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arXiv2026

Approximating CDTW Distance of Piecewise Algebraic Curves

Curves as input data naturally arise in a variety of fields including finance, seismology, medicine, spatio-temporal data mining, malicious activity detection, and more. A common way to analyze these data sets is to do similarity matching or clustering. The most common metrics used for measuring similarity of curves are Dynamic Time Warping (DTW) and Fréchet distance. These metrics are sensitive to sampling rate and outliers respectively, and do not yield robust outcomes. Continuous Dynamic Time Warping (CDTW) is a more robust distance metric that improves upon DTW and Fréchet distances. Existing algorithms for CDTW are either exact algorithms that focus on non-Euclidean norms and piecewise linear curves, or approximation algorithms limited to piecewise linear curves. We present an approximation algorithm for computing the CDTW distance under Euclidean norm between piecewise (higher degree) algebraic curves. That is, we present a fully polynomial-time approximation scheme (FPTAS) of multiplicative error $\varepsilon$, with $O \left( (m+n)^{\frac{19}{6}} (\frac{1}{\varepsilon})^{\frac{10}{3}} \log \left( \frac{ (m+n) }{\varepsilon^2} \right) \right)$ complexity, where $m$ and $n$ are the number of pieces of the two input curves.

cs.CG↗

arXiv2026

Sampling for Region-Aggregated Spatial Scan Statistics

Anomaly detection in geospatial data is a crucial tool in geographic information science (GIS), with applications ranging from national security to public-health surveillance to the study of societal disparities. This work focuses on spatial scan statistics and addresses a key mismatch: spatial counts are typically aggregated into predefined regions (census tracts, zip codes, counties), whereas the most efficient scan algorithms operate on spatial point data. The standard remedy -- collapsing each region to its centroid, as in widely used tools such as SaTScan -- is convenient but, as we show, discards the region's spatial extent and causes a significant loss in statistical power. To resolve this, we propose a simple yet scalable fix: replace each spatial region with 20-50 points sampled uniformly from its geometry, and divide the region's measured and baseline counts evenly among them. This approach improves statistical power while maintaining computational tractability. A convergence analysis explains why so few samples per region suffice. We recommend this sampling-based conversion as the default way to apply point-based spatial scan statistics to region-aggregated data for anomaly detection.

stat.AP↗

arXiv2026

A unified geometric design framework for kirigami structures

In recent years, kirigami metamaterials have been widely studied and applied in science and engineering. While various two- and three-dimensional kirigami design methods have been developed, most of them are only applicable to a limited class of kirigami structures. In this work, we develop a unified framework for kirigami design that encompasses a wide range of 2D-to-2D, 2D-to-3D, and 3D-to-3D shape-morphing effects, as well as additional geometric and physical properties such as compact reconfigurability and rigid deployability. In particular, by reformulating the design task as a length-based constrained optimization problem and solving it simultaneously for multiple target states of the kirigami structure, our unified design framework enables greater design flexibility and stronger theoretical support. Experimental results with a wide range of shape-morphing effects are presented to demonstrate the effectiveness of our framework. We further present a rigorous theoretical analysis of several key aspects of kirigami design, covering inertia transposition, aspect-ratio law, and angle defects, thereby elucidating important design rules and limitations. Altogether, our work paves a new way for the design of shape-morphing mechanical metamaterials.

cond-mat.soft↗

arXiv2026

On the volumes of the elliptope, metric polytope, and cut polytope

In this paper, we investigate the relationships between the volumes of four convex bodies: the cut polytope, metric polytope, rooted metric polytope, and elliptope, defined on graphs with n vertices. After an affine change of coordinates for the elliptope, the cut polytope is contained in each of the other three, which, for optimization purposes, provide polynomial-time relaxations. It is therefore of interest to see how tight these relaxations are. Worst-case ratio bounds are well known, but these are limited to objective functions with non-negative coefficients. Volume ratios, pioneered by Jon Lee with several co-authors, give global bounds and are the subject of this paper. For the rooted metric polytope over the complete graph, we show that for large n its volume is much greater than that of the elliptope. For the metric polytope, for small values of n, we show that its volume is smaller than that of the elliptope; however, for large values, we prove that the converse is true. Volume estimates place the crossover near n = 13. We also give exact formulae for the volumes of several families of sparse cut polytopes. In particular, we give an exact formula for the volume of the elliptope of every cycle and show that its volume ratio with the corresponding cut polytope converges rapidly to one.

