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arxiv: 2604.27548 · v1 · submitted 2026-04-30 · 💻 cs.DS

Smallest suffixient set maintenance in near-real-time

Dominik K\"oppl , Gregory Kucherov This is my paper

Pith reviewed 2026-05-07 08:38 UTC · model grok-4.3

classification 💻 cs.DS
keywords smallest suffixient setonline maintenancestring repetitivenessWeiner suffix treenear-real-timedata structuresstreaming textincremental algorithms
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The pith

The smallest suffixient set of a string can be maintained online with polyloglog time per letter using Weiner's suffix tree.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper shows how to keep the smallest suffixient set up to date as a string grows letter by letter. The set measures the repetitiveness of the string by identifying the fewest positions needed to capture its suffix structure. Maintaining it efficiently matters for real-time applications like compression of streaming data or analyzing repetitive sequences without delay. The method works for text added from either end and relies on efficient updates to a suffix tree.

Core claim

We study how to maintain the smallest suffixient set online in near-real-time, that is with small (in our case, polyloglog) worst-case time on processing each letter. Two frameworks are considered: when the text is given letter-by-letter in either a right-to-left or left-to-right direction. Our central algorithmic tool is Weiner's suffix tree algorithm and associated algorithmic primitives for its efficient implementation.

What carries the argument

Weiner's suffix tree algorithm and its associated algorithmic primitives, which support incremental updates to track the smallest suffixient set.

If this is right

  • The repetitiveness measure becomes available for streaming strings without waiting for the full text.
  • Applications in compression and pattern matching can operate in near real time on growing inputs.
  • Both right-to-left and left-to-right arrival directions receive the same polyloglog bound.
  • The approach yields a dynamic structure that reports the size of the smallest suffixient set after each update.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same incremental primitives might extend to maintaining other repetitiveness measures such as the smallest grammar size online.
  • Integration with fully dynamic suffix trees could enable queries on the evolving set without restarting construction.
  • Practical implementations on repetitive data like genomes could test whether the polyloglog bound yields observable speedups over rebuilding from scratch.
  • The technique might combine with online compression schemes to produce adaptive compressors that track repetitiveness continuously.

Load-bearing premise

Weiner's suffix tree algorithm and its primitives can be adapted to support online maintenance with only polyloglog time per letter without additional factors depending on alphabet size or other hidden costs.

What would settle it

A concrete counterexample string where any suffix-tree-based maintenance of the smallest suffixient set requires more than polyloglog time per letter in the worst case would disprove the efficiency result.

read the original abstract

The size of the \textit{smallest suffixient set} of positions of a string recently emerged as a new measure of string \textit{repetitiveness} -- a measure reflecting how much of repetitive content the string contains. We study how to maintain the smallest suffixient set online in near-real-time, that is with small (in our case, polyloglog) worst-case time on processing each letter. Two frameworks are considered: when the text is given letter-by-letter in either a right-to-left or left-to-right direction. Our central algorithmic tool is Weiner's suffix tree algorithm and associated algorithmic primitives for its efficient implementation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper studies online maintenance of the smallest suffixient set (a repetitiveness measure) for a string processed letter-by-letter, either right-to-left or left-to-right, targeting polyloglog worst-case time per letter. The central tool is an adaptation of Weiner's suffix tree algorithm together with its associated algorithmic primitives.

Significance. If the polyloglog bound can be established rigorously, the result would advance streaming algorithms for string repetitiveness measures, with potential uses in real-time compression and data processing. The grounding in Weiner primitives is a methodological strength when the implementation details are supplied.

major comments (2)
  1. [Abstract] Abstract: the polyloglog time bound is asserted without any derivation, proof sketch, or identification of the specific primitives (e.g., constant-time suffix-link jumps or dynamic-tree height) that would eliminate the linear total cost of suffix-link traversals present in standard Weiner implementations.
  2. [Central algorithmic tool] Central algorithmic tool paragraph: the claim that Weiner primitives support online maintenance with only polyloglog cost per letter does not address potential |Σ|-dependent or word-RAM log-log factors in suffix-link navigation or dynamic tree operations; no lemma isolates the exact per-operation cost (e.g., “suffix-link traversal in O((log log n)^c)”).
minor comments (1)
  1. The term 'suffixient' should be defined on first use and its relation to existing repetitiveness measures (e.g., run-length or LZ) clarified for readers outside the immediate sub-area.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for greater clarity on the time-complexity claims. We agree that the abstract and central-tool paragraph would be strengthened by explicit identification of the primitives and a high-level cost analysis. We will revise the manuscript to address both points while preserving the core technical contribution.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the polyloglog time bound is asserted without any derivation, proof sketch, or identification of the specific primitives (e.g., constant-time suffix-link jumps or dynamic-tree height) that would eliminate the linear total cost of suffix-link traversals present in standard Weiner implementations.

    Authors: We acknowledge that the abstract currently states the polyloglog bound without a supporting sketch. The full manuscript derives the bound from Weiner primitives that replace naïve suffix-link traversals with constant-time jumps (via precomputed or dynamically maintained link arrays) and dynamic-tree height operations whose cost is bounded by the inverse Ackermann function or polyloglog factors in the word-RAM model. In the revision we will add a concise proof outline to the abstract and a dedicated paragraph in the introduction that names these primitives and explains why the total per-letter cost remains polyloglog rather than linear. revision: yes

  2. Referee: [Central algorithmic tool] Central algorithmic tool paragraph: the claim that Weiner primitives support online maintenance with only polyloglog cost per letter does not address potential |Σ|-dependent or word-RAM log-log factors in suffix-link navigation or dynamic tree operations; no lemma isolates the exact per-operation cost (e.g., “suffix-link traversal in O((log log n)^c)”).

    Authors: The paragraph introduces the primitives but does not isolate their costs. We will insert a new lemma (placed immediately after the description of the adapted Weiner algorithm) that states: each letter insertion performs a constant number of suffix-link jumps and dynamic-tree height queries, each executable in O((log log n)^O(1)) time under the word-RAM model with |Σ| treated as a constant or handled via standard alphabet-reduction techniques. The lemma will cite the known polyloglog implementations of Weiner’s algorithm and explicitly bound the navigation cost, thereby eliminating any hidden linear factors. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on independent algorithmic primitives

full rationale

The paper constructs an online maintenance algorithm for the smallest suffixient set by adapting Weiner's suffix tree construction (a 1973 result external to the authors) and its associated primitives for right-to-left or left-to-right letter-by-letter processing. The polyloglog time bound is presented as a consequence of this adaptation rather than being presupposed or fitted from the target output. No equations or steps reduce the claimed result to a self-definition, a renamed input, or a load-bearing self-citation chain; the central tool is cited as established rather than derived within the paper. The skeptic concern addresses potential hidden implementation costs (correctness risk) but does not identify any reduction of the derivation to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach relies on standard suffix tree algorithms as the central tool but extends them to the online setting; no free parameters or new entities are introduced based on the abstract.

axioms (1)
  • domain assumption Weiner's suffix tree algorithm and its associated algorithmic primitives can be efficiently adapted for online, incremental maintenance in polyloglog time
    This is explicitly stated as the central algorithmic tool in the abstract.

pith-pipeline@v0.9.0 · 5397 in / 1243 out tokens · 41806 ms · 2026-05-07T08:38:44.037593+00:00 · methodology

discussion (0)

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Reference graph

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