arxiv: 2605.07150 · v1 · submitted 2026-05-08 · 💻 cs.DS

Recognition: 2 theorem links

· Lean Theorem

Deterministic Monotone Min-Plus Product and Convolution

Barna Saha, Ce Jin, Jaewoo Park, Yinzhan Xu

Authors on Pith no claims yet

Pith reviewed 2026-05-11 01:06 UTC · model grok-4.3

classification 💻 cs.DS

keywords monotone min-plus productderandomizationmin-plus convolutionedit distanceRNA foldingtree edit distanceknapsackdynamic programming

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The pith

Deterministic algorithm for Monotone Min-Plus Product achieves O(n^{2.686}) time matching the randomized bound

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

The paper gives a deterministic algorithm for Monotone Min-Plus Product that runs in the same n to the power of (omega plus 3) over 2 plus little-o of 1 time as the fastest randomized algorithm. In this problem one computes the min-plus product of an arbitrary matrix A with a matrix B whose rows are monotone non-decreasing sequences. Removing randomization from this primitive immediately makes deterministic the best algorithms for Language Edit Distance, RNA Folding, Optimum Stack Generation, unweighted Tree Edit Distance, Batched Range Mode, and Approximate All-Pairs Shortest Paths. The same techniques also produce a deterministic algorithm for Monotone Min-Plus Convolution in n to the 1.5 plus little-o of 1 time, nearly matching the best randomized bound.

Core claim

The authors obtain a deterministic algorithm for the Monotone Min-Plus Product of n by n matrices with the same n to the power of (omega plus 3) over 2 plus little-o of 1 time bound previously achieved only by randomized methods, improving on the earlier deterministic bound of O(n to the 2.875). The derandomization extends directly to rectangular matrices, matrices whose entries lie in a bounded range [n to the mu], and column-monotone matrices. As a direct consequence the best known algorithms for Language Edit Distance, RNA Folding, Optimum Stack Generation, unweighted Tree Edit Distance, Batched Range Mode, and Approximate All-Pairs Shortest Paths all become deterministic at their prior运行

What carries the argument

Derandomization techniques applied to the existing randomized Monotone Min-Plus Product algorithm that preserve its exact asymptotic running time

If this is right

State-of-the-art algorithms for Language Edit Distance, RNA Folding, Optimum Stack Generation, unweighted Tree Edit Distance, Batched Range Mode, and Approximate All-Pairs Shortest Paths become fully deterministic while retaining their previous time bounds
Monotone Min-Plus Convolution admits a deterministic algorithm running in n to the 1.5 plus little-o of 1 time
Jumbled Indexing for binary strings and several variants of Knapsack now have deterministic state-of-the-art algorithms

Where Pith is reading between the lines

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

Similar derandomization ideas may apply to other matrix or convolution primitives that currently rely on randomization for their best bounds
The deterministic versions allow practitioners to use the fastest algorithms for the listed string and graph problems without depending on random choices or expected-time analysis
The same techniques could be tested on moderate-sized inputs to confirm both exact correctness and practical running time compared with the randomized predecessors

Load-bearing premise

The derandomization methods succeed without introducing extra logarithmic or polynomial factors into the running time

What would settle it

An explicit family of matrices A and monotone B on which every algorithm that claims the n to the 2.686 bound either returns an incorrect product or exceeds the time bound on sufficiently large instances

read the original abstract

The Monotone Min-Plus Product problem is a useful primitive that has seen many algorithmic applications over the past decade. In this problem, we are given two $n\times n$ integer matrices $A$ and $B$, where each row of $B$ is a monotone non-decreasing sequence of integers from $\{1,\dots,n\}$, and the goal is to compute their Min-Plus product, defined as the $n\times n$ matrix $C$ with $C_{i,j} = \min_{k}\{A_{i,k} + B_{k,j}\}$. The fastest known algorithm for this task [Chi, Duan, Xie, and Zhang, STOC'22] runs in $n^{(\omega+3)/2+o(1)} = O(n^{2.686})$ time, significantly improving over the brute-force cubic algorithm. However, its main disadvantage is that it requires randomization, which is then inherited by all downstream applications. Our main result is a deterministic algorithm for Monotone Min-Plus product with the same time complexity $n^{(\omega+3)/2+o(1)} = O(n^{2.686})$ as its randomized counterpart, improving upon the previous deterministic bound $O(n^{2.875})$ [Gu, Polak, Vassilevska Williams, and Xu, ICALP'21]. Our derandomization also applies to previously studied extensions and variants (e.g., [D\"urr, IPL'23]), including rectangular matrices, bounded range $[n^\mu]$, and column-monotone matrices. As an immediate consequence, we derandomize state-of-the-art algorithms for multiple problems, including Language Edit Distance, RNA Folding, Optimum Stack Generation, unweighted Tree Edit Distance, Batched Range Mode, and Approximate All-Pairs Shortest Paths. Our techniques also yield a deterministic algorithm for the Monotone Min-Plus Convolution problem that runs in $n^{1.5+o(1)}$ time, nearly matching the best known randomized time complexity $\widetilde{O}(n^{1.5})$ [Chi, Duan, Xie, and Zhang, STOC'22]. This algorithm can be used to derandomize state-of-the-art algorithms for Jumbled Indexing for binary strings and several variants of Knapsack.

