Search and Rescue on the Plane

Danny Krizanc; Evangelos Kranakis; Jared Coleman; Oscar Morales-Ponce

REVIEW 3 major objections 5 minor 20 references

Search and Rescue on the Plane

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read For every starting point in the plane, one of two routes is optimal for search-and-rescue, with the switch at about 15.6 degrees.

desk verdict A clean, mostly self-contained closed-form solution to a natural planar search-and-rescue variant; the optimality theorem has a fixable gap around negative-axis first hits, but the result looks right. read the letter →

arxiv 2608.12039 v1 pith:UTCRNBIQ submitted 2026-08-12 cs.DM

Jared Coleman , Evangelos Kranakis , Danny Krizanc , Oscar Morales-Ponce This is my paper · ORCID

classification cs.DM MSC 68W2790B40

keywords searchandrescuecompetitiveratioonlinealgorithmgeometriccheckpointphasetransitionmobileagentx-axis

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper establishes the optimal online strategy for a search-and-rescue agent that starts anywhere in the plane, must find an object at an unknown distance along the positive x-axis, and must carry it to the origin. The optimal strategy is always one of two shapes: either go straight to the origin and then search outward, or first land on a specific checkpoint on the x-axis computed from the starting angle, then return to the origin, then search. The paper derives closed-form competitive ratios for both shapes and shows that the better one depends only on the angle of the starting point, not on its distance from the origin. The crossover happens at a critical angle $\theta^* \approx 15.6^\circ$, so an agent starting within that forward cone should use a nonzero checkpoint while an agent starting outside the cone should head straight for the origin. If correct, this reduces a continuous search-and-delivery planning problem to a simple two-option decision rule with an explicit formula.

What carries the argument

The central object is the one-parameter family of algorithms $A_s$, indexed by the checkpoint distance $s$ on the x-axis, with $\rho(\theta,s)=\sqrt{1-2s\cos\theta+s^2}$ as the distance from the starting point to the checkpoint. The argument turns on two identities: the root $k_\theta=\frac{5+2\cos\theta+\cos 2\theta+\sqrt{2\cos^2\theta\,(11+4\cos\theta+\cos 2\theta)}}{8}$ where the two candidate worst cases $CR_{d\le s}(s,\theta)$ and $CR(s,s,\theta)$ cross, and the angle equation $4\cos^3\theta+8\cos^2\theta-11=0$ whose unique root in $(0,\pi/2)$ is the critical angle. The derivative analysis of the ratio curve $CR(s,d,\theta)$ identifies $d_-(s,\theta)$ as the unique worst-case object distance and proves it is decreasing in $s$, which forces the global minimum of $CR(A_s,\theta)$ to lie at one of the two endpoints $s=0$ or $s=k_\theta$.

What would settle it

Take a specific angle inside the cone, say $\theta=10^\circ$, and simulate the non-canonical strategy that enters the x-axis at some $s$, sweeps partway back toward the origin, then returns to the origin and searches outward; if any choice of $s$ and sweep length gives a worst-case ratio smaller than the paper's value $CR(A_{k_\theta},\theta)$ from equations (9) and (11), the canonical-form theorem and the claimed optimality fail.

Watch

Extended reading notes

Core claim

After rescaling the starting radius $r$ away, the paper proves that every optimal algorithm can be put in the canonical form $A_s$: fly directly to a point $(s,0)$ on the x-axis, walk back to the origin, then search away from the origin along the positive x-axis until the object is found and deliver it. For this family the worst-case competitive ratio is the maximum of two closed-form expressions, one for objects found before reaching the origin and one for objects found later; a derivative analysis locates the worst object distance $d_-(s,\theta)$ and shows the ratio is minimized only at $s=0$ or at $s=k_\theta$, where $k_\theta$ is an explicit function of $\cos\theta$. Comparing the two candidates gives the phase transition: the checkpoint at $(r\cdot k_{|\theta|},0)$ is optimal for $|\theta|<\theta^*$, and the origin is optimal for $|\theta|\ge\theta^*$, with $\theta^* = \arccos\!\big(\tfrac{1}{12}\big(-8 + \sqrt[3]{1864-312\sqrt{33}} + 2\sqrt[3]{233+39\sqrt{33}}\big)\big) \approx 0.2716$ rad $\approx 15.6^\circ$. The resulting competitive ratio is independent of $r$ and recovers the one-dimensional bound $1+\sqrt{2}$ at $\theta=0$.