cs.DM↗

arXiv2026

Bellman--Shoreline Search in Arbitrary Dimension: Exponential Vector Oscillators, Active Memory, Precession, and Effective Computability

We study online search for an unknown affine hyperplane in $\mathbb{R}^D$, for arbitrary fixed finite dimension. Building on a companion self-similar cell reduction and support-function formulation, we ask how the mechanism changes as the normal space grows from $\mathbb{S}^0$ to $\mathbb{S}^{D-1}$. In $D=1$, alternation and productivity yield an equal-ripple principle and the exact stationary constant $9$. In $D=2$, the analogous relative equilibrium is a logarithmic spiral whose bottleneck chord imposes tangency and selects the pitch. For exponential orbits $Γ(σ)=e^{κσ}ω(σ)$, we develop log-directional geometry, exponentially discounted memory, gauges, and recursive hyperspherical parametrizations. Without a shape ansatz, the bottleneck admits a certificate supported by at most $D$ historical suppliers, and at globally worst phases the current point lies on the active face. Within regular chambers we derive exact variation, tangency, pitch, age, and, in $D=3$, delay-system identities. Odd-dimensional obstructions, antipodal subclasses, and harmonic towers provide constraints and explicit candidate families but are not claimed globally optimal. Finally, the N-COMP theorem shows that $C_D^*$ is a computable real for every fixed finite $D$ and that algebraic polygonal $\varepsilon$-optimal cells can in principle be synthesized. Numerical screening through $D=10$ is kept separate from the proved results.

cs.CG↗

arXiv2026

Condorcet-Winning Sets and Peer Selection in Planar Metric Elections

In ranked-choice voting, a Condorcet-winning set is a group of candidates for which no outside candidate is preferred to every member of the group by a majority of voters. We study Condorcet-winning sets in planar metric elections, where voters rank candidates according to their distance under a given norm. We formulate general metric elections, peer selection, and the one-round Voronoi game as instances of a two-player Stackelberg game and place these problems in a common hierarchy. We also introduce a new variant, which we call strong peer selection. Our main result concerns peer selection under the $\ell_1$ and $\ell_\infty$ norms. We prove that every planar instance admits a Condorcet-winning set of size at most three, even under strong peer selection. This follows from a new result for strong rectangular $\varepsilon$-nets. We show that, for every set of points in the plane, one can choose at most three input points that intersect every axis-parallel rectangle containing more than half of the points, improving the previous threshold of $9 / 16$ due to Ashok et al. Under the $\ell_2$ norm, we prove that every planar metric election admits a Condorcet-winning set of size at most four, improving on the general bound of five due to Song et al. Finally, we give new norm-independent bounds for the one-round Voronoi game.

cs.GT↗

arXiv2026

AnyGS2Mesh: Feed-Forward Mesh Reconstruction from 3D Gaussian Splatting with Arbitrary-Resolution Views

Existing 3D mesh reconstruction methods from Gaussian scene representations predominantly rely on iterative optimization, resulting in slow inference and limited scalability to high-resolution inputs. In this paper, we present AnyGS2Mesh, the first feed-forward framework for directly reconstructing 3D meshes from 3D Gaussian Splatting representations with support for arbitrary input image resolutions. Our approach incorporates a Gaussian-Guided Transformer architecture that exploits explicit 3D geometric priors for efficient mesh generation. We introduce three key components: (1) a Gaussian-Guided Spatial Reasoning Transformer represents Gaussian primitives as structured 3D tokens and jointly reasons over Gaussian and image features; (2) a Streaming and Patchwise Geometry Encoder processes native-resolution views sequentially and aggregates information across variable-length view sets; (3) a Scale-Aligned Hybrid Depth Refiner uses a PatchFusion-style encoder--decoder to fuse RGB-conditioned predicted depth with Gaussian-rendered metric depth, combining fine local structures with globally consistent metric scale. The refined depth maps are integrated through TSDF fusion, followed by Marching Cubes for deterministic mesh extraction. Extensive experiments show that AnyGS2Mesh achieves state-of-the-art reconstruction quality while significantly reducing inference time compared with optimization-based baselines, enabling near-real-time, high-quality mesh reconstruction. Our results demonstrate the potential of combining Gaussian representations and feed-forward Transformer architectures for scalable 3D geometry reconstruction. The code will be made publicly available upon acceptance.

cs.CG↗

arXiv2026

Bellman Search in Arbitrary Finite Dimension: A Self-Similar Cell Theorem and Effective Computability of Planar Shoreline Search

