Box Filled With Extreme Values

A novel square tiling method is proposed, which is based on extremal length theory. This method converts the solving of a square tiling to a quadratic programming with convex constraints, and then it gets an efficient and robust result. The method maps a given planar graph to a square tessellation of a rectangle, such that each node is represented by a square, two adjacent nodes are mapped to two squares in contact. The method is based on the extremal length theory in conformal geometry. The algorithm uses discrete Ricci flow method to get an initial estimation, and then the square tilling is computed by optimizing a quadratic energy with convex constraints.The embedding has many merits for graph visualization:each node is mapped to a square, so it is convenient to add labels or images to the node; the edges are encoded by the cell adjacency relation, so the embedding has no edge-crossings, and is more intuitive to visualize graphs.The application of square tiling which is a new method of graph visualization is promising. It overcomes the drawback that the traditional irregular box cannot contain messages in regular appearance. Also, it owns a dominant position to the visualization of relationship between nodes. We construct two examples which show the advantages of square tiling, it establishes the foundation of the theory and application about square tiling in the future.