Rm Log Research Articles

Total progeny in killed branching random walk

We consider a branching random walk for which the maximum position of a particle in the n’th generation, Rn, has zero speed on the linear scale: Rn/n → 0 as n → ∞. We further remove (“kill”) any particle whose displacement is negative, together with its entire descendence. The size Z of the set of un-killed particles is almost surely finite (Gantert and Muller in Markov Process. Relat. Fields 12:805–814, 2006; Hu and Shi in Ann. Probab. 37(2):742–789, 2009). In this paper, we confirm a conjecture of Aldous (Algorithmica 22:388–412, 1998; and Power laws and killed branching random walks) that E [Z] < ∞ while \({{\mathbf E}\left[Z\,{\rm log}\, Z\right]=\infty}\) . The proofs rely on precise large deviations estimates and ballot theorem-style results for the sample paths of random walks.

On the dimension of invariant measures of endomorphisms of $${\mathbb{C}\mathbb{P}^k}$$

Let f be an endomorphism of \({\mathbb{C}\mathbb{P}^k}\) and ν be an f-invariant measure with positive Lyapunov exponents (λ1, . . . , λk). We prove a lower bound for the pointwise dimension of ν in terms of the degree of f, the exponents of ν and the entropy of ν. In particular our result can be applied for the maximal entropy measure μ. When k = 2, it implies that the Hausdorff dimension of μ is estimated by \({{\rm dim}_\mathcal{H}\mu \geq {{\rm log} d \over \lambda_1} + {{\rm log} d \over \lambda_2}}\), which is half of the conjectured formula. Our method for proving these results consists in studying the distribution of the ν-generic inverse branches of fn in \({\mathbb{C}\mathbb{P}^k}\) . Our tools are a volume growth estimate for the bounded holomorphic polydiscs in \({\mathbb{C}\mathbb{P}^k}\) and a normalization theorem for the ν-generic inverse branches of fn.

Top-k queries on temporal data

The database community has devoted extensive amount of efforts to indexing and querying temporal data in the past decades. However, insufficient amount of attention has been paid to temporal ranking queries. More precisely, given any time instance t, the query asks for the top-k objects at time t with respect to some score attribute. Some generic indexing structures based on R-trees do support ranking queries on temporal data, but as they are not tailored for such queries, the performance is far from satisfactory. We present the Seb-tree, a simple indexing scheme that supports temporal ranking queries much more efficiently. The Seb-tree answers a top-k query for any time instance t in the optimal number of I/Os in expectation, namely, $${O\left({\rm log}_B\,\frac{N}{B}+\frac{k}{B}\right)}$$ I/Os, where N is the size of the data set and B is the disk block size. The index has near-linear size (for constant and reasonable k max values, where k max is the maximum value for the possible values of the query parameter k), can be constructed in near-linear time, and also supports insertions and deletions without affecting its query performance guarantee. Most of all, the Seb-tree is especially appealing in practice due to its simplicity as it uses the B-tree as the only building block. Extensive experiments on a number of large data sets, show that the Seb-tree is more than an order of magnitude faster than the R-tree based indexes for temporal ranking queries.

The structure of $${\phi}$$ -stable minimal hypersurfaces in manifolds of nonnegative P-scalar curvature

Suppose (N n , g) is an n-dimensional Riemannian manifold with a given smooth measure m. The P-scalar curvature is defined as $${P(g)=R^m_\infty(g)=R(g)-2\Delta_g{\rm log}\,\phi-|\nabla_g{\rm log}\,\phi|_g^2}$$ , where $${dm=\phi\,dvol(g)}$$ and R(g) is the scalar curvature of (N n , g). In this paper, under a technical assumption on $${\phi}$$ , we prove that $${\phi}$$ -stable minimal oriented hypersurface in the three-dimensional manifold with nonnegative P-scalar curvature must be conformally equivalent to either the complex plane $${\mathbb{C}}$$ or the cylinder $${\mathbb{R}\times\mathbb{S}^1}$$ .

Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds

Let M be a complete noncompact Riemannian manifold. We consider gradient estimates for the positive solutions to the following nonlinear parabolic equation $$ \frac{\partial u}{\partial t} = \Delta _{f}u +au\,{\rm log}\, u + bu$$ on \({M \times [0, + \infty)}\), where a, b are two real constants, f is a smooth real-valued function on M and \({\Delta_f = \Delta - \nabla f \nabla}\). Under the assumption that the N-Bakry-Emery Ricci tensor is bounded from below by a negative constant, we obtain a gradient estimate for positive solutions of the above equation. As an application, we obtain a Harnack inequality and a Gaussian lower bound of the heat kernel of such an equation.