A shoreline-search path starts at the origin and must meet an unknown affine line, without knowing either its normal or its distance. We first establish a self-similar reduction theorem for homogeneous search problems whose historical information is a record profile updated by pointwise maximum. Two quasi-returns of the normalized state delimit a block that renews the required profile by itself; a short connector closes this block into a cell. Every finite-ratio path can therefore be approximated, with arbitrarily small loss, by repetitions of a single cell at all scales. The main chain is then made effective. A finite coding of the state space computably bounds the scale factor and normalized length of a nearly optimal cell. For planar Shoreline search, the support function of the convex hull gives an exact cell functional. A one-sided polygonalization then reduces the problem to a computable number of vertices, after which quantifier elimination decides whether a polygonal cell exists below a rational threshold. It follows that the optimal deterministic planar Shoreline value $C_2^*$ is a computable real: for every rational $ε>0$, an algorithm terminates with a rational interval of width at most $ε$ containing $C_2^*$. Additional results---sliding memory, Bellman transitions, deadlines, geometric filters, and relative equilibria---are presented separately as a toolbox for certified computation and for the study of spiral rigidity; they are not used in the computability proof.

cs.CG↗

arXiv2026

Collision-based logic in Lenia and its composition boundary

Continuous cellular automata such as Lenia spontaneously produce lifelike, self-propelling patterns, including the Orbium glider, which travels in a straight line while pulsing through a fixed breathing cycle. Collision-based logic, where moving patterns compute by colliding, is established in discrete cellular automata and continuous physical media. Within continuous cellular automata, computation has so far been trained into the rule rather than emerging from collisions, and whether a fixed-rule automaton like Lenia can support general collision-based computation remains open. This paper constructs an INHIBIT gate from collisions of the Orbium glider. Of the patterns searched across four continuous-CA rule types, the Orbium glider is the only one shown to survive a collision with both copies intact. A control glider deflects a signal glider off its output line, so the output carries a signal only when no control is present. The gate blocks across all twenty-four phases of the breathing cycle and nine integer offsets of the control. Two such gates in series, with one signal line and two controls, compose into an AND-NOT chain, correct on all eight input combinations. By contrast, routing a signal beyond that single chain is undemonstrated. A deflected signal is not restored to a fixed landing position, and no reusable absorber for the surviving gliders was found. The immediate open question for collision-based computation in Lenia therefore narrows from whether a gate exists to whether a deflected signal can be delivered to a downstream gate, the next requirement for composing the gate beyond a single straight chain.

cs.ET↗

arXiv2026

Blended Chart Surfaces: A Seamless Explicit Representation for Smooth Surface Fitting

A surface representation suitable for geometry processing should be compact and explicit, provide global smoothness guarantees, support a wide range of surface topologies, and offer reliable access to differential quantities such as normals and surface energies, while remaining compatible with modern differentiable optimization. Existing neural representations typically sacrifice one or more of these properties: implicit fields typically require iso-surfacing for downstream use, while explicit neural maps are constrained by canonical-domain parametrizations or exhibit seam artifacts between local charts. We introduce Blended Chart Surfaces, a compact, network-free, explicit representation that is smooth by construction and anchored to user-provided topology. Given a coarse proxy mesh encoding the intended surface topology and approximate geometry, Blended Chart Surfaces jointly optimize for a polynomial map at each proxy vertex using an off-the-shelf optimizer to fit to an implicit target shape, avoiding the need for an input parametrization. Neighboring maps are fused using a smooth 'one-ring coordinate' blending scheme, decoupling topology and coarse geometry (carried by the proxy) from geometric details (carried by the local patches). The surface is globally smooth, fully differentiable, and enables stable evaluation of derivatives, making differential quantities and surface energies directly accessible. Additionally, our construction is equivariant to rigid motions and scaling of the proxy mesh. We evaluate Blended Chart Surfaces on various topologies and geometric complexity, and compare against explicit alternatives including interpolating-function baselines and mesh-displacement MLPs. Across these, Blended Chart Surfaces achieve a favorable trade-off among compactness, simplicity, access to differential quantities, and expressivity while remaining smooth across patch boundaries.

