Source-linked AI summary
Constrained Maximum Entropy Contiguous Aggregations
Roberto Bruno, Ugo Vaccaro
TL;DR
The paper studies maximum-entropy contiguous aggregations under an upper bound R on Shannon entropy. It gives an exact dynamic programming algorithm and two time-efficient greedy algorithms with close-to-optimal guarantees. The problem is motivated by information-theoretic and data-processing applications requiring alphabet or entropy reduction while preserving statistical information.
Problem
The problem is to compute a maximum-entropy contiguous aggregation of p whose Shannon entropy is at most R.
Method
The paper develops an exact dynamic programming algorithm and two time-efficient greedy approximation algorithms for the constrained optimization problem.
Results
The exact algorithm solves the problem, while the greedy algorithms provide additive and multiplicative approximation guarantees.
Takeaways & Limitations
The problem is relevant to alphabet partitioning, alphabetic coding, scalar quantization, and histogram segmentation, where contiguity can be important.
Takeaways & Limitations
The authors conjecture that the maximization problem is NP-hard and identify a polynomial-time approximation scheme as an open problem.
Abstract
from arXiv · showhide
Given a probability distribution $p = (p_1, \dots, p_n)$ and an integer $1\leq m \leq n$, a contiguous aggregation of $p$ is a probability distribution $q = (q_1, \dots, q_m)$ such that each $q_i$ is a sum of consecutive elements of $p$. Given $p$ and a positive number $R$, we consider the problem of computing a maximum entropy contiguous aggregation $q$ of $p$, under the constraint that its Shannon entropy $H(q)$ is at most $R$. We devise a dynamic programming algorithm that solves the problem exactly, and two time-efficient greedy algorithms that provide close-to-optimal solutions. We discuss a few scenarios where our problem arises.
I. INTRODUCTION
The paper formalizes probability distributions, Shannon entropy, and contiguous aggregations, where each output component sums a consecutive segment of the input. It studies such aggregations through the set CA(p).
- A probability distribution p lies in the (n −1)-dimensional probability simplex, with nonnegative components summing to one.
- Shannon entropy is defined as H(p) = −∑ p_i log2 p_i for a probability distribution p.
- A contiguous aggregation q of p is formed by partitioning p into consecutive segments and summing each segment.
- CA(p) denotes the set of all probability distributions that are contiguous aggregations of p.
A. Our Results
The paper presents an exact dynamic programming algorithm for the constrained optimization problem and two time-efficient greedy approximation algorithms. The greedy methods come with additive and multiplicative approximation analyses.
- The dynamic programming approach computes the optimization problem’s solution exactly but has Θ(2^n)-time complexity in the worst case.
- The first greedy algorithm returns a value within h(p↓⌊2R⌋+1) of the optimum, where p↓⌊2R⌋+1 is the corresponding largest probability.
- The second greedy algorithm returns a value within 1/(⌊(⌊2R⌋−1)/2⌋+ 1) of the optimum.
- Both greedy algorithms are also analyzed using multiplicative approximation factors, which are more useful when the optimum is very small.
II. MOTIVATIONS AND RELATED WORK
Informative contiguous aggregations arise in information theory and data processing, where alphabet or entropy reduction must preserve statistical information. Contiguity matters in applications including alphabetic coding, scalar quantization, and histogram segmentation.
- Contiguous aggregation supports reducing source-alphabet size or source entropy while preserving as much statistical information as possible.
- Alphabet Partitioning groups symbols into clusters to speed coding algorithms for large alphabets, but prior work generally ignores symbol contiguity.
- Preserving contiguity is crucial in applications such as alphabetic coding, scalar-quantizer design, and histogram segmentation.
III. A DYNAMIC PROGRAMMING ALGORITHM
The paper develops a dynamic programming algorithm that tracks feasible entropy values for contiguous partitions, correcting a natural but insufficient maximum-only approach. Its worst-case time and space complexity is Θ(2^n).
- A natural dynamic program that stores only the maximum-entropy aggregation for each segment count can miss the constrained optimum.An inadmissible higher-entropy aggregation may dominate the stored aggregation while another aggregation with the same segment count satisfies H(q) ≤ R.
- Prefix sums P_i allow each segment contribution to be computed as E(j, i) = −(P_i − P_j) log2(P_i − P_j).P_i represents the cumulative mass through position i, so P_i − P_j gives the mass of the segment from j+1 through i.
- The corrected state DP[k, i] stores entropy values achievable by partitioning the prefix p_1, …, p_i into k segments.The recurrence is built from segment contributions and retains values relevant to the entropy bound R.
- The base case for a one-segment prefix retains E(0, i) when it is at most R and otherwise stores the empty set.This initializes feasible entropy values for prefixes containing a single contiguous segment.
- Θ(2^n) worst-case time and space complexity arises because the DP state may contain a distinct entropy value for every partition.The exponential bound occurs when different partitions generate different entropy values and this unfavorable case occurs often.
IV. EFFICIENT GREEDY APPROXIMATION ALGORITHMS
The paper presents two greedy algorithms for constrained entropy maximization, each iteratively merging contiguous probabilities until the entropy bound is met. They offer efficient computation with approximation guarantees, but neither algorithm uniformly dominates the other.
- Setup: The algorithms assume R < log2 n, because for R ≥ log2 n the original distribution p is optimal.Both methods construct aggregations of decreasing entropy and stop at the first one with entropy at most R.
- Algorithm 1: Algorithm 1, AvoidMaxAggregation, processes probabilities using the largest ⌊2R⌋ values and runs in O(n) time.It avoids merging the ⌊2R⌋ largest probabilities with one another and computes the relevant order statistics and merges by linear scans.
- Algorithm 1: Algorithm 1 returns q with OPT(p, R) − H(q) ≤ h(p↓⌊2R⌋+1), where p↓⌊2R⌋+1 is the corresponding order statistic.The bound uses the binary entropy function h and the (⌊2R⌋ + 1)-th largest probability in p.
- Guarantees: Both algorithms also receive multiplicative approximation analyses, which are more useful than additive bounds when the optimum is very small.The paper states this explicitly for Algorithm 1 and presents an analogous multiplicative guarantee for Algorithm 2.
- Algorithm 2: Algorithm 2 repeatedly merges the consecutive pair with the smallest sum and runs in O(n log n) time.A heap supports each smallest-sum search in O(log n), while entropy updates use only the entropy decrease.
- Comparison: The two greedy algorithms are not directly comparable: Algorithm 1 performs better on one distribution, whereas Algorithm 2 performs better on a uniform distribution.For R = 0.9, their reported entropies are approximately 0.7415 and 0; for n = 10 and R = 3, they are approximately 2.846 and 2.921.
V. CONCLUSIONS AND OPEN PROBLEMS
The paper combines an exact but inefficient dynamic program with efficient approximation algorithms for constrained entropy maximization. It conjectures NP-hardness and identifies a polynomial-time approximation scheme as an open direction.
- Contributions: The paper derives an exact but inefficient algorithm and efficient approximation algorithms with good guarantees for the constrained entropy maximization problem.It also describes scenarios where the problem is relevant.
- Open problems: The authors conjecture that the maximization problem is NP-hard.
- Open problems: A proposed extension is a polynomial-time approximation scheme based on suitably rounding input probabilities in the dynamic programming algorithm.