An Optimal Randomized Cell Probe Lower Bound for Approximate Nearest Neighbor Searching

We consider the approximate nearest neighbor search problem on the Hamming cube $\{0,1\}^d$. We show that a randomized cell probe algorithm that uses polynomial storage and word size $d^{O(1)}$ requires a worst case query time of $\Omega({\rm log}\,{\rm log}\,d/{\rm log}\,{\rm log}\,{\rm log}\,d)$. The approximation factor may be as loose as $2^{{\rm log}^{1-\eta}d}$ for any fixed $\eta>0$. Our result fills a major gap in the study of this problem since all earlier lower bounds either did not allow randomization [A. Chakrabarti et al., A lower bound on the complexity of approximate nearest-neighbor searching on the Hamming cube, in Discrete and Computational Geometry, Springer, Berlin, 2003, pp. 313–328; D. Liu, Inform. Process. Lett., 92 (2004), pp. 23–29] or did not allow approximation [A. Borodin, R. Ostrovsky, and Y. Rabani, Proceedings of the 31st Annual ACM Symposium on Theory of Computing, 1999, pp. 312–321; O. Barkol and Y. Rabani, Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, 2000, pp. 388–396; T. S. Jayram et al., J. Comput. System Sci., 69 (2004), pp. 435–447]. We also give a cell probe algorithm that proves that our lower bound is optimal. Our proof uses a lower bound on the round complexity of the related communication problem. We show, additionally, that considerations of bit complexity alone cannot prove any nontrivial cell probe lower bound for the problem. This shows that the “richness technique” [P. B. Miltersen et al., J. Comput. System Sci., 57 (1998), pp. 37–49] used in a lot of recent research around this problem would not have helped here. Our proof is based on information theoretic techniques for communication complexity, a theme that has been prominent in recent research [A. Chakrabarti et al., Proceedings of the 42nd Annual IEEE Symposium on Foundations of Computer Science, 2001, pp. 270–278; Z. Bar-Yossef et al., Proceedings of the 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002, pp. 209–218; P. Sen, Proceedings of the 18th Annual IEEE Conference on Computational Complexity, 2003, pp. 73–83; R. Jain, J. Radhakrishnan, and P. Sen, Proceedings of the 30th International Colloquium on Automata, Languages and Programming, 2003, pp. 300–315].

Electrical transport properties of amorphous CNx/p‐Si heterostructures

AbstractThe electrical transport properties of amorphous CNx/p‐Si heterostructures fabricated using reactive facing‐target sputtering were investigated systematically. The obvious rectifying effect is observed in the I–V curves for the heterostructures fabricated at the nitrogen partial pressure ($P_{{\rm N}_2 }$) of 20%, and the resistance of the heterostructures in both the forward and reverse voltage ranges increases with the decrease of the current and temperature. At low‐voltage range, the resistance satisfies the relation of ${\rm log}\,R \propto {\rm log}\,I$, and the slope of the ${\rm log}\,R \propto {\rm log}\,I$ plots increases with the decrease of temperature. The electrical transport characteristic of the heterostructures can be affected by the changes of the number and cluster size of sp2 C and the ratio of N–C(sp2)/N–C(sp3) by adjusting $P_{{\rm N}_2 }$ significantly, and the good rectifying effect of the heterostructures fabricated at a certain condition makes them useful in the field of electronic devices.

