Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.18740

Selecting the 7th Smallest of 16 Elements Takes Exactly 28 Comparisons

Jonathan Cremer
View original source ↗Read source document ↗Export metadata ↓

Abstract

Let $V_i(n)$ denote the minimum worst-case number of comparisons needed to find the $i$-th smallest of $n$ distinct elements. We construct algorithms showing $V_7(16)\le28$ and $V_8(16)\le29$, improving the previously reported upper bounds of 30 and 31. Together with known lower bounds, these results establish $V_7(16)=28$ and leave $V_8(16)\in\{28,29\}$. For 17 elements, we establish $V_5(17)=27$ and improve the upper bounds for ranks 6 through 9 to 29, 30, 31, and 31, respectively. Every new upper bound has an explicit decision-graph certificate verified by a static checker developed independently of the search. We also derive a lower bound for selection from a partially ordered state by combining the Fussenegger-Gabow weight with its order-reversed counterpart. All new upper bounds and this lower bound are formally verified in Lean 4. For $V_8(16)$, we show that a lower bound based solely on the total number of decision-tree leaves cannot establish $V_8(16)\ge29$. The algorithms were found using fixed-budget search, information-guided comparison ordering, and proof-number search. Proof-number search was decisive in finding the 28-comparison algorithm; in our experiments, refuting a comparison budget remained substantially more expensive than finding an algorithm within it.

Explore related subjects

cs.DSmath.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Jonathan Cremer. 2026-10-13. Selecting the 7th Smallest of 16 Elements Takes Exactly 28 Comparisons. https://arxiv.org/abs/2610.18740

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Adding an Edge in a $P_4$-sparse Graph

For a graph family~${\cal F}$, the minimum ${\cal F}$-completion problem asks for the minimum number of non-edges to be added to a given graph~$G$ so that the resulting graph belongs to the family~${\cal F}$. This problem is NP-complete for many subclasses of perfect graphs, while polynomial solutions are available for minimal completion sets or restricted input. We study the minimum $P_4$-sparse completion problem that arises after adding a prescribed non-edge to a \(P_4\)-sparse graph \(G\), with the resulting graph required to remain \(P_4\)-sparse. For the case in which $G$ consists of two connected components, each containing one endpoint of the added non-edge, we characterize the possible forms of an optimal solution. Combining this characterization with the solution of the other cases identified by the structural decomposition of \(P_4\)-sparse graphs, we obtain an algorithm for the general problem that runs in $O(n^2)$ time and $O(n)$ space.

cs.DS↗
arXiv2026

Optimal Extrapolation Bounds for Sparse Fourier Sums

We prove a sharp extrapolation bound for $k$-sparse Fourier sums with arbitrary real frequencies and no separation assumption. For every $g(t)=\sum_{j=1}^k v_je^{iλ_jt}$ with $λ_j\in\mathbb{R}$ and every $x\geq 1$, \begin{align*} |g(x)|\le k\exp(O(k\operatorname{arcosh} x))\|g\|_{L^2[-1,1]}. \end{align*} Near the endpoint at $x=1+δ$, this gives an exponential factor of $\exp(O(k\sqrtδ))$, improving on the $\exp(O(k^2\log k\cdotδ))$ estimate of Chen and Price (ICALP 2019), and the exponent is optimal up to a constant factor. Our theorem has algorithmic applications to recovering gapless Fourier-sparse signals: it improves the resolution of cluster-location algorithms and the sample complexity of Fourier interpolation algorithms. In addition, it gives sharp bounds on the error transfer of sparse Fourier models just outside the training interval.

cs.DS↗
arXiv2026

Stochastic Network Speed Scaling with Competitive Ratios Independent of the Network

In network speed scaling, jobs arrive over time at a network of servers whose speeds can be tuned, every job must be processed by the servers along one of its allowed routes, and the goal is to minimize the total flow time plus the total energy. For stochastic arrivals, the competitive ratio of Vaze and Nair depends on the network, through the lengths of the routes it uses. We show that this dependence can be removed: we give an algorithm, which routes the jobs by solving a convex program and runs every server at a fixed speed, whose competitive ratio depends only on the power functions; for $P(s)=s^2$, it is at most the golden ratio $φ\approx1.618$. The key idea is a lower bound on the optimal cost which, like the cost of our algorithm, is a sum over the servers of a function of each server's load, so the analysis reduces to a single server. We also show that the routing rule of Vaze and Nair can be a factor $Ω(L)$ away from optimal, where $L$ is the length of the longest route.

cs.DS↗