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On the Cyclic Assumption of the Cow-Path Search Algorithm
Yuan Ma, Yiqun Lisa Yin
TL;DR
The paper addresses whether the lower bound for cyclic cow-path algorithms applies to all algorithms. It proves that an optimal algorithm can be converted into a cyclic one, so the least expected competitive ratio is attained cyclically. The argument’s extension to a truncated HD distribution remains uncertain under the stated smoothness conditions.
Problem
The paper addresses whether the lower bound proved for cyclic algorithms also holds for all algorithms, as claimed in Fact 4 of.
Source: §6 · p. 9Method
Results
The least expected competitive ratio from F0 can be attained by a cyclic algorithm.
Source: §6 · p. 9Takeaways & Limitations
The lower bound established for cyclic algorithms in [3] is a lower bound for all algorithms, so the cyclic assumption may be dropped.
Source: §6 · p. 9The argument uses positive, strictly decreasing, smooth ϕ, while truncated HD has ϕ vanish beyond D; numerical results for w = 3 only suggest cyclic optimality there.
Source: §7 · p. 9
Abstract
from arXiv · showhide
In the cow-path problem, a cow must find a goal lying at an unknown distance on one of $w$ paths connected only at the origin, and performance is measured by competitive ratio. Kao, Reif and Tate designed an efficient randomized algorithm in which the cow visits the paths in a fixed cyclic order. They proved the algorithm is optimal for $w=2$, and subsequently Kao, Ma, Sipser and Yin proved its optimality for all $w$, with a claim that no algorithm does better than the best cyclic one. This note provides a detailed proof of that claim.
1 Introduction
The cow-path problem asks how to find a goal at an unknown distance on one of w paths while minimizing travel, measured by competitive ratio. Prior work established optimal cyclic algorithms, but the claim that cyclic algorithms suffice for all w needed a detailed proof.
1 Introduction
- The cow-path problem places an unknown-distance goal on one of w paths and measures search performance by competitive ratio.The cow starts at the common origin and learns the goal’s path and distance only by reaching it.Source: §1 · p. 1
- The optimal deterministic competitive ratio is 1+2ww/(w−1)w−1, equal to 9 when w = 2.Source: §1 · p. 1
- Kao, Reif and Tate [2] introduced a randomized algorithm, proved it optimal for w = 2, and conjectured optimality for w > 2; Kao, Ma, Sipser and Yin proved the conjecture.The randomized algorithm searches paths in a fixed cyclic order, visiting all w paths in its first w visits and repeating that order.Source: §1 · p. 1
2 Preliminaries
The preliminaries formalize search schedules, the input distribution, and a state-based cost framework used to compare algorithms. The framework makes each step’s contribution depend on the current frontiers and supports minimizing expected competitive ratio from the initial state.
2.1 Algorithms
- An algorithm is a goal-independent sequence of path sweeps; finite expected competitive ratio requires finite sweeps and unbounded searching on every path.Each unsuccessful sweep costs a round trip, while finding the goal ends the search on the outward leg.Source: §2.1 · p. 2
2.2 The distribution and competitive ratio
- Yao’s principle lower-bounds randomized algorithms by analyzing deterministic algorithms under a fixed input distribution, with the goal path chosen uniformly.The distance density is ϵx^−(1+ϵ) for x ≥ 1, with ϵ > 0 fixed.Source: §2.2 · p. 2
- After deleting steps that cover no new ground, the expected competitive ratio separates the final trip to the goal from unsuccessful round trips.Minimizing that expectation is equivalent to minimizing the total contribution from the initial state F0.Source: §2.2 · pp. 2–3, §2.3 · p. 4
2.3 States, remaining weights, and contributions
- A state records all path frontiers, with the cow understood to be at the origin unless stated otherwise.The remaining weight ϕ(x) is probability mass beyond x normalized by h; it is positive, strictly decreasing, and smooth, while κ cancels from comparisons.Source: §2.3 · p. 3
- A full round-trip step contributes 2ℓtΦt+1, and a continuation’s contribution depends on the past only through the current state.The total remaining weight is symmetric under path relabelling; therefore, minimizing expected competitive ratio from F0 reduces to minimizing total contribution.Source: §2.3 · pp. 3–4
3 Proposition 1: prioritize a least-frontier path
Proposition 1 establishes that an algorithm can be improved whenever it sweeps a path whose frontier is not least. The proof redirects that sweep to a least-frontier path and relabels the continuation.
3 Proposition 1: prioritize a least-frontier path
- Sweeping a path with a frontier strictly larger than the least frontier can be modified to give strictly lower contribution.Source: §3 · p. 4
- The modification redirects the sweep to a lower-frontier path and swaps the labels of those paths in the remaining schedule.It skips prescribed sweeps on the other path that do not extend its current frontier; only finitely many are skipped because its sweep distances are unbounded.Source: §3 · p. 4
- The redirected sweep leaves less total remaining weight because ϕ is strictly decreasing, while the relabelled continuation contributes no more than before.Combining the comparisons yields 1 + C∗2 < C1 + C2 = EF(X).Source: §3 · p. 5
4 Proposition 2: each sweep advances its path by strictly more than a factor β
A stationary algorithm that advances a path by at most the threshold β admits a step deletion that strictly lowers its contribution. Thus an algorithm with no improving deletion must advance each sweep by more than β.
