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In this video, we describe an analytic unification of two actions frequently used in graph reduction: the deletion of edges, often used in graph sparsification algorithms, and the contraction of edges (that is, the merging of two adjacent nodes), which is often used in graph coarsening algorithms. Prior to this work, sparsification and coarsening were treated as separate algorithmic primitives, with different objective functions. What you are seeing now is our graph reduction algorithm in action, which uses this analytic unification to simultaneously sparsify and coarsen a graph. The actions of edge deletion and edge contraction are, in fact, dual operations. One manifestation of this duality is seen by considering a planar graph, shown here in blue. The planar dual of this graph, shown here in red, is created by first placing its red vertices in the regions formed by the planar embedding of the original graph. Then, one adds a red edge between each of these red vertices that share a