Load-bearing premise

The optimality result hinges on the assumption that an optimal agent goes straight to a checkpoint on the x-axis, returns to the origin, and then searches outward monotonically; the paper compresses any early zigzagging into a first-hit point and a farthest point, but does not directly analyze pre-origin paths that reverse direction or search before reaching the origin.

Editorial extensions

If this is right

At $\theta=0$ the optimal checkpoint is $((2+\sqrt{2})/2)\,r$ and the competitive ratio is $1+\sqrt{2}\approx 2.414$, recovering the one-dimensional search-and-rescue bound.
For every starting angle with $|\theta|\ge\theta^*$, the direct-to-origin strategy is optimal; its ratio decreases from $5/4$ at $\theta=\pi/2$ to $1$ as $\theta$ approaches $\pi$.
For every starting angle with $|\theta|<\theta^*$, the optimal strategy places the first checkpoint at $(r\,k_{|\theta|},0)$ on the x-axis before any search begins.
Since the ratio is independent of $r$, the same angular decision rule applies at every distance: the checkpoint distance simply scales linearly with the starting distance.
No online algorithm can beat $\min\{CR(A_0,\theta), CR(A_{k_\theta},\theta)\}$ for any starting point, so the two-option rule is exactly optimal.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper's machinery implies a testable extension the authors do not state: if the search is confined to a bounded interval rather than the full ray, the same ratio curve should acquire a third boundary case, and the optimal checkpoint would become a function of the interval length.
I infer from the structure that a two-agent version would split the outward sweep at the checkpoint, and the $15.6^\circ$ threshold computed here would serve as the baseline for deciding whether splitting is ever worthwhile.
Another unstated consequence: because the formulas are continuous in $\theta$ except at the phase boundary, the competitive-ratio landscape has a sharp but non-smooth kink at $\theta^*$, which could make the threshold visible in simulated trial data before any theoretical derivation.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This paper does something real: it takes the one-dimensional search-and-rescue problem, moves the start to an arbitrary point in the plane, and produces exact competitive ratios, a closed-form checkpoint k_|θ|, and a phase transition at θ* ≈ 15.6°. The θ=0 limit recovers the known 1+√2 line result, the figures match the formulas, and the algebra in Lemmas 3.1–3.7 is careful. The reduction to min{A0, A_kθ} via monotonicity is clever, and the CAS-supported closed form for θ* is a genuine contribution. I believe the result, not just the paper, is new.

The soft spots are real but, I think, repairable. Theorem 2.2, the canonical-form lower bound, is under-specified in exactly the place the stress-test flags: it parametrizes the first x-axis hit by distance s without signing it, and for θ > π/2 a first hit on the negative axis has a smaller first-leg cost than ρ(θ,s) assumes. This is not fatal—going to any negative point before the origin is dominated by going straight to the origin, so the A0 conclusion for those angles survives—but the proof needs to say that explicitly. The smax argument also compresses arbitrary pre-origin motion into an s → smax → O path; the cost that path incurs is indeed the cheapest way to explore [0,smax] and end at O, but the proof never says why reversals or interleaved searches cannot do better. The gap is in the exposition, not the math, but Theorem 2.2 is load-bearing, so a referee should push on it.

Two smaller things. Lemma 3.2's displayed constant C is missing a -4αγ term; the discriminant and subsequent d− formula use the correct C, so it is a typo, but a confusing one. And the θ* derivation relies on a CAS factorization and spurious-root rejection that are checked by substitution but not fully machine-certified; the linked script helps, but the paper should spell out the sign checks more.

Who is this for? Search theorists and competitive-analysis people who want a clean benchmark or an extension exercise. It is specialized, and the idealized model limits practical reach, but within the subfield it is a solid, citable result. I would bring it to a reading group and I would cite it if I worked on geometric search. It deserves a serious referee: my verdict would be conditional, not reject. Send it to review, with the request that the authors fix the negative-axis case in Theorem 2.2, tighten the smax argument, and correct the C typo.

