Implicit representations via the polynomial method

TL;DR

This work develops sublinear adjacency labeling schemes for graphs defined by semialgebraic predicates by leveraging polynomial partitioning to build balanced biclique decompositions. The authors prove that $d$-dimensional semialgebraic graph families admit $O(n^{1-2/(d+1)+\varepsilon})$-bit labels, with $\varepsilon>0$ arbitrarily small, and provide concrete corollaries such as $O(n^{1/3+\varepsilon})$-bit labels for unit disk and segment-intersection graphs. They further show that semilinear graphs have $O(\log n)$-bit labels and that polygon visibility graphs achieve $O(\log^3 n)$-bit labels (with $O(\log^2 n)$ for capped graphs), by decomposing graphs into (two-dimensional) comparability graphs and exploiting partition trees and duality. The techniques yield improved encoding sizes over previous bounds and establish a versatile framework for compact representations across several geometric graph families, with implications for universal graphs and efficient adjacency testing in geometric settings.

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 $O(\log^3 n)$.

Figures (8)

• Figure 1: Example of an adjacency labeling of a graph (left). Here two vertices $u$ and $v$ are adjacent, hence $A(\ell (u), \ell(v))$ is true, if and only if the Hamming distance between the two labels $\ell(u)$ and $\ell(v)$ is exactly equal to one. The graph is therefore an induced subgraph of the cube (shown in red, right).
• Figure 2: Example of an adjacency labeling of a graph with 16 edges (right) obtained from a biclique decomposition of size 14 (left). The label $\ell(v)$ of a vertex $v$ consists of the list of bicliques $v$ is contained in, from the set $\{R,G,B\}$, together with an additional bit indicating on which side of the biclique the vertex $v$ lies.
• Figure 3: The Perles configuration of 9 points and 9 lines, every realization of which requires at least one irrational coordinate. This provides an example of a semialgebraic family, the bipartite point-line incidence graphs, for which the natural implicit representation by real numbers cannot be turned into an adjacency labeling scheme with labels using a bounded number of bits.
• Figure 4: A schematic description of the partition tree used in the proof of Theorem \ref{['thm:main']}. The top tree corresponds to the first phase, in which we repeatedly apply Lemma \ref{['lem:aaez']}, and at the end of which we are left with leaves each containing at most $N = n^{d/(d+1)}$ elements of $P$. A dual partition tree is constructed from each leaf $v$ by repeatedly applying Lemma \ref{['lem:mp']}. The blue and red subtrees depict the nodes in which, respectively, some element of $P$ (that is, of $P^*$) and some element of $S$ appear.
• Figure 5: The visibility graph of a simple polygon.

Theorems & Definitions (32)

• Theorem 1
• Corollary 1
• Corollary 2
• Theorem 2
• Theorem 3
• Theorem 4: Alon MR4800729
• Theorem 5
• proof
• Lemma 1
• proof