There are many representations of planar graphs, but few are as elegant as Turán's (1984): it is simple and practical, uses only 4 bits per edge, can handle self-loops and multi-edges, and can store any specified embedding. Its main disadvantage has been that “it does not allow efficient searching” (Jacobson, 1989). In this paper we show how to add a sublinear number of bits to Turán's representation such that it supports fast navigation while retaining simplicity. As a consequence of the inherited simplicity, we offer the first efficient parallel construction of a compact encoding of a planar graph embedding. Our experimental results show that the resulting representation uses about 6 bits per edge in practice, supports basic navigation operations within a few microseconds, and can be built sequentially at a rate below 1 microsecond per edge, featuring a linear speedup with a parallel efficiency around 50% for large datasets.

平面图的表示形式很多，但很少有像Turán（1984）那样优雅的：简单实用，每个边缘仅使用4位，可以处理自环和多边缘，并且可以存储任何指定的嵌入。它的主要缺点是“它不允许有效的搜索”（Jacobson，1989）。在本文中，我们展示了如何在Turán的表示中添加亚线性位数，以使其在保持简单性的同时支持快速导航。由于继承的简单性，我们提供了平面图嵌入的紧凑编码的第一个有效并行构造。我们的实验结果表明，实际上，结果表示每个边缘使用大约6位，在几微秒内支持基本的导航操作，