A Compact Encoding of Rectangular Drawings with Edge Lengths

摘要

A rectangular drawing is a plane drawing of a graph in which every face is a rectangle. Rectangular drawings have an application for floorplans, which may have a huge number of faces, so compact code to store the drawings is desired. The most compact code for rectangular drawings needs at most 4f - 4 bits, where f is the number of inner faces of the drawing. The code stores only the graph structure of rectangular drawings, so the length of each edge is not encoded. A grid rectangular drawing is a rectangular drawing in which each vertex has integer coordinates. To store grid rectangular drawings, we need to store some information for lengths or coordinates. One can store a grid rectangular drawing by the code for rectangular drawings and the width and height of each inner face. Such a code needs 4f - 4 + f log W + f log H + o(f) + o(W) + o(H) bits~*, where W and H are the maximum width and the maximum height of inner faces, respectively. In this paper we design a simple and compact code for grid rectangular drawings. The code needs 4f-4+(f+l) log L+o(f)+o(L) bits for each grid rectangular drawing, where L is the maximum length of edges in the drawing. Note that L ≤ max (W, H) holds. Our encoding and decoding algorithms run in O(f) time.