cs.GR↗

arXiv2026

Curves of constant width and Lebesgue's covering problem

A universal cover is a convex set in the plane that contains a congruent copy of every planar set of diameter one. Lebesgue asked in 1914 for one of least area, and the value is not known. We prove that every convex universal cover has area at least 0.8344, improving on 0.832, published in 2005, and 0.833, in a 2026 preprint, both of which come from a disc together with an equilateral triangle and a regular pentagon. Our test sets are instead curves of constant width: the disc, the Reuleaux triangle and the Reuleaux pentagon. Each contains the regular polygon it is built on, so the family is strictly larger at the same number of bodies and the same number of placement parameters, and we show that the classical configuration admits an arrangement whose hull has area below 0.8336, so no bound drawn from those three sets by this argument reaches ours. Curves of constant width were proposed for this role, and explored numerically, by Gibbs in 2014; what is added here is a proof. It consists of an analytic reduction followed by one finite computation. The reduction bounds the hull area from below over an entire box of placements at once, by eroding each Reuleaux polygon to a fixed set contained in every placement that box allows. The computation is an exhaustive subdivision of the resulting five-dimensional space, recorded as a certificate of 486,799,600 nodes and checked by a verifier independent of the search, with a rigorous bound on its floating point error some thousands of times smaller than the margin the verification attains.

cs.CG↗

arXiv2026

Sequential Euclidean connections with exponential memory: distributional performance and adversarial robustness

Points in the unit ball of $\mathbb R^d$ are processed sequentially. Each new point $p_i$ is connected to a state $x_{i-1}$ that summarizes earlier observations, after which $x_i=γx_{i-1}+(1-γ)p_i$, with $0\leqγ\leq1$. The cost is the sum of the $α$-powers of the connection lengths. This constant-gain rule interpolates between the input-order path and the star centered at the initial point. For independent uniform points, we establish the stationary insertion-length distribution and prove that it decreases in stochastic order as $γ$ increases. If $d+α>2$, or if $(d,α)=(1,1)$, the optimal constant parameter satisfies $1-γ_N^*=Θ(N^{-1/2})$, with an explicit asymptotic constant and closed bounds. For $α=1$, its leading expected tree length equals that of the center star and is eventually smaller than the expected lengths of both endpoint constructions. For $α=2$, the optimizer is unique and characterized exactly. For the same $N$, choosing $1-γ_N$ as a fixed positive multiple of $N^{-1/2}$ gives a sharp two-term expansion of the expected uniform-input cost and a maximal adversarial mean cost of $1+O(N^{-1/2})$. For every fixed $0\leqγ<1$ and $0<α\leq3$, the exact asymptotic adversarial value is $(2/(1+γ))^α$. When $d\geq2$, exponential weighting is within a factor smaller than $1.161^α$ of the best fixed nonnegative weighted rule with the same average look-back, for $0<α\leq3$. Comparison with the running mean highlights its time-homogeneous update, stationary coefficient profile, and fixed effective memory.

cs.CG↗

arXiv2026

Improved $\ell_0$-Isoperimetry for Convex Bodies via Mass Transport

We study $\ell_0$ isoperimetry for a convex body $K\subset \mathbb{R}^n$, $n\ge2$. For a Borel set $S\subset K$, let $\partial_0^K S$ be the set of points in $K \setminus S$ that can be reached from $S$ by changing at most one coordinate (i.e. the $\ell_0$ boundary of $S$). Suppose that, for some unconditional convex body $Q \subset \mathbb{R}^n$, numbers $r,R>0$, and possibly different centers $x_0,y_0$, \[ x_0+rQ \subset K\subset y_0+RQ. \] Writing $s=\text{vol}(S)/\text{vol}(K)$, we prove that whenever $0 0$ is an absolute constant. Consequently, the associated $\ell_0$-isoperimetric coefficient is at least $cr/(n^2R)$. Previous direct lower bounds were only known for $\ell_2$ and $\ell_\infty$ regularity whereas our lower bound holds directly for any $Q$-regularity, where $Q$ is an unconditional convex body. Compared to $\ell_2$ and $\ell_\infty$ regularity, our lower bound result improves upon the previously best known lower bounds, for any $s$, by a factor of $n$. As an application of our result, we give improved mixing time bounds for the Coordinate Hit and Run walk (CHAR). Our proof of the lower bound is based on a modification of the method of canonical paths applied to a continuous Hamming graph over our convex body. Our construction of canonical paths can be viewed as a suitable coordinate discretization of certain mass transport maps from $S$ to $S^c$. We also give complementary upper-bounds for any $Q$-regularity, with an overall factor of $n$ gap between the two.

math.FA↗

arXiv2026

The Prime Clockwork: A Dynamic Representation of Modular and Multiplicative Arithmetic