Distributed algorithms for ultrasparse spanners and linear size skeletons

We present efficient algorithms for computing very sparse low distortion spanners in distributed networks and prove some non-trivial lower bounds on the tradeoff between time, sparseness, and distortion. All of our algorithms assume a synchronized distributed network, where relatively short messages may be communicated in each time step. Our first result is a fast distributed algorithm for finding an \({O(2^{{\rm log}^{*} n} {\rm log} n)}\) -spanner with size O(n). Besides being nearly optimal in time and distortion, this algorithm appears to be the first that constructs an O(n)-size skeleton without requiring unbounded length messages or time proportional to the diameter of the network. Our second result is a new class of efficiently constructible (α, β)-spanners called Fibonacci spanners whose distortion improves with the distance being approximated. At their sparsest Fibonacci spanners can have nearly linear size, namely \({O(n(\log \log n)^{\phi})}\) , where \({\phi = (1 + \sqrt{5})/2}\) is the golden ratio. As the distance increases the multiplicative distortion of a Fibonacci spanner passes through four discrete stages, moving from logarithmic to log-logarithmic, then into a period where it is constant, tending to 3, followed by another period tending to 1. On the lower bound side we prove that many recent sequential spanner constructions have no efficient counterparts in distributed networks, even if the desired distortion only needs to be achieved on the average or for a tiny fraction of the vertices. In particular, any distance preservers, purely additive spanners, or spanners with sublinear additive distortion must either be very dense, slow to construct, or have very weak guarantees on distortion.

Mean-Field Conditions for Percolation on Finite Graphs

Let {G n } be a sequence of finite transitive graphs with vertex degree d = d(n) and |G n | = n. Denote by p t(v, v) the return probability after t steps of the non-backtracking random walk on G n . We show that if p t(v, v) has quasi-random properties, then critical bond-percolation on G n behaves as it would on a random graph. More precisely, if $$\mathop {\rm {lim\, sup\,}} \limits_{n} n^{1/3} \sum\limits_{t = 1}^{n^{1/3}} {t{\bf p}^t(v,v) < \infty ,}$$then the size of the largest component in p-bond-percolation with \({p =\frac{1+O(n^{-1/3})}{d-1}}\) is roughly n 2/3. In Physics jargon, this condition implies that there exists a scaling window with a mean-field width of n −1/3 around the critical probability \({p_c =\frac{1}{d-1}}\).A consequence of our theorems is that if {G n } is a transitive expander family with girth at least \({\left( {\frac{2}{3} + \epsilon} \right) {\rm log}_{d - 1}n}\) then {G n } has the above scaling window around \({p_c =\frac{1}{d-1}}\). In particular, bond-percolation on the celebrated Ramanujan graph constructed by Lubotzky, Phillips and Sarnak [LuPS] has the above scaling window. This provides the first examples of quasi-random graphs behaving like random graphs with respect to critical bond-percolation.

Bond percolation with attenuation in high dimensional Voronoĭ tilings

AbstractLet $\cal{P}$ be the set of points in a realization of a uniform Poisson process in ℝn. The set $\cal{P}$ determines a Voronoĭ tiling of ℝn. Construct an infinite graph $\cal{G}$ with vertex set $\cal{P}$ and edges joining vertices when the corresponding Voronoĭ cells share a (n − 1)‐dimensional boundary face. We consider bond percolation models on $\cal{G}$ obtained by declaring each edge xy of $\cal{G}$ open independently with probability p(∥x − y∥), depending only on the Euclidean distance ∥x − y∥ between the vertices. We give some sufficient conditions on p(t) that ensures that an infinite connected component (i.e., percolation) occurs, or does not occur. In particular, we show that for p(t) = p is a constant, there is a phase transition at a critical probability p = pc(n), where 2−n(5nlog n)−1 ≤ pc(n) ≤ C2−n $\sqrt{n}\;{\rm log} \; n$. We also show that if p(t) = e−λt then there is a phase transition at a critical parameter λ = λc(n), where λc(n) = (log e2 + o(1))n/2rn, where rn is the radius of the n‐dimensional sphere that, on average, contains a single point of $\cal{P}$. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2010

Constructions of Low-degree and Error-Correcting ε-Biased Generators

In this work we give two new constructions of e-biased generators. Our first construction significantly extends a result of Mossel et al. (Random Structures and Algorithms 2006, pages 56-81), and our second construction answers an open question of Dodis and Smith (STOC 2005, pages 654-663). In particular we obtain the following results: For every k = o(log n) we construct an e-biased generator $$G : \{0, 1\}^{m} \rightarrow \{0, 1\}^n$$that is implementable by degree k polynomials (namely, every output bit of the generator is a degree k polynomial in the input bits). For any constant k we get that $$n = \Omega(m/{\rm log}(1/ \epsilon))^k$$, which is nearly optimal. Our result also separates degree k generators from generators in NC0k, showing that the stretch of the former can be much larger than the stretch of the latter. The problem of constructing degree k generators was introduced by Mossel et al. who gave a construction only for the case of k = 2. We construct a family of asymptotically good binary codes such that the codes in our family are also e-biased sets for an exponentially small e. Our encoding algorithm runs in polynomial time in the block length of the code. Moreover, these codes have a polynomial time decoding algorithm. This answers an open question of Dodis and Smith. The paper also contains an appendix by Venkatesan Guruswami that provides an explicit construction of a family of error correcting codes of rate 1/2 that has efficient encoding and decoding algorithms and whose dual codes are also good codes.

