[论文解读] Implicit representations via the polynomial method

Jean Cardinal, Micha Sharir|arXiv (Cornell University)|Feb 11, 2026

Computational Geometry and Mesh Generation被引用 0

一句话总结

论文展示了使用多项式划分的子线性相邻接标注方案，适用于半代数图，得到 O(n^{1-2/(d+1)+ε}) 位标签和高效查询，针对单位圆盘、线段相交、半线性和多边形可视性图有专门结果。

ABSTRACT

Semialgebraic graphs are graphs whose vertices are points in $\mathbb{R}^d$, and adjacency between two vertices is determined by the truth value of a semialgebraic predicate of constant complexity. We show how to harness polynomial partitioning methods to construct compact adjacency labeling schemes for families of semialgebraic graphs. That is, we show that for any family of semialgebraic graphs, given a graph on $n$ vertices in this family, we can assign a label consisting of $O(n^{1-2/(d+1) + \varepsilon})$ bits to each vertex (where $\varepsilon > 0$ can be made arbitrarily small and the constant of proportionality depends on $\varepsilon$ and on the complexity of the adjacency-defining predicate), such that adjacency between two vertices can be determined solely from their two labels, without any additional information. We obtain for instance that unit disk graphs and segment intersection graphs have such labelings with labels of $O(n^{1/3 + \varepsilon})$ bits. This is in contrast to their natural implicit representation consisting of the coordinates of the disk centers or segment endpoints, which sometimes require exponentially many bits. It also improves on the best known bound of $O(n^{1-1/d}\log n)$ for $d$-dimensional semialgebraic families due to Alon (Discrete Comput. Geom., 2024), a bound that holds more generally for graphs with shattering functions bounded by a degree-$d$ polynomial. We also give new bounds on the size of adjacency labels for other families of graphs. In particular, we consider semilinear graphs, which are semialgebraic graphs in which the predicate only involves linear polynomials. We show that semilinear graphs have adjacency labels of size $O(\log n)$. We also prove that polygon visibility graphs, which are not semialgebraic in the above sense, have adjacency labels of size $O(\log^3 n)$.

研究动机与目标

• 推动对由半代数谓词定义的图的紧凑、分布式编码。
• 为 d 维半代数图族开发子线性标签大小的邻接标注方案。
• 应用多项式划分来界定每个顶点的双图参与度 ν(n) 并推导标签大小。
• 为半线性和可视性图族导出简明的隐式表示。
• 为具体图类（单位圆盘、线段、球、以及多边形可视性）给出推论。

提出的方法

• 以固定复杂度的半代数谓词对邻接关系进行建模。
• 通过多项式划分构造双图分解以界定顶点参与度 ν(n)。
• 在原始/对偶设定中应用划分树以平衡相容关系并控制标签长度。
• 利用 Milnor–Thom 型界来界定单元数和穿越，实现子线性编码。
• 由平衡的双图分解导出明确的标签大小定理（定理 1 及推论）。
• 对特殊情况（半线性和多边形可视性图）给出定制分解和已知几何结果的处理。

实验结果

研究问题

• RQ1 半代数图族是否可以在标签大小低于线性 n 的条件下实现子线性邻接标注方案？
• RQ2 使用多项式划分在 d 维半代数图中可达到的最小标签大小是多少？
• RQ3 结果如何对单位圆盘、任意半径圆盘、单位球、球体和线段相交图等具体图族进行特化？
• RQ4 相较于一般半代数图，半线性图是否能实现更小（对数级）的邻接标签？
• RQ5 对于多边形可视性图和相关地形，可以得到哪些标注界？

主要发现

• d 维半代数族存在 O(n^{1-2/(d+1)+ε}) 位的邻接标注方案（ε>0 任意小）。
• 单位圆盘图具有 O(n^{1/3+ε}) 位标签；任意半径圆盘图具有 O(n^{1/2+ε}) 位标签。
• R^d 中的单位球图具有 O(n^{1-2/(d+1)+ε}) 位标签；球图具有 O(n^{1-2/(d+2)+ε}) 位标签。
• 线段相交图具有 O(n^{1/3+ε}) 位标签（在实践中将参数维度降至 2）。
• 半线性图 admit O(log n) 位邻接标注方案。
• 多边形可视性图 admit O(log^{3} n)-位邻接标注方案（对封顶图为 O(log^{2} n)）。