We present the first almost-linear time algorithm for constructing linear-sized spectral sparsification for graphs. This improves all previous constructions of linear-sized spectral sparsification, which requires time.
A key ingredient in our algorithm is a novel combination of two techniques used in literature for constructing spectral sparsification: Random sampling by effective resistance, and adaptive constructions based on barrier functions.
Apologies to the paper authors for my liberties with their title: Constructing Linear-Sized Spectral Sparsification in Almost-Linear Time but I wanted to capture eyes that might glaze past their more formal title.
The PR release where I saw this article reads as follows:
In the second paper, Constructing linear-sized spectral sparsification in almost-linear time, Dr He Sun, Lecturer in Computer Science in the University’s Department of Computer Science and Yin Tat Lee, a PhD student from MIT, have presented the first algorithm for constructing linear-sized spectral sparsifiers that runs in almost-linear time.
More and more applications from today’s big data scenario need to deal with graphs of millions of vertices. While traditional algorithms can be applied directly in these massive graphs, these algorithms are usually too slow to be practical when the graph contains millions of vertices. Also, storing these practical massive graphs are very expensive.
Dr He Sun said: “Over the past decade, there have been intensive studies in order to overcome these two bottlenecks. One notable approach is through the intermediate step called spectral sparsification, which is the approximation of any input graph by a very sparse graph that inherits many properties of the input graph. Since most algorithms run faster in sparse graphs, spectral sparsification is used as a key intermediate step in speeding up the runtime of many practical graph algorithms, including finding approximate maximum flows in an undirected graph, and approximately solving linear systems, among many others.”
Using spectral sparsification, the researchers ran many algorithms in a sparse graph, and obtained approximately the correct results as well. This general framework allowed them to speed up the runtime of a wide range of algorithms by a magnitude. However, to make the overall approach practical, a key issue was to find faster constructions of spectral sparsification with fewer edges in the resulting sparsifiers. There have been many studies looking at this area in the past decade.
The researchers have proved that, for any graph, they can construct in almost-linear time its spectral sparsifier, and in the output sparsifier every vertex has only constant number of vertices. This result is almost optimal respect to time complexity of the algorithm, and the number of edges in the spectral sparsifier.
Very heavy sledding in the paper but you don’t have to be able to originate the insight in order to take advantage of the technique.