[Paper Review] Early Pruning in the Restricted Postage Stamp Problem

Jukka Kohonen|arXiv (Cornell University)|Mar 11, 2015

Algorithms and Data Compression6 references7 citations

TL;DR

This paper presents an improved algorithm for solving the restricted postage stamp problem by leveraging early pruning through element-wise lower bounds derived from mirroring and range constraints. By applying these bounds during prefix construction, the authors drastically reduce search space, enabling the enumeration of all extremal restricted bases up to length k=47—previously unknown—using optimized pruning and the meet-in-the-middle approach with Challis's algorithm for prefix generation.

ABSTRACT

A set of non-negative integers is an additive basis with range $n$, if its sumset covers all consecutive integers from 0 to $n$, but not $n+1$. If the range is exactly twice the largest element of the basis, the basis is restricted. Restricted bases have important special properties that facilitate efficient searching. With the help of these properties, we have previously listed the extremal restricted bases up to length $k = 41$. Here, with a more prudent use of the properties, we present an improved search algorithm and list all extremal restricted bases up to $k = 47$.

Motivation & Objective

• To extend the known extremal restricted additive bases beyond k=41 by improving search efficiency.
• To develop a more effective pruning strategy that leverages structural properties of restricted bases.
• To compute all extremal restricted bases for lengths k=42 through k=47.
• To validate that the bases found are complete and symmetric, without prior assumption of symmetry.

Proposed method

• Utilizes the mirroring property: if A is a restricted basis, then a_k - A is also a restricted basis with the same range.
• Applies element-wise lower bounds derived from Theorem 4: a_j ≥ a_k - n_2(k-j-1) - 1, which are enforced early in prefix construction.
• Employs range bounds from Corollary 5: n_2(A_j) ≥ a_k - n_2(k-j-2) - 2, used to filter prefixes early.
• Integrates Challis’s algorithm for generating admissible prefixes, enhanced with real-time checks against lower bounds to prune invalid paths.
• Uses a meet-in-the-middle strategy: compute valid prefixes up to midpoint, then extend and merge with suffixes.
• Starts from upper bounds on n_2^*(k) and decreases range in steps of 2 until a valid restricted basis is found.

Experimental results

Research questions

• RQ1 Can early pruning using element-wise lower bounds significantly reduce the search space in the restricted postage stamp problem?
• RQ2 What is the complete set of extremal restricted additive bases for lengths k=42 to k=47?
• RQ3 Are all extremal restricted bases of length k ≤ 47 symmetric, and can this be established without prior assumption?
• RQ4 How does the availability of n_2(k) values up to k=24 affect the feasibility of computing n_2^*(k) for larger k?

Key findings

• The authors successfully computed all extremal restricted additive bases for lengths k=42 through k=47, extending the previous limit of k=41.
• The search effort was reduced by orders of magnitude: for k=30, the modified algorithm visited only 1.9×10^6 prefixes (vs. 4.0×10^8) and ran 300× faster.
• Only 4,020 valid 21-prefixes were generated for k=47, indicating highly effective pruning via early range and element-wise bounds.
• All computed extremal restricted bases of lengths k=42 to k=47 are symmetric, confirming the results of Challis and Robinson’s preamble-amble construction without assuming symmetry a priori.
• The method demonstrates that early pruning using lower bounds derived from known n_2 values (up to k=24) enables scalable computation of n_2^*(k) up to k=47.