We present an exact and efficient algorithm for computing the Reeb space of a bivariate PL map. The Reeb space is a topological structure that generalizes the Reeb graph to the setting of multiple scalar-valued functions defined over a shared domain, a situation that frequently arises in practical applications. While the Reeb graph has become a standard tool in computer graphics, shape analysis, and scientific visualization, the Reeb space is still in the early stages of adoption. Although several algorithms for computing the Reeb space have been proposed, none offer an implementation that is both exact and efficient, which has substantially limited its practical use. To address this gap, we introduce singular arrange and traverse, a new algorithm built upon the arrange and traverse framework. Our method exploits the fact that, in the bivariate case, only singular edges contribute to the structure of Reeb space, allowing us to ignore many regular edges. This observation results in substantial efficiency gains on datasets where most edges are regular, which is common in many numerical simulations of physical systems. We provide an implementation of our method and benchmark it against the original arrange and traverse algorithm, showing performance gains of up to four orders of magnitude on real-world datasets.

翻译：本文提出了一种精确且高效的算法，用于计算双变量分段线性映射的Reeb空间。Reeb空间是一种拓扑结构，它将Reeb图推广到在共享域上定义多个标量值函数的情形，这种情形在实际应用中经常出现。尽管Reeb图已成为计算机图形学、形状分析和科学可视化领域的标准工具，但Reeb空间仍处于应用的早期阶段。虽然已有多种计算Reeb空间的算法被提出，但尚未出现兼具精确性与高效性的实现方案，这极大限制了其实际应用。为填补这一空白，我们提出了奇异排列遍历算法——一种基于排列遍历框架的新算法。我们的方法利用了双变量情形下仅奇异边对Reeb空间结构有贡献的特性，从而可以忽略大量正则边。这一观察结果在多数边为正则边的数据集上带来了显著的效率提升，而这种情况在许多物理系统的数值模拟中十分常见。我们提供了该方法的实现，并与原始排列遍历算法进行基准测试，结果表明在实际数据集上性能提升可达四个数量级。