Referee Report

3 major / 5 minor

Summary. The paper studies a planar search-and-rescue problem in which an agent starting at P_{\theta,r}=(r\cos\theta,r\sin\theta) must find an object at an unknown distance on the positive x-axis and deliver it to the origin. The authors prove scale invariance, propose a one-parameter family A_s of algorithms that move directly to a checkpoint (s,0), then to the origin, then search outward, and derive closed-form competitive ratios for this family. They further show that the optimal checkpoint is either the origin or (r k_{|\theta|},0), where k_{|\theta|} is given explicitly, and they identify a critical angle \theta^*\approx 15.6^\circ separating the two regimes. The main theorem (Theorem 1.1) asserts that Algorithm 1, built from this analysis, is optimal among all online algorithms, with the lower-bound half resting on the canonical-form reduction in Theorem 2.2.

Significance. If the result is fully established, it provides a complete optimal competitive analysis for a natural planar online search-and-delivery problem, including a phase transition determined by the starting angle. The paper has real strengths: the scale-invariance reduction is clean, the derivation of the competitive ratio for the family A_s is mostly self-contained, the limit \theta=0 correctly recovers the known one-dimensional bound 1+\sqrt{2}, and the claims are accompanied by numerical figures and a referenced Mathematica script. The phase-transition statement is sharp and falsifiable. However, the proof of Theorem 2.2 contains a gap that is load-bearing for the optimality claim, and the proof of Lemma 3.2 contains an algebraic error in displayed coefficients. Both issues are local in nature, and the underlying results appear plausible, but the manuscript in its current form does not yet establish optimality over all online algorithms.

major comments (3)

[Section 2.2, Theorem 2.2] The proof never excludes a first contact with the negative x-axis. All cost formulas in the paper use \rho(\theta,s)=\sqrt{1-2s\cos\theta+s^2}, the distance from P_\theta to the positive point (s,0), and the explored region is described as the interval [0,s_{\max}]. A strategy whose first x-axis contact is at (-s,0) has first-leg cost \sqrt{1+2s\cos\theta+s^2}, which for \cos\theta<0 is strictly smaller than the positive-point formula, and the explored interval before reaching the origin would be [s_{\max},0] or a signed interval, not [0,s_{\max}]. Since Theorem 2.2 is the lower-bound half of Theorem 1.1, the optimality claim is not proven over all online algorithms until a lemma rules out negative checkpoints or the analysis is extended to signed s.
[Section 3.1, Lemma 3.2] The displayed coefficients A, B, C are inconsistent. From C=\alpha^2-(2+\alpha\gamma)^2 one obtains -4-4\alpha\gamma+\alpha^2(1-\gamma^2), not -(4-\alpha^2(1-\gamma^2)); the -4\alpha\gamma term is dropped. With the stated A, B, and C, the stated discriminant \Delta=16\alpha^2(1-\gamma^2)(1+\alpha\gamma) does not follow. The final formula for d_- is correct if one uses the standard polynomial with C'=(2+\alpha\gamma)^2-\alpha^2 and A'=(\alpha+2\gamma)^2-\alpha^2, but as written the proof of Lemma 3.2, which supplies the critical point used throughout Section 3, is invalid.
[Section 3.3, Theorem 3.9] The derivation of the closed-form \theta^* depends on a degree-11 factorization, shown in Equation (12), and on rejecting three of its factors as spurious. The text delegates the elimination and the substitution checks to a CAS call in an accompanying Mathematica script without giving the eliminated polynomial or the detailed verification of each factor against the unsquared equation. Because the exact form of \theta^* is a central advertised contribution, the reader cannot verify from the manuscript alone that the factorization is correct and that no genuine root is discarded. The paper should either include the eliminated polynomial and the spurious-root checks, or make the script's verification steps part of the proof.

minor comments (5)