REVISITING THE “FUNDAMENTAL PLANE” OF BLACK HOLE ACTIVITY AT EXTREMELY LOW LUMINOSITIES

We investigate the origin of the X-ray emission in low-luminosity AGNs (LLAGNs). Yuan & Cui (2005) predicted that the X-ray emission should originate from jets rather than from an advection-dominated accretion flow (ADAF) when the X-ray luminosity $L_{\rm X}$ of the source is below a critical value of $L_{\rm X,crit} \approx 10^{-6}L_{\rm Edd}$. This prediction implies that the X-ray spectrum in such sources should be fitted by jets rather than ADAFs. Furthermore, below $L_{\rm X,crit}$ the correlation between radio ($L_{\rm R}$) and X-ray ($L_{\rm X}$) luminosities and the black hole mass ($M$)--the so-called fundamental plane of black hole activity--should deviate from the general correlation obtained by Merloni, Heinz & Di Matteo (2003) and become steeper. The Merloni et al. correlation is described by ${\rm log}L_{\rm R} =0.6{\rm log}L_{\rm X}+0.78{\rm log}M+7.33$, while the predicted correlation is ${\rm log}L_{\rm R}=1.23{\rm log}L_{\rm X} +0.25{\rm log}M-13.45$. We collect data from the literature to check the validity of these two expectations. We find that among the 16 LLAGNs with good X-ray and radio spectra, 13 are consistent with the Yuan & Cui prediction. For the 22 LLAGNs with $L_{\rm X} < L_{\rm X,crit}$, the fundamental plane correlation is described by ${\rm log}L_{\rm R}=1.22{\rm log}L_{\rm X}+0.23{\rm log}M-12.46 $, also in excellent agreement with the prediction.

Long-Time Asymptotics for the Focusing NLS Equation with Time-Periodic Boundary Condition on the Half-Line

We consider the focusing nonlinear Schrödinger equation on the quarter plane. The initial data are vanishing at infinity while the boundary data are time- periodic, of the form \({a{\rm e}^{\i\alpha} {\rm e}^{2\i\omega t}}\) . The goal of this paper is to study the asymptotic behavior of the solution of this initial-boundary-value problem. The main tool is the asymptotic analysis of an associated matrix Riemann–Hilbert problem. We show that for \({\omega < -3a^2}\) the solution of the IBV problem has different asymptotic behaviors in different regions. In the region \({x > 4bt}\) , where \({b\mathop{:=} \sqrt{(a^2-\omega)/2} > 0}\) , the solution takes the form of the Zakharov-Manakov vanishing asymptotics. In a region of type \({4bt-\frac{N+1}{2a} {\rm log} t < x < 4bt}\) , where N is any integer, the solution is asymptotic to a train of asymptotic solitons. In the region \({4(b-a\sqrt2)t < x < 4bt}\) , the solution takes the form of a modulated elliptic wave. In the region \({0 < x < 4(b-a\sqrt2)t}\) , the solution takes the form of a plane wave.

Approximate Inclusion-Exclusion for Arbitrary Symmetric Functions

Let $$A_{1}, \ldots, A_{n}$$ be events in a probability space. The approximate inclusion-exclusion problem, due to Linial and Nisan (1990), is to estimate $${\bf P} [A_{1} \cup \ldots \cup A_{n}]$$ given $${\bf P} [\bigcap_{i\in S} A_{i}]$$ for |S| ≤ k. Kahn et al. (1996) solved this problem optimally for each k. We study the following more general question: estimate $${\bf P} [f(A_{1}, \ldots, A_{n})]$$ given $${\bf P} [\bigcap_{i\in S} A_{i}]$$ for |S| ≤ k, where f : {0, 1} n → {0, 1} is a given symmetric function. We solve this general problem for every f and k, giving an algorithm that runs in polynomial time and achieves an approximation error that is essentially optimal. We prove this optimal error to be $$2^{-\tilde\Theta(k^{2}/n)}$$ for k above a certain threshold, and $$\Theta(1)$$ otherwise. As part of our solution, we analyze, for every nonconstant symmetric f : {0, 1} n → {0, 1} and every $$\epsilon \in [2^{-n}, 1/3]$$ , the least degree $${\rm deg}_{\epsilon}(f)$$ of a polynomial that approximates f pointwise within $$\epsilon$$ . We show that $${\rm deg}_{\epsilon}(f) = \tilde\Theta({\rm deg}_{1/3}(f) + \sqrt{n {\rm log}(1/\epsilon))}$$ , where deg1/3(f) is well-known for each f. Previously, the answer for vanishing $$\epsilon$$ was known only for f = OR. We construct the approximating polynomial explicitly for all f and $$\epsilon$$ .