4 Proposition 2: each sweep advances its path by strictly more than a factor β
- A one-step length variation improves the contribution unless the current sweep balances the remaining weight against accumulated sweeps until that path is revisited.The feasible replacement length lies between the path's current frontier and its next sweep length.Source: §4 · pp. 5–6
- For a stationary algorithm with advance factor 1 < ℓ_t/p ≤ β, deleting step t strictly lowers its contribution.The deleted sweep's saving exceeds the extra contribution incurred before that path is swept again.Source: §4 · p. 6
- The deletion argument compares the removed step's saving with the added cost while its path remains at frontier p.The step's saving is 2ℓ_tΦ_t+1; deleting it leaves the path at p until the next sweep.Source: §4 · p. 6
5 Proposition 3: step lengths are nondecreasing
The section proves that an algorithm with no strictly better alternative must have nondecreasing sweep lengths. The proof exchanges adjacent decreasing lengths and uses path relabelling to compare the resulting contributions.
5 Proposition 3: step lengths are nondecreasing
- Exchanging adjacent lengths on distinct paths preserves the continuation after relabelling, so only the two exchanged steps affect the contribution difference.The exchange lowers contribution under the stated frontier condition min(u, v) > max(a, b).Source: §5 · pp. 7–8
- An algorithm from F0 with no strictly lower-contribution alternative has nondecreasing step lengths.The proof rules out an adjacent decrease by constructing a strictly better exchange.Source: §5 · p. 8
- The exchange proof uses the least-frontier rule and the advance-factor condition to show that an adjacent decrease can be improved.For the decrease u > v, the threshold condition v > β establishes the sign needed for the exchange.Source: §5 · p. 8
6 Main result
The least expected competitive ratio is attained by a cyclic algorithm. The proof converts an optimal algorithm into a cyclic one while preserving its contribution.
6 Main result
- An optimal algorithm attaining the least competitive ratio can be chosen cyclic, extending the lower bound for cyclic algorithms to all algorithms [3].The proof first establishes that an optimal algorithm exists, then converts it to a cyclic one.Source: §6 · p. 9
- Breaking least-frontier ties in favor of the least recently swept path preserves the multiset of frontiers and the contribution.The construction interchanges tied path labels from that step onward.Source: §6 · p. 9
- Because every sweep length exceeds 1, the first w steps visit distinct paths, and least-recently-swept tie-breaking continues that cyclic order.At each induction step, the least frontier belongs to the least recently swept path.Source: §6 · p. 9
- The resulting cyclic algorithm attains the least competitive ratio.This completes the conversion from an optimal algorithm to a cyclic one.Source: §6 · p. 9
7 Discussion
A truncated HD distribution offers an alternative lower-bound proof, but the note's arguments for cyclic optimality do not directly carry over to it.
7 Discussion
- Schuierer's alternative, simpler proof of the lower bound in still relies on the cyclic assumption [5].The note asks whether cyclic optimality can be proved under Schuierer's truncated HD distribution.Source: §7 · p. 9
- Under truncated HD, ϕ vanishes at and beyond D, unlike the positive, strictly decreasing, smooth function required by these arguments.Each path is also swept to D in finitely many steps, making modifications like the note's difficult.Source: §7 · p. 9
- Numerics for w = 3 suggest the optimal algorithm may still be cyclic under truncated HD.The passage presents this as a numerical suggestion, not a proof.Source: §7 · p. 9
A Existence of an optimal algorithm
The appendix proves that an algorithm attains the infimum contribution from the initial state. It does so by bounding step lengths, extracting a limiting sequence, and verifying that the limit remains a valid search algorithm.
A Existence of an optimal algorithm
- For w ≥ 2, an algorithm starting from F0 attains the infimum contribution V(F0).The appendix concludes that the constructed algorithm has contribution V0.Source: A Existence of an optimal algorithm · pp. 10, 12
- The proof selects algorithms whose contributions approach V0, then bounds each fixed step length uniformly across the sequence.The step-length bounds are established by induction.Source: A Existence of an optimal algorithm · p. 10
- Repeated subsequence selection fixes each step's path label and makes its sweep lengths converge, defining a limiting sequence of steps.Bounded step lengths permit the convergent subsequence selections.Source: A Existence of an optimal algorithm · p. 11
- For every finite T, continuity of ϕ lets the limiting sequence's first T steps have contribution no greater than the corresponding limiting bound.Taking T to infinity extends the bound to the full sequence.Source: A Existence of an optimal algorithm · p. 11
- Every path in the limiting algorithm is swept without bound, and no step merely repeats an existing frontier.A bounded frontier would force a positive contribution at every later step; a no-advance step could be deleted to reduce contribution.Source: A Existence of an optimal algorithm · p. 12
- The limiting algorithm's contribution equals V0, establishing attainment rather than only approximation of the infimum.The proof combines the contribution bound with the definition of V0.Source: A Existence of an optimal algorithm · p. 12
B Terminology and notation
The notation defines the input distribution and the remaining-weight quantities used to express the search algorithm's contribution.
B Terminology and notation
- The goal lies on a uniformly chosen path, with conditional distance density ϵx^−(1+ϵ) for x ≥ 1.The remaining-weight exponent is α = 1 + ϵ.Source: B Terminology and notation · p. 12
- ϕ(x) = κx^−α is the remaining weight on a path with frontier x, and Φ(F) is the total remaining weight in state F.κ is a positive scaling constant; α is the exponent in the remaining-weight function.Source: B Terminology and notation · p. 12
- A step t sweeps path Pt for length ℓt, and its contribution to expected competitive ratio is 2ℓtΦt+1.Wt denotes one-way sweep length from step t up to, but excluding, the next sweep of Pt.Source: B Terminology and notation · p. 12