[Section 2.1] The reduction assumes \theta\in[0,\pi) by reflection about the x-axis, but the abstract and Algorithm 1 state the result for -\pi\le\theta\le\pi. It should be stated explicitly that all formulas are even in \theta and that reflection maps the negative-angle case to the positive-angle case.
[Section 3.1, Lemma 3.2] In the case \gamma>0, the proof says that d_+ > (2+\alpha\gamma)/(\alpha+2\gamma) > d_- can be verified from the explicit formulas, but the verification is not shown. This inequality is needed to reject d_+ as spurious and should be stated with a short derivation or moved to an appendix.
[Section 3.2, Theorem 3.8] The claim that CR(s,s,\theta) is unimodal is justified by a derivative calculation, but the derivative is given in compressed form; writing the algebra for \partial/\partial s \,[1+2s/(\rho+s)] would make the monotonicity argument easier to check.
[Section 3.3, Theorem 3.9] The statement that CR(A_0,\theta) is decreasing in \theta is correct pointwise, but the proof phrase "since \rho(\theta,d) is increasing in \theta for each fixed d>0" should also note that this pointwise monotonicity is preserved by the supremum over d; a one-sentence justification would remove ambiguity.
[Algorithm 1] In the pseudocode, k_{|\theta|} is defined only through a displayed formula; for readability, the formula should be labeled or repeated inside the algorithm box, since the reader must otherwise cross-reference Equation (9).

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the checkpoint k_theta and critical angle theta* are derived from the same closed-form ratio expressions, not fitted, and the only self-citation is a non-load-bearing boundary check.

full rationale

The paper's central quantities are obtained by minimization and equation-solving, not by fitting or by defining the answer into the input. k_theta is defined in Lemma 3.7 as the unique positive root of CR_{d<=s}(s,theta) = CR(s,s,theta), and theta* is defined in Theorem 3.9 as the angle where the closed-form expressions CR(A_0,theta) and CR(A_{k_theta},theta) cross. Neither formula is tuned to a data subset; the competitive ratios are evaluated against the offline optimum T*(d,theta)=rho(theta,d)+d throughout. The only self-citation is the line result [10], used as a consistency check at theta=0 ('the problem reduces to search and rescue on a ray [10]'); the values quoted there (k_0=(2+sqrt(2))/2 and ratio 1+sqrt(2)) are also directly produced by the paper's own formulas (9) and (11), so the citation is not load-bearing. The optimality claim does depend on Theorem 2.2, which asserts that any optimal algorithm has the canonical form A_s. That proof is an analytic reduction, not a circular one: it lower-bounds arbitrary algorithms by summarizing pre-origin motion with s and s_max and then shows the worst-case bound is minimized at s_max=s. I would note one genuine gap rather than circularity: Claim 3 in Section 2.2 writes the first x-axis contact as S=(s,0) with s>=0 and asserts 'the explored region is always the interval [0,s_max]', which silently excludes a first hit on the negative x-axis; that is an unproved reduction and a correctness risk for universal optimality, but it does not make the derivation equivalent to its input. There is also an apparent misprint in Lemma 3.2's displayed C coefficient (the printed expansion drops a -4*alpha*gamma term), while the printed discriminant matches the correct algebra; that is an internal algebra slip, not circularity. Overall, the derivation is self-contained and no step reduces by construction to its own inputs.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted: k_|θ|, ℓθ, and θ* are solved in closed form from the competitive-ratio equations, and r scales out by Lemma 2.1. The model introduces no new physical entities. The proof depends on standard competitive-analysis background and on an unverified CAS elimination step for the critical angle.

assumptions (4)

domain assumption The agent moves in the Euclidean plane at unit speed, can turn instantly, and the object and destination both lie on the positive x-axis.
This is the problem model stated in Section 1.1; the optimality proof and checkpoint formulas depend on these kinematic and geometric assumptions.
domain assumption The competitive ratio is measured against an offline optimum that knows the object distance d, while the online agent knows only the starting point Pθ,r.
This is the standard competitive-analysis benchmark introduced in Section 1.1 and used throughout the lower-bound arguments.
standard math In the worst case, an optimal deterministic strategy never benefits from revisiting already-explored points or from randomized behavior.
This underpins the canonical-form reduction in Theorem 2.2, where any pre-origin motion is summarized by the first hit point and farthest point reached before visiting the origin.
ad hoc to paper The degree-11 polynomial factorization in Equation (12) and the rejection of spurious roots in Theorem 3.9 are correct as implemented in the accompanying Mathematica script.
The paper asserts the factorization and the spurious-root checks without a machine-checkable certificate or a commit-hashed script, so the closed form for θ* rests on an unverified symbolic computation.

