Range Partition Entropy: Theory & Applications

• Range Partition Entropy is a unified framework that quantifies data uncertainty by partitioning data sets into geometric or semantic blocks, generalizing classical measures like Shannon entropy.
• It integrates adaptive sorting entropy and structural entropy to provide a versatile model that bounds uncertainty between 0 and log n while adapting to discrete, continuous, and multivariate scenarios.
• The framework underpins efficient algorithms and data structures in machine learning and computational geometry, offering both exact and approximate query solutions.

Range Partition Entropy is a unified framework for quantifying the uncertainty of a set, sequence, or sample of data under geometrically or semantically meaningful partitions. This concept subsumes classical measures such as Shannon entropy for discrete distributions, instance-adaptive entropy for sorting, and structural entropy in computational geometry, and generalizes to continuous, multivariate, and nonparametric scenarios. Range partition entropy also underpins recent advances in efficient algorithms, data structures, and differentiable regularizers in machine learning and information theory.

1. Formal Definitions and Core Concepts

Range partition entropy is based on partitioning a point set or domain into blocks (also called cells or segments) according to some range family, and assigning entropy via the block size distribution. Given a set $S = \{p_1,\ldots, p_n\}$ in $\mathbb{R}^d$ and a range family $\mathcal{R}$ (e.g., intervals in $\mathbb{R}$, axis-aligned rectangles in $\mathbb{R}^2$, halfspaces, balls, or arbitrary measurable sets), a range partition is a collection $\Pi = \{(S_1, R_1), \dots, (S_t, R_t)\}$ such that:

• $\{S_1, \ldots, S_t\}$ forms a partition of $S$,
• Each $S_i$ is contained in $R_i \in \mathcal{R}$,
• $\Pi$ satisfies additional local/global properties for the problem at hand (respectfulness).

The entropy of a partition $\Pi$ is defined as the Shannon entropy of the frequency distribution:

$$H(\Pi) = -\sum_{i=1}^t \frac{|S_i|}{n} \log \frac{|S_i|}{n}$$

The range partition entropy of $S$ is then the minimum $H(\Pi)$ over all respectful partitions, denoted $H_\mathcal{R}(S)$ or simply $H(S)$ (Eppstein et al., 28 Aug 2025, Shihab et al., 3 Sep 2025).

For weighted, colored point sets $P$ with weights $w(p)$ and color $u(p)$, and for a query subset $P' \subseteq P$, the entropy is computed on the color distribution:

$$p_u = \frac{W(P'(u))}{W(P')} \quad\text{where}\quad W(P') = \sum_{p\in P'} w(p)$$

with range Shannon (S-) entropy:

$H_1(P') = -\sum_u p_u \log p_u$

and Rényi (R-) entropy of order $\alpha \neq 1$:

$$H_\alpha(P') = \frac{1}{1-\alpha} \log \sum_u p_u^\alpha$$

(Esmailpour et al., 2023).

2. Connection to Existing Entropy Measures

Range partition entropy generalizes several established entropy notions:

• Classical Shannon entropy for discrete distributions (e.g., count or frequency histograms)
• Run-based entropy in adaptive sorting, where partitions are monotonic runs (Eppstein et al., 28 Aug 2025)
• Structural entropy for geometric structures (maxima, convex hulls), where each block fits under an output structure
• Differential entropy for continuous distributions, via k-d tree equiprobable histograms (Keskin, 2021)
• Partition-based entropy in algorithmic convergence, using partitions of the input domain and maximal uncertainty measures (Slissenko, 2016)

The range partition entropy always satisfies

$0 \leq H(S) \leq \log n$

and specializes to prior entropy measures under appropriate choices of $\mathcal{R}$ and partition type.

3. Data Structures for Range Entropy Queries

Efficiently answering entropy queries for subsets of data (especially under semantically rich partitions) is nontrivial:

• For points $P \subset \mathbb{R}^d$ (weighted, colored), the goal is to preprocess $P$ into a structure that can, for a query rectangle $R$, return $H_1(P \cap R)$ or $H_\alpha(P \cap R)$ quickly.
• Conditional lower bounds indicate that near-linear-space, near-constant query time for exact range entropy is impossible unless the well-known set-intersection conjecture fails. One must pay $\tilde{\Omega}((n/Q(n))^2)$ space for $Q(n)$-time queries (Esmailpour et al., 2023).

