered \emph{balanced} if every triangle either consists of three
positive edges (three mutual ``friends''), or one positive edge and two
negative edges (two ``friends'' with a common ``enemy''). From a computational
perspective, structural balance turns out to be a special case of correlation
clustering with the number of clusters at most two. The two main algorithmic
problems of interest are: detecting whether a given graph is balanced, or
finding a partition that approximates the \emph{frustration index},
i.e., the minimum number of edge flips that turn the graph balanced.
We study these problems in the streaming model where edges are given one by
one and focus on \emph{memory efficiency}. We provide randomized single-pass
algorithms for: determining whether an input graph is balanced with
memory, and finding a partition that induces a -approximation to the frustration index with memory. We further provide several new lower bounds,
complementing different aspects of our algorithms such as the need for
randomization or approximation.
To obtain our main results, we develop a method using pseudorandom generators
(PRGs) to sample edges between independently-chosen \emph{vertices} in graph
streaming. Furthermore, our algorithm that approximates the frustration index
improves the running time of the state-of-the-art correlation clustering with
two clusters (Giotis-Guruswami algorithm [SODA 2006]) from
to time for
-approximation. These results may be of independent interest
Local Routing in Sparse and Lightweight Geometric Graphs
Online routing in a planar embedded graph is central to a number of fields and has been studied extensively in the literature. For most planar graphs no O(1)-competitive online routing algorithm exists. A notable exception is the Delaunay triangulation for which Bose and Morin showed that there exists an online routing algorithm that is O(1)-competitive. However, a Delaunay triangulation can have Omega(n) vertex degree and a total weight that is a linear factor greater than the weight of a minimum spanning tree. We show a simple construction, given a set V of n points in the Euclidean plane, of a planar geometric graph on V that has small weight (within a constant factor of the weight of a minimum spanning tree on V), constant degree, and that admits a local routing strategy that is O(1)-competitive. Moreover, the technique used to bound the weight works generally for any planar geometric graph whilst preserving the admission of an O(1)-competitive routing strategy
Local Routing in Sparse and Lightweight Geometric Graphs
Online routing in a planar embedded graph is central to a number of fields and has been studied extensively in the literature. For most planar graphs no O(1)-competitive online routing algorithm exists. A notable exception is the Delaunay triangulation for which Bose and Morin (SIAM J Comput 33(4):937–951, 2004) showed that there exists an online routing algorithm that is O(1)-competitive. However, a Delaunay triangulation can have Ω(n) vertex degree and a total weight that is a linear factor greater than the weight of a minimum spanning tree. We show a simple construction, given a set V of n points in the Euclidean plane, of a planar geometric graph on V that has small weight (within a constant factor of the weight of a minimum spanning tree on V), constant degree, and that admits a local routing strategy that is O(1)-competitive. Moreover, the technique used to bound the weight works generally for any planar geometric graph whilst preserving the admission of an O(1)-competitive routing strategy