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Cite this review

Pith. "Pith review of Search and Rescue on the Plane." pith.science (2026). https://pith.science/paper/UTCRNBIQ

@misc{pith2026260812039,
  author       = {Pith},
  title        = {Pith review of: Search and Rescue on the Plane},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UTCRNBIQ}},
  note         = {Machine review of arXiv:2608.12039}
}

abstract

We study a planar variant of the search and rescue problem whereby an agent starting at an arbitrary position $P_{\theta,r} = (r\cos\theta, r\sin\theta)$ in the plane must locate an object at an unknown position on the positive $x$-axis and deliver it to the origin. Our main contribution is to characterize the optimal form of any competitive algorithm, derive closed-form expressions for the competitive ratio, and identify a critical angle $\theta^* \approx 15.6^\circ$ which yields a phase transition to optimal competitive search and delivery in the following sense. For each angle $-\pi \leq \theta \leq \pi$ we compute a checkpoint (landing position on the $x$-axis) where the agent must go first prior to initiating a search on the $x$-axis in order to optimize the competitive ratio of search and delivery. We show that if $|\theta| \geq \theta^*$ then the checkpoint is at the origin, while if $|\theta| < \theta^*$ then the agent should land at the checkpoint $(r \cdot k_{|\theta|}, 0)$ on the $x$-axis, where $k_{|\theta|}$ is a real number given by an explicit formula we present.

Figures

Figures reproduced from arXiv: 2608.12039 by the authors.

**Figure 1.** An agent starting at a point Pθ,r is searching for a lost object at an unknown location (d, 0). The goal is to design a strategy leading to an “optimal” landing position (s, 0) on the x-axis, so as to find the object and deliver it to the destination at O. search algorithm with optimal worst-case performance that finds the object and delivers it to the origin O. As in the one-dimensional case, we measure an algorith… view at source ↗

**Figure 2.** Competitive ratio curves as functions of [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** Optimal algorithm on the plane, with the object located on the positive [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

Springer (2003)

Alpern, S., Gal, S.: The theory of search games and rendezvous. Springer (2003)

work page 2003

[2] [2]

Informa- tion and computation106(2), 234–252 (1993)

Baezayates, R.A., Culberson, J.C., Rawlins, G.J.: Searching in the plane. Informa- tion and computation106(2), 234–252 (1993)

work page 1993

[3] [3]

Israel Journal of Mathematics2(4), 221– 228 (1964)

Beck, A.: On the linear search problem. Israel Journal of Mathematics2(4), 221– 228 (1964)

work page 1964

[4] [4]

European journal of operational research202(1), 8–15 (2010) 16 J

Berbeglia, G., Cordeau, J.F., Laporte, G.: Dynamic pickup and delivery problems. European journal of operational research202(1), 8–15 (2010) 16 J. Coleman, E. Kranakis, D. Krizanc, and O. Morales-Ponce

work page 2010

[5] [5]

Algorithmica82(11), 3250–3281 (2020)

Bouchard, S., Dieudonné, Y., Pelc, A., Petit, F.: Deterministic treasure hunt in the plane with angular hints. Algorithmica82(11), 3250–3281 (2020). https://doi. org/10.1007/s00453-020-00724-4, https://doi.org/10.1007/s00453-020-00724-4

work page doi:10.1007/s00453-020-00724-4 2020

[6] [6]

In: International Symposium on Fundamentals of Computation Theory

Carvalho, I.A., Erlebach, T., Papadopoulos, K.: An efficient algorithm for the fast delivery problem. In: International Symposium on Fundamentals of Computation Theory. pp. 171–184. Springer (2019)

work page 2019

[7] [7]

Carvalho, I.A., Erlebach, T., Papadopoulos, K.: On the fast delivery problem with one or two packages. J. Comput. Syst. Sci.115, 246–263 (2021). https://doi.org/ 10.1016/j.jcss.2020.09.002, https://doi.org/10.1016/j.jcss.2020.09.002

work page doi:10.1016/j.jcss.2020.09.002 2021

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In: International Symposium on Algorithms and Experiments for Sensor Systems, Wireless Networks and Distributed Robotics

Chalopin, J., Das, S., Mihal’ák, M., Penna, P., Widmayer, P.: Data delivery by energy-constrained mobile agents. In: International Symposium on Algorithms and Experiments for Sensor Systems, Wireless Networks and Distributed Robotics. pp. 111–122. Springer (2013)

work page 2013