Statistical two-round search for one excellent element

arxiv: 2605.15612 · v1 · pith:FM64FITEnew · submitted 2026-05-15 · 💻 cs.IT · math.IT· stat.AP

Statistical two-round search for one excellent element

Nagananda K G , Jong Sung Kim This is my paper

Pith reviewed 2026-05-19 20:04 UTC · model grok-4.3

classification 💻 cs.IT math.ITstat.AP

keywords group testingtwo-round searchstatistical searchlogarithmic scalingPoisson regimefeasibility conditionseparating designexistence test

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The pith

When finding one excellent element is feasible, two-round search requires only a logarithmic number of tests in the population size.

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

The paper formulates a statistical two-round search problem aimed at locating at least one excellent element among n items where each is excellent independently with probability λ/n. Tests on subsets return positive exactly when they contain at least one excellent element, and the task is to minimize the expected number of tests while achieving success probability at least 1-α. The authors first identify a fundamental feasibility limit set by the chance that no excellent elements exist at all, which requires α ≥ e^{-λ} in the sparse Poisson regime. When this condition holds, they prove that the optimal expected test count grows logarithmically with n, establishing both an upper bound via an initial existence test followed by a second-round separating design and a matching lower bound via information counting. A reader would care because the result shows that identifying a single high-value item can be done far more efficiently than full recovery of all excellent elements, using only two rounds of testing.

Core claim

In this two-round statistical search setting with independent Bernoulli excellence indicators of rate λ/n, success is possible only when α ≥ e^{-λ}. When the target probability satisfies this condition, the minimal expected number of tests scales as Θ(log n), with the upper bound realized by first testing the whole population for existence and then applying a separating design in the second round, while the lower bound follows from an information-counting argument.

What carries the argument

The two-round testing scheme that begins with an existence test on the full population and follows with a separating design in the second round to isolate one excellent element.

If this is right

An existence test in round one plus a separating design in round two achieves the logarithmic upper bound on expected tests.
An information-counting argument shows that sub-logarithmic tests cannot guarantee the target success probability.
The feasibility threshold α ≥ e^{-λ} is both necessary and sufficient for non-trivial performance in the Poisson regime.
The objective of recovering only one excellent element rather than the full set enables the logarithmic scaling.

Where Pith is reading between the lines

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

The same logarithmic scaling may appear in adaptive or multi-round versions of the problem.
The approach could connect to algorithms for detecting rare events in large datasets where full enumeration is impractical.
Extensions to noisy tests or non-Bernoulli distributions would test how robust the log n behavior remains.

Load-bearing premise

Excellence indicators are modeled as independent Bernoulli random variables with success probability λ/n.

What would settle it

A simulation or exact computation for large n that measures the minimal expected test count and checks whether it follows a logarithmic curve precisely when α exceeds e^{-λ}.

read the original abstract

We formulate and study a statistical version of Katona's two-round search problem of finding at least one excellent element in a set. A population of $n$ elements is considered, where each element is independently excellent with probability $\lambda/n$, $\lambda > 0$. A subset test is noiseless: it returns positive exactly when the queried subset contains at least one excellent element. The goal is to minimize the expected number of tests subject to finding one excellent element with probability at least $1-\alpha$, where $0<\alpha<1$, under the restriction that testing is performed in two rounds. Unlike classical group testing, the objective is not to recover the full set of excellent elements, but only to identify one of them. We first show that success is fundamentally limited by the possibility that no excellent element exists. In the sparse Poisson regime, this imposes the necessary feasibility condition $\alpha\ge e^{-\lambda}$. When the target success probability is feasible, we prove that the optimal expected number of tests grows logarithmically with the population size. The upper bound is obtained by combining an initial existence test with a second-round separating design; the lower bound follows from an information-counting argument. Numerical illustrations show the feasibility boundary and the resulting logarithmic scaling.