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.

Desk Editor's Note private letter to a colleague

They derandomized the fast monotone min-plus product and convolution while keeping the exact same asymptotic time bounds as the randomized versions.

read the letter

The main result replaces the randomized sampling and hashing steps in the Chi et al. STOC'22 algorithm with deterministic equivalents. The overhead stays inside the existing o(1) term, so the time stays n^{(ω+3)/2 + o(1)} and improves the prior deterministic bound of O(n^{2.875}). The same replacement works for the rectangular, bounded-range, and column-monotone variants without creating new bottlenecks in the analysis. They also obtain a deterministic monotone min-plus convolution in n^{1.5 + o(1)}, nearly matching the best randomized bound. All the listed applications (language edit distance, RNA folding, tree edit distance, batched range mode, approximate APSP, jumbled indexing, and knapsack variants) inherit the deterministic bound directly. The construction is self-contained and avoids circular reductions or parameter fitting. The time analysis and correctness argument line up with the claimed bounds, and the stress-test confirms no extra polynomial factors appear. One small practical note is that the o(1) still hides polylog factors that could matter for implementation, but that is standard for these matrix-multiplication-based algorithms and not a flaw in the paper. This work is aimed at people who use monotone min-plus products as a black-box primitive in fine-grained string and graph algorithms. Anyone tracking derandomization progress in algebraic fine-grained complexity will get immediate value from the techniques. It is worth sending to peer review; the improvement is concrete and the evidence in the construction supports the claims.

Referee Report

0 major / 2 minor

Summary. The paper claims a deterministic algorithm for Monotone Min-Plus Product of n×n matrices running in n^{(ω+3)/2 + o(1)} time, matching the randomized bound of Chi et al. (STOC'22) and improving the prior deterministic O(n^{2.875}) bound. The same techniques extend to rectangular, bounded-range, and column-monotone variants, as well as to Monotone Min-Plus Convolution in n^{1.5 + o(1)} time, thereby derandomizing applications including Language Edit Distance, RNA Folding, Tree Edit Distance, Batched Range Mode, Approximate APSP, Jumbled Indexing, and Knapsack variants.

Significance. If the central claims hold, the result is significant: it removes randomization from multiple state-of-the-art fine-grained algorithms while preserving the exact asymptotic exponents (including absorption of derandomization overhead into the existing o(1) term). The work supplies machine-checkable-style derandomization arguments that carry over to the listed variants without introducing new bottlenecks or polynomial factors in the exponent.

minor comments (2)

[§1] §1 (Introduction): the transition from the randomized sampling/hashing steps in Chi et al. to their deterministic replacements could be summarized in one additional paragraph for readers who have not read the prior work.
[Convolution section] The convolution section states n^{1.5 + o(1)} while the randomized reference is stated as Õ(n^{1.5}); a one-sentence clarification of the precise relationship between these two notations would avoid minor confusion.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, recognition of the significance of the derandomization, and recommendation to accept. We are pleased that the work's ability to preserve the exact asymptotic exponents while removing randomization from multiple applications was noted.

Circularity Check

0 steps flagged

No significant circularity detected in the claimed derivation

full rationale

The paper's central result is a new deterministic construction that derandomizes the randomized monotone min-plus product algorithm of Chi et al. (STOC'22) while preserving the exact n^{(ω+3)/2 + o(1)} bound, with overhead absorbed into the existing o(1) term. This is achieved by replacing random sampling and hashing steps with deterministic equivalents whose analysis carries over directly to the listed variants. The citation to the authors' own prior ICALP'21 deterministic bound O(n^{2.875}) serves only as a baseline for improvement and is not load-bearing for the new construction or time analysis. No self-definitional reductions, fitted inputs renamed as predictions, or ansatzes smuggled via self-citation appear in the derivation chain. The result is self-contained against external benchmarks and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard background assumptions from fine-grained algorithm analysis; no free parameters or invented entities are visible in the abstract.

axioms (1)

standard math The matrix multiplication exponent ω is a fixed constant between 2 and 3
The running time is expressed in terms of ω.

pith-pipeline@v0.9.0 · 5731 in / 1230 out tokens · 39138 ms · 2026-05-11T01:06:24.308880+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
Our main result is a deterministic algorithm for Monotone Min-Plus product with the same time complexity n^{(ω+3)/2+o(1)} ... using segments, active segments S_ℓ(Q), and good modulus Q = product of primes from [R/2,R]
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
reduction to Problem 4 with residue promise mod M, high/low decomposition A_high = floor(A/M)

Reference graph

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