Disjointness is Hard in the Multiparty Number-on-the-Forehead Model

We show that disjointness requires randomized communication $$\Omega(\frac{n^{1/(k+1)}}{2^{2^k}})$$ in the general k-party number-on-the-forehead model of complexity. The previous best lower bound for k ≥ 3 was $$\frac{{\rm log} \, n}{k-1}$$ . Our results give a separation between nondeterministic and randomized multiparty number-on-the-forehead communication complexity for up to k = log log n − O(log log log n) many players. Also, by a reduction of Beame, Pitassi, and Segerlind, these results imply subexponential lower bounds on the size of proofs needed to refute certain unsatisfiable CNFs in a broad class of proof systems, including tree-like Lovasz–Schrijver proofs.

Gibbs rapidly samples colorings of G(n, d/n)

Gibbs sampling also known as Glauber dynamics is a popular technique for sampling high dimensional distributions defined on graphs. Of special interest is the behavior of Gibbs sampling on the Erdős–Rényi random graph G(n, d/n), where each edge is chosen independently with probability d/n and d is fixed. While the average degree in G(n, d/n) is d(1−o(1)), it contains many nodes of degree of order (log n) / (log log n). The existence of nodes of almost logarithmic degrees implies that for many natural distributions defined on G(n, d/n) such as uniform coloring (with a constant number of colors) or the Ising model at any fixed inverse temperature β, the mixing time of Gibbs sampling is at least n 1+Ω(1 / log log n) with high probability. High degree nodes pose a technical challenge in proving polynomial time mixing of the dynamics for many models including coloring. Almost all known sufficient conditions in terms of number of colors needed for rapid mixing of Gibbs samplers are stated in terms of the maximum degree of the underlying graph. In this work we consider sampling q-colorings and show that for every d < ∞ there exists q(d) < ∞ such that for all q ≥ q(d) the mixing time of the Gibbs sampling on G(n, d/n) is polynomial in n with high probability. Our results are the first polynomial time mixing results proven for the coloring model on G(n, d/n) for d > 1 where the number of colors does not depend on n. They also provide a rare example where one can prove a polynomial time mixing of Gibbs sampler in a situation where the actual mixing time is slower than npolylog(n). In previous work we have shown that similar results hold for the ferromagnetic Ising model. However, the proof for the Ising model crucially relied on monotonicity arguments and the “Weitz tree”, both of which have no counterparts in the coloring setting. Our proof presented here exploits in novel ways the local treelike structure of Erdős–Rényi random graphs, block dynamics, spatial decay properties and coupling arguments. Our results give the first polynomial-time algorithm to approximately sample colorings on G(n, d/n) with a constant number of colors. They extend to much more general families of graphs which are sparse in some average sense and to much more general interactions. In particular, they apply to any graph for which there exists an α > 0 such that every vertex v of the graph has a neighborhood N(v) of radius O(log n) in which the induced sub-graph is the union of a tree and at most O(1) edges and where each simple path Γ of length O(log n) satisfies $${\sum_{u \in \Gamma}\sum_{v \neq u}\alpha^{d(u,v)} = O({\rm log} n)}$$ . The results also generalize to the hard-core model at low fugacity and to general models of soft constraints at high temperatures.