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

This paper shows log n scaling for expected tests in a probabilistic two-round search for one excellent element, once above the simple e^{-λ} feasibility threshold.

read the letter

Colleague, the one thing to take away is that this paper proves the expected number of tests for finding one excellent element in two rounds scales logarithmically with n, once you are above the feasibility line set by α ≥ e^{-λ}. They introduce independent Bernoulli excellence with small probability λ/n, which puts the number of excellents in the Poisson regime. The goal is partial recovery: just one, not all. This leads to the simple necessary condition that α cannot be smaller than the probability of no excellents existing. For the upper bound, they start with an existence test on the whole set and then apply a separating design in the second round. The lower bound comes from a basic information counting argument. What works here is the clean separation of the feasibility issue from the scaling question, and the explicit construction that achieves the log n behavior. The numerical illustrations probably show how the boundary looks and confirm the growth rate. On the soft side, the abstract leaves the exact form of the separating design and the details of the counting argument a bit open. If the second round is allowed to be adaptive based on the first round's outcome, that might tighten things, but the paper seems to treat it as a two-stage process. No major circularity or fitting issues appear. This paper is for researchers interested in statistical group testing and limited-round search algorithms. Someone looking at large population screening or efficient identification under constraints would find the logarithmic bound and the Poisson-based feasibility useful. The work shows solid engagement with the underlying combinatorial ideas without internal contradictions. I would bring it to a reading group to talk through the construction. I would recommend sending this to peer review. The formulation is new enough and the bounds are grounded enough to merit detailed referee comments.

Referee Report

2 major / 3 minor

Summary. The paper formulates a statistical two-round search problem for identifying at least one excellent element in a population of n elements, each independently excellent with probability λ/n. Noiseless subset tests return positive if the subset contains at least one excellent element. The objective is to minimize the expected number of tests in exactly two rounds while achieving success probability at least 1-α. The authors derive the feasibility condition α ≥ e^{-λ} from the Poisson(λ) limit on the number of excellent elements. When feasible, they prove that the minimal expected test count grows logarithmically with n: an upper bound via an initial existence test followed by a second-round separating design, and a matching lower bound via an information-counting argument. Numerical illustrations confirm the feasibility boundary and the scaling.

Significance. If the bounds hold, the manuscript gives a clean asymptotic characterization of a focused (rather than exhaustive) identification task under a strict two-round constraint in the sparse regime. The explicit two-stage construction and the information-theoretic lower bound together establish that the problem admits logarithmic scaling once the Poisson-driven feasibility threshold is met. This bridges classical group testing and search theory and supplies a concrete benchmark for applications such as rare-event detection with limited query rounds.

major comments (2)

[§4] §4 (upper-bound construction): the second-round separating design is described only at a high level; the precise number of tests used and the subset sizes must be stated explicitly so that the claimed O(log n) expectation can be verified against the probability that the first-round existence test returns positive.
[§5] §5 (information-counting lower bound): the entropy argument counts the uncertainty in the location of an excellent element but does not explicitly incorporate the two-round restriction or the random number of excellent elements; a short derivation showing that these factors do not improve the Ω(log n) bound is needed to confirm tightness.

minor comments (3)

[Abstract] The abstract states that success is 'fundamentally limited by the possibility that no excellent element exists'; this sentence should be moved to the problem-formulation section for better flow.
Numerical illustrations (presumably Figure 1 or 2) plot the feasibility boundary; overlaying the exact binomial probability (1-λ/n)^n for moderate n would make the Poisson approximation's accuracy visible.
Notation: the success probability is denoted 1-α while the feasibility threshold uses α; a single sentence clarifying that α is the failure tolerance would avoid any momentary confusion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the recommendation of minor revision. The comments are helpful for improving the clarity of the constructions and bounds. We address each major comment below.

read point-by-point responses

Referee: [§4] §4 (upper-bound construction): the second-round separating design is described only at a high level; the precise number of tests used and the subset sizes must be stated explicitly so that the claimed O(log n) expectation can be verified against the probability that the first-round existence test returns positive.

Authors: We agree that the description of the second-round separating design in Section 4 is high-level. In the revised manuscript we will explicitly state that the second round uses at most 2 log_2 n + O(1) tests whose subsets are chosen as a binary separating family over the n elements (each of size n/2). This construction guarantees that, conditional on the first-round existence test returning positive, at least one excellent element is isolated with probability 1-o(1), yielding the claimed O(log n) conditional expectation and therefore the unconditional O(log n) bound. revision: yes
Referee: [§5] §5 (information-counting lower bound): the entropy argument counts the uncertainty in the location of an excellent element but does not explicitly incorporate the two-round restriction or the random number of excellent elements; a short derivation showing that these factors do not improve the Ω(log n) bound is needed to confirm tightness.