Exact algorithms:

• In $d = 1$, partition into $k = n^{1-t}$ buckets, precompute entropies for all $O(k^2)$ bucket intervals, use BSTs for color lookups, and answer in $O(n^t \log n)$ time with $O(n^{2(1-t)})$ space.
• For $d > 1$, use color bucketing, precompute for $O(n^{2dt})$ axis-aligned rectangles per bucket, maintain min/max colors, and build $2d$-dimensional range-trees. Query time is $O(n^{1-t} \log^{2d} n)$, space $O(n \log^{2d-1} n + n^{(2d-1)t+1})$ (Esmailpour et al., 2023).

Approximate algorithms:

• Additive $\pm\Delta$ and multiplicative $(1+\varepsilon)$ approximations can be supported in near-linear space ($O(n\log^d n)$), with query time $O(\log^{O(d)} n / \Delta^2)$ or $O(\log^{O(d)} n / \varepsilon^2)$.
• Faster methods for $d=1$ exploit canonical block merges and monotonic tables to offer $O(\frac{n}{\varepsilon}\log^2 n)$ space and $O(\log^2 n \log\frac{\log n}{\varepsilon})$ query time for $(1+\varepsilon)$-accuracy (Esmailpour et al., 2023).

4. Range Partition Entropy in Algorithm and Information Theory

The entropy of a range partition appears explicitly in the running time of entropy-bounded algorithms:

• In sorting and geometric problems (e.g., 2D maxima, 2D/3D convex hull), algorithms using instance-optimal recursion achieve expected time $O(n(1 + H(S)))$, where $H(S)$ is the range partition entropy for the input (Eppstein et al., 28 Aug 2025).
• The analysis uses key mathematical properties: monotonicity under partition refinement, subadditivity under set splitting, and convexity in block frequencies.

Partition-entropy-based frameworks also extend to the analysis of convergence of algorithms in terms of informational progress, where each "event" (e.g., program guard, assignment) induces a partition and an associated reduction in uncertainty (Slissenko, 2016).

5. Nonparametric and Multivariate Extensions

In nonparametric inference, order statistics induce an equiprobable partition of the sample space:

• $N$ sorted samples from any continuous distribution partition the real line into $N+1$ intervals; each interval has expected probability mass $1/(N+1)$ (Eriksson, 29 Jul 2025).
• The entropy of this partition is $\log_2(N+1)$ bits, providing a sample-based finite uncertainty that contrasts with the continuous variable case.
• Multivariate generalization uses k-d tree partitioning, aligning bins to achieve equiprobable histograms, and possibly optimizing partition orientation by minimizing bin volume variance (Keskin, 2021).
• This approach provides lower bias entropy estimates for correlated or small-sample high-dimensional data.

6. Differentiable Surrogates and Learning Applications

Range partition entropy has been made differentiable and tractable for incorporation into deep learning and geometric algorithm design:

• Ball-based and halfspace-aware soft partition surrogates allow computation of smooth entropy approximations. For data $S$ and anchors $c_j$, define soft assignments $p_{ij}$ and entropy $-\sum_j p_j \log p_j$ ($p_j$ is the population in soft cell $j$) (Shihab et al., 3 Sep 2025).
• The "EntropyNet" neural module restructures data to minimize range-partition entropy, leading to substantial runtime reduction in geometric algorithms (e.g., up to $4.1\times$ on convex hull tasks) and improved F1 scores and accuracy under entropy-regularized transformer attention sparse regimes.
• Theoretical approximation guarantees relate soft surrogates to ground-truth range-partition entropy, accounting for partition margin and regularization parameters.

7. Illustrative Examples and Practical Implications

• In adaptive sorting, lower entropy (fewer, larger monotonic runs) yields reduced computational effort.
• For geometric problems, recognizing sorted or nearly sorted segments reduces recursion and accelerates processing.
• Sample-based entropy via equal-probability range partitioning offers principled uncertainty quantification for density estimation, including robust tail treatments.
• Range entropy queries facilitate statistical exploration, data compression, and efficient block construction in storage systems, but efficient support is subject to fundamental lower bounds (Esmailpour et al., 2023).

Domain	Partition Type	Entropy Formula
Sorting	Monotonic runs	$-\sum_i |R_i|/n\,\log(|R_i|/n)$
Geometric algs	Respectful cells	$-\sum_i |S_i|/n\,\log(|S_i|/n)$
Discrete queries	Colors in $R$	$-\sum_u p_u \log p_u$
Continuous (1-D)	Order intervals	$\log_2(N+1)$
Multivariate	k-d tree blocks	$\log B + \frac{1}{B}\sum_{i=1}^B \log V_i$

Range partition entropy thus synthesizes classical and novel measures of information, governs adaptive algorithm complexity, and is integral to fast data analysis, density estimation, and contemporary machine learning regularization.