The way numbers are represented strongly influences which arithmetic structures are easy to see. The \emph{prime clockwork} is a recursively growing discrete dynamical system: a list of autonomous two-hand clocks driven by one common $+1$ signal. No primes or primality labels are supplied. Starting empty, the process appends a clock of period $n$ whenever none already present rings; the primes are generated internally as its growth times. For each installed prime $p$, the seconds reading $R_p$ advances through $0,\ldots,p-1$, and each return to zero increments the minutes reading $M_p$, which counts completed $p$-cycles. The hands use only increment, comparison, reset, and carry, without explicit \texttt{mod} or \texttt{div} operations. At time $n$, $n=pM_p(n)+R_p(n)$. The valuation readout $V_p(n)=ν_p(n)$ is generated locally: it is zero when the seconds counter is non-zero (silent state) and otherwise (when the p-clock rings) one plus the earlier valuation addressed by the current minutes reading. The valuation vector gives the integer in unique prime-factorized form. Its coordinates add and subtract under multiplication and division, representing every positive rational uniquely; divisibility becomes weak componentwise order, and unique factorization is natural in this representation. Finite seconds arrays form Cartesian-product state spaces whose common orbit visits every joint state once before repeating; this \emph{grand cycle} is the order-sensitive dynamical counterpart of the Chinese remainder theorem. The same coordinates expose gcd, lcm, perfect powers, Bézout's identity, and Euler's totient. Rational valuation levels reach certain positive algebraic irrationalities, but not algebraic numbers in general.

math.HO↗

arXiv2026

Sierpiński--Knopp Wasserstein Distance for Persistence Diagrams and Applications to 2-Wasserstein Approximation

This paper introduces the Sierpiński-Knopp (SK) Wasserstein distance, a fast metric between persistence diagrams. The SK-Wasserstein distance, denoted $d_{\mathrm{SK}}$, maps diagram points and their diagonal projections to the unit interval via the Sierpiński-Knopp space-filling curve on the upper diagonal triangle. The encoded point sets are then efficiently matched via one-dimensional optimal assignment, in $O(N\log N)$ steps, yielding an explicit diagonal-aware point assignment between the two input persistence diagrams. We show that the SK-Wasserstein distance controls the classical $2$-Wasserstein distance between diagrams, admits an explicit isometric embedding into a Hilbert space, and induces a positive-definite Gaussian kernel, making the resulting geometry directly compatible with Euclidean and kernel-based learning methods. A tighter surrogate dissimilarity, noted $W_Γ$, is also introduced based on the point assignments along the curve. Experiments on 12 scientific collections comprising 227 diagrams show median per-collection speedup of $d_{\mathrm{SK}}$ over state-of-the-art approximations of $W_2$ is $626\times$, while the aggregate speedup over the full benchmark is $2100\times$. Average-linkage partitions obtained from $d_{\mathrm{SK}}$ and $W_Γ$ each exactly match the corresponding $W_2$ partition on 8 of the 12 collections. Hilbert $k$-means and Gaussian spectral clustering, both based on $d_{\mathrm{SK}}$, achieve mean adjusted Rand indices (ARI) of $0.756$ and $0.800$, respectively, with respect to the benchmark reference partitions, compared to $0.750$ obtained by average linkage on $W_2$. The Gaussian $d_{\mathrm{SK}}$ kernel supports other kernel-based analysis tasks, as illustrated by its use for contiguous segmentation of ordered diagram collections in our experiments.

cs.CG↗

arXiv2026

A Dimension-Reducing Fréchet Simplification Oracle

Let $P$ be a polygonal curve with $n$ vertices in the plane. We construct a data structure of size $O(n \log n)$ suited for simplification queries of the following kind. Given a query line $\ell$ and an integer $k\ge1$, find a curve $Q$ on $\ell$ with at most $k$ vertices that minimizes the discrete Fréchet distance to $P$, among all such curves. Using our data structure, a query can be handled in $O(k^2 \log^3 n + k\log^4 n)$ time. More generally, a geometric tree $T$ on $n$ vertices in the plane can be preprocessed into a near-linear-size structure so that, given a pair $u$, $v$ of its vertices, a line $\ell$, and an integer $k\ge1$, one can find a curve $Q$ on $\ell$ with at most $k$ vertices that minimizes the discrete Fréchet distance to the path from $u$ to $v$ in $T$, in time $O(k^2 \mathop{polylog} n)$. For the general dimension-reduction problem, where $P$ is a curve in $\mathbb{R}^d$ ($d \ge 3$), $0 < \varepsilon_0 < 1$ is a real parameter, and a query specifies a $g$-flat $h$ ($1 \le g \le d-1$) and an integer $k \ge 1$, we construct a data structure of size $O(n\log n + f(\varepsilon_0) n)$, where $f(\varepsilon_0)=(1+1/\varepsilon_0)^{(d-1)/2}$, that allows us to find a curve $Q$ on $h$ with at most $k$ vertices, whose discrete Fréchet distance to $P$ is at most $1+\varepsilon_0$ times the distance of $Q^*$ to $P$, where $Q^*$ is such a curve that minimizes the distance to $P$. The query handling time is $O(f(\varepsilon_0) k^2 \log^2 n)$.