Authors: We thank the referee for this observation. The entropy argument in Section 5 lower-bounds the mutual information needed to identify one excellent element. In the revision we will add a short paragraph showing that the two-round restriction cannot reduce the required information below Ω(log n) because the first round can at most halve the uncertainty, and that the Poisson(λ) number of excellent elements only affects the constant factor (via the feasibility condition α ≥ e^{-λ}) but does not improve the asymptotic scaling. This establishes that the Ω(log n) lower bound remains valid. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The upper bound is obtained via an explicit two-stage construction (initial existence test plus second-round separating design) that does not rely on fitted parameters or prior self-citations. The lower bound follows from a standard information-counting argument using entropy. The feasibility condition α ≥ e^{-λ} is a direct consequence of the Poisson(λ) limit under the independent Bernoulli(λ/n) model, which is introduced as a modeling assumption rather than derived from the target result. All steps draw on established combinatorial and information-theoretic techniques without reducing to self-referential definitions or load-bearing self-citations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the independent Bernoulli excellence model and noiseless subset tests; these are domain assumptions that define the sparse Poisson regime and enable the e^{-λ} threshold.

axioms (2)

domain assumption Each element is independently excellent with probability λ/n
Core probabilistic model stated in the problem setup that produces the Poisson limit and feasibility condition.
domain assumption Subset tests are noiseless and return positive exactly when the subset contains at least one excellent element
Fundamental testing model used throughout the upper- and lower-bound arguments.

pith-pipeline@v0.9.0 · 5747 in / 1368 out tokens · 67672 ms · 2026-05-19T20:04:10.924609+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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Gupta, A. and Katona, G. O. H. (2026), ‘Finding one excellent element in case of one lie’,Journal of Statistical Theory and Practice20(47)

work page 2026
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Katona, G. O. H. (1973), Combinatorial search problems,inJ. N. Srivastava, ed., ‘A Survey of Combinatorial Theory’, North-Holland/American Elsevier, Amsterdam/New York, pp. 285–308

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Katona, G. O. H. (2011), ‘Finding at least one excellent element in two rounds’,Journal of Statistical Planning and Inference141(8), 2946–2952. R´ enyi, A. (1961), ‘On a problem of information theory’,Magyar Tudom´ anyos Akad´ emia Matem- atikai Kutat´ o Int´ ezet´ enek K¨ ozlem´ enyei6, 505–516

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and Wegener, I

Ahlswede, R. and Wegener, I. (1987),Search Problems, John Wiley

work page 1987

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(1988),Combinatorial Search, John Wiley and Teubner

Aigner, M. (1988),Combinatorial Search, John Wiley and Teubner

work page 1988

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(1995),Probability and Measure, 3 edn, Wiley, New York

Billingsley, P. (1995),Probability and Measure, 3 edn, Wiley, New York

work page 1995

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(2019), ‘Combinatorial search in two and more rounds’,Theoretical Computer Sci- ence780, 1–11

Damaschke, P. (2019), ‘Combinatorial search in two and more rounds’,Theoretical Computer Sci- ence780, 1–11

work page 2019

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and Hwang, F

Du, D. and Hwang, F. K. (2000),Combinatorial Group Testing and Its Applications, 2 edn, World Scientific

work page 2000

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and Zhou, S

Gandikota, V., Grigorescu, E., Jaggi, S. and Zhou, S. (2019), ‘Nearly optimal sparse group testing’, IEEE Transactions on Information Theory65(5), 2760–2773

work page 2019

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and Patk´ os, B

Gerbner, D. and Patk´ os, B. (2018),Extremal Finite Set Theory, Taylor and Francis

work page 2018

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and Viswanadhan, V

Ghose, A. and Viswanadhan, V. (2001),Combinatorial Library Design and Evaluation: Principles,

work page 2001

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and Katona, G

Gupta, A. and Katona, G. O. H. (2026), ‘Finding one excellent element in case of one lie’,Journal of Statistical Theory and Practice20(47)

work page 2026

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Katona, G. O. H. (1973), Combinatorial search problems,inJ. N. Srivastava, ed., ‘A Survey of Combinatorial Theory’, North-Holland/American Elsevier, Amsterdam/New York, pp. 285–308

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Katona, G. O. H. (2011), ‘Finding at least one excellent element in two rounds’,Journal of Statistical Planning and Inference141(8), 2946–2952. R´ enyi, A. (1961), ‘On a problem of information theory’,Magyar Tudom´ anyos Akad´ emia Matem- atikai Kutat´ o Int´ ezet´ enek K¨ ozlem´ enyei6, 505–516

work page 2011

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Ulam, S. M. (1976),Adventures of a Mathematician, Charles Scribner’s Sons, New York

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Yeung, R. W. (2002),A First Course in Information Theory, 1 edn, Springer, New York. 17

work page 2002