Fast and Efficient Graph Processing Techniques

In this paper, we propose a novel approach to graph simplification, which aims to minimize the complexity measure of aspect ratio. The aspect ratio is the multiplication factor between the heaviest and lightest edge weights in a graph. By reducing the aspect ratio, we can improve the efficiency of algorithms that operate on the simplified graph.
To achieve this goal, we introduce a two-stage procedure: (1) generating a collection of edges with at least some weight w, and (2) combining these edges into a single collection of weight w, while ensuring that each path has at least ℓ edges. We then show how to construct a collection of not-shortest paths with the same endpoints as the shortest paths, but with fewer edges. By analyzing the weights of these paths, we can determine whether they remain not-shortest after reweighting, and if so, by how much their edge weights must be multiplied to ensure this property.
Our approach is based on the idea of expanders, which are graphs that have a high expansion parameter. By decomposing a graph into a collection of expanders, we can reduce its complexity measure and make it more suitable for efficient algorithm design. Our method has applications in various fields, including network analysis, web search, and data compression.
In summary, this paper presents a new paradigm for graph simplification that minimizes the aspect ratio, making it easier to develop efficient algorithms for solving problems on simplified graphs. By using expanders to decompose graphs, we can reduce their complexity measure and improve their suitability for algorithm design. This approach has broad implications in various areas of computer science and can lead to significant improvements in computational efficiency.