cs.CG↗

arXiv2026

Parameter optimization for restarted mixed precision iterative sparse solver

The problem of optimal precision switching for the conjugate gradient (CG) method applied to sparse linear systems is considered. A sparse matrix is defined as an $n\!\times\!n$ matrix with $m\!=\!O(n)$ nonzero entries. The algorithm first computes an approximate solution in single precision with tolerance $\varepsilon_1$, then switches to double precision to refine the solution to the required stopping tolerance $\varepsilon_2$. Based on estimates of system matrix parameters -- computed in time which does not exceed $1\%$ of the time needed to solve the system in double precision -- we determine the optimal value of $\varepsilon_1$ that minimizes total computation time. This value is obtained by classifying the matrix using the $k$-nearest neighbors method on a small precomputed sample. Classification relies on a feature vector comprising: the matrix size $n$, the number of nonzeros $m$, the pseudo-diameter of the matrix sparsity graph, and the average rate of residual norm decay during the early CG iterations in single precision. We show that, in addition to the matrix condition number, the diameter of the sparsity graph influences the growth of rounding errors during iterative computations. The proposed algorithm reduces the computational complexity of the CG -- expressed in equivalent double-precision iterations -- by more than $17\%$ on average across the considered matrix types in a sequential setting. The resulting speedup is at most $1.5\%$ worse than that achieved with the optimal (oracle) choice of $\varepsilon_1$. While the impact of matrix structure on Krylov subspace method convergence is well understood, the use of the sparsity graph diameter as a predictive feature for rounding error growth in mixed-precision CG appears to be novel. To the best of our knowledge, no prior work employs graph diameter to guide precision switching in iterative linear solvers.

math.NA↗

arXiv2026

Quantum Query Complexity of Persistence Statistics in Graph Zigzags

We study the query complexity of estimating scalar summaries of zigzag bar lifetimes from snapshot-adjacency bits. For graphs $G_1,\ldots,G_m$ on $n$ labeled vertices, let $\ell_b$ be the snapshot lifetime of a degree-one bar $b$ of the intersection zigzag. For a probability generating function $ϕ(x)=\mathbb{E}[x^R]$, the statistic $F_ϕ=\sum_bϕ(\ell_b/m)$ includes normalized degree-$r$ total persistence and the mean generalized rank over a uniform time window. An exact identity underlies our algorithm: sample $R$ uniform times; the expected generalized rank between their minimum and maximum equals $F_ϕ$. For graphs that rank is the circuit rank of an intersection graph, so a nonlinear barcode functional becomes an average of edge and component counts, and no barcode is computed. Without spectral-gap, homology-state, or QRAM assumptions, this gives a quantum estimator with additive error $\varepsilon n$ and $\widetilde O(\sqrt{m(K+n)}/\varepsilon)$ queries when a bound $K\ge F_ϕ$ is supplied, against $\widetilde O(m\min\{n^2,(K+n)/\varepsilon^2\})$ classically, and an adaptive quantum variant with the same instance dependence. These estimators are optimal in two regimes. For every fixed power weight $x^r$, $r\ge2$, and for the uniform-window mean, the worst-case complexities are $\widetildeΘ(n\sqrt m/\varepsilon)$ quantum and $Θ(n^2m)$ classical. On sparse instances, under an explicit split-leakage promise met by power and binomial weights of logarithmic degree and the promise $F_ϕ\le K$, they are $\widetildeΘ(\sqrt{mK}/\varepsilon)$ and $\widetildeΘ(m\min\{n^2,K/\varepsilon^2\})$. The classical lower bounds hold against fully adaptive algorithms, and fewer than $m$ such statistics cannot determine the positive-lifetime histogram. All bounds concern snapshot access; with an explicit update stream, near-linear full-barcode algorithms are known.

quant-ph↗