INTERNAL EXTINCTION IN THE SLOAN DIGITAL SKY SURVEY LATE-TYPE GALAXIES

We study internal extinction of late-type galaxies in the Sloan Digital Sky Survey. We find that the degree of internal extinction depends on both the concentration index c and K_s-band absolute magnitude M_K. We give simple fitting functions for internal extinction. In particular, we present analytic formulae giving the extinction-corrected magnitudes from the observed optical parameters. For example, the extinction-corrected r-band absolute magnitude can be obtained by M_{r,0}=-20.77 +(-1+\sqrt{1+4\Delta (M_{r,obs}+20.77+4.93\Delta)})/2\Delta, where \Delta = 0.236\{1.35(c-2.48)^2-1.14\}\log(a/b), c=R_{90}/R_{50} is the the concentration index, and a/b is the isophotal axis ratio of the 25 mag/arcsec^2 isophote in the i-band. The 1\sigma error in M_{r,0} is 0.21{\rm log}(a/b). The late-type galaxies with very different inclinations are found to trace almost the same sequence in the (u-r)-$M_r$ diagram when our prescriptions for extinction correction are applied. We also find that (u-r) color can be a third independent parameter that determines the degree of internal extinction.

Algorithms for Singleton Attractor Detection in Planar and Nonplanar AND/OR Boolean Networks

Singleton attractor (also called fixed point) detection is known to be NP-hard even for AND/OR Boolean networks (AND/OR BNs in short, i.e., BNs consisting of AND/OR nodes), where BN is a mathematical model of genetic networks and singleton attractors correspond to steady states. In our recent paper, we developed an O(1.787n) time algorithm for detecting a singleton attractor of a given AND/OR BN where n is the number of nodes. In this paper, we present an O(1.757n) time algorithm with which we succeeded in improving the above algorithm. We also show that this problem can be solved in \(2^{O(({\rm log} \, n)\sqrt{n})}\) time, which is less than O((1 + ∈)n) for any positive constant ∈, when a BN is planar.

Boundary proximity of SLE

This paper examines how close the chordal SLEκ curve gets to the real line asymptotically far away from its starting point. In particular, when κ ϵ (0, 4), it is shown that if β > βκ := 1/(8/κ − 2), then the intersection of the SLEκ curve with the graph of the function y = x/(log x)β, x > e, is a.s. bounded, while it is a.s. unbounded if β = βκ. The critical SLE4 curve a.s. intersects the graph of \(y=x^{{-({\rm log\,log\,x})}^{\alpha}}, x > e^e\), x > ee, in an unbounded set if α ≤ 1, but not if α > 1. Under a very mild regularity assumption on the function y(x), we give a necessary and sufficient integrability condition for the intersection of the SLEκ path with the graph of y to be unbounded. When the intersection is bounded a.s., we provide an estimate for the probability that the SLEκ path hits the graph of y. We also prove that the Hausdorff dimension of the intersection set of the SLEκ curve and the real axis is 2 − 8/κ when 4 < κ < 8.

An LLL Algorithm with Quadratic Complexity

The Lenstra-Lenstra-Lovasz lattice basis reduction algorithm (called LLL or ${\rm L}^3$) is a fundamental tool in computational number theory and theoretical computer science, which can be viewed as an efficient algorithmic version of Hermite's inequality on Hermite's constant. Given an integer $d$-dimensional lattice basis with vectors of Euclidean norm less than $B$ in an $n$-dimensional space, the ${\rm L}^3$ algorithm outputs a reduced basis in $O(d^3n\,{\rm log}\,B\cdot\mathcal{M}(d\,{\rm log}\,B))$ bit operations, where $\mathcal{M}(k)$ denotes the time required to multiply $k$-bit integers. This worst-case complexity is problematic for applications where $d$ or/and ${\rm log}\,B$ are often large. As a result, the original ${\rm L}^3$ algorithm is almost never used in practice, except in tiny dimension. Instead, one applies floating-point variants where the long-integer arithmetic required by Gram-Schmidt orthogonalization is replaced by floating-point arithmetic. Unfortunately, this is known to be unstable in the worst case: the usual floating-point ${\rm L}^3$ algorithm is not even guaranteed to terminate, and the output basis may not be ${\rm L}^3$-reduced at all. In this article, we introduce the ${\rm L}^2$ algorithm, a new and natural floating-point variant of the ${\rm L}^3$ algorithm which provably outputs ${\rm L}^3$-reduced bases in polynomial time $O(d^2n(d+{\rm log}\,B)\,{\rm log}\,B\cdot\mathcal{M}(d))$. This is the first ${\rm L}^3$ algorithm whose running time (without fast integer arithmetic) provably grows only quadratically with respect to ${\rm log}\,B$, like Euclid's gcd algorithm and Lagrange's two-dimensional algorithm.