Global roundings of sequences

For a given sequence a=(a₁,…,an) of numbers, a global rounding is an integer sequence b=(b₁,…,bn) such that the rounding error |_∑i∈I(ai−bi)| is less than one in all intervals I⊆{1,…,n}. We give a simple characterization of the set of global roundings of a. This allows to compute optimal roundings in time O(nlogn) and generate a global rounding uniformly at random in linear time under a non-degeneracy assumption and in time O(nlogn) in the general case.     

On the cubicity of bipartite graphs

A unit cube in k-dimension (or a k-cube) is defined as the Cartesian product R₁×R₂×?×Rk, where each Ri is a closed interval on the real line of the form [ai,ai+1]. The cubicity of G, denoted as cub(G), is the minimum k such that G is the intersection graph of a collection of k-cubes. Many NP-complete graph problems can be solved efficiently or have good approximation ratios in graphs of low cubicity. In most of these cases the first step is to get a low dimensional cube representation of the given graph.It is known that for a graph G, . Recently it has been shown that for a graph G, cub(G)?4(Δ+1)lnn, where n and Δ are the number of vertices and maximum degree of G, respectively. In this paper, we show that for a bipartite graph G=(A∪B,E) with |A|=n₁, |B|=n₂, n₁?n₂, and Δ^′=min{ΔA,ΔB}, where ΔA=maxa∈Ad(a) and ΔB=maxb∈Bd(b), d(a) and d(b) being the degree of a and b in G, respectively, cub(G)?2(Δ^′+2)⌈lnn₂⌉. We also give an efficient randomized algorithm to construct the cube representation of G in 3(Δ^′+2)⌈lnn₂⌉ dimensions. The reader may note that in general Δ^′ can be much smaller than Δ.     

Validation of SeaWiFS chlorophyll a concentrations in the Southern Ocean: A revisit

Surface chlorophyll a concentrations (C_a, mg m^− 3) in the Southern Ocean estimated from SeaWiFS satellite data have been reported in the literature to be significantly lower than those measured from in situ water samples using fluorometric methods. However, we found that high-resolution (∼ 1 km²/pixel) daily SeaWiFS C_a (C_a^SWF) data (SeaDAS4.8, OC4v4 algorithm) was an accurate measure of in situ C_a during January-February of 1998-2002 if concurrent in situ data measured by HPLC (C_a^HPLC) instead of fluorometric (C_a^Fluor) measurements were used as ground truth. Our analyses indicate that C_a^Fluor is 2.48 ± 2.23 (n = 647) times greater than C_a^HPLC between 0.05 and 1.5 mg m^− 3 and that the percentage overestimation of in situ C_a by fluorometric measurements increases with decreasing concentrations. The ratio of C_a^SWF/C_a^HPLC is 1.12 ± 0.91 (n = 96), whereas the ratio of C_a^SWF/C_a^Fluor is 0.55 ± 0.63 (n = 307). Furthermore, there is no significant bias in C_a^SWF (12% and − 0.07 in linear and log-transformed C_a, respectively) when C_a^HPLC is used as ground truth instead of C_a^Fluor. The high C_a^Fluor/C_a^HPLC ratio may be attributed to the relatively low concentrations of chlorophyll b (C_b/C_a = 0.023 ± 0.034, n = 482) and relatively high concentrations of chlorophyll c (C_c/C_a = 0.25 ± 0.59, n = 482) in the phytoplankton pigment composition when compared to values from other regions. Because more than 90% of the waters in the study area, as well as in the entire Southern Ocean (south of 60° S), have C_a^SWF between 0.05 and 1.5 mg m^− 3, we consider that the SeaWiFS performance of C_a retrieval is satisfactory and for this C_a range there is no need to further develop a “regional” bio-optical algorithm to account for the previous SeaWiFS “underestimation”.     

The interval-merging problem

A closed interval is an ordered pair of real numbers [x, y], with x ? y. The interval [x, y] represents the set {i ∈ R∣x ? i ? y}. Given a set of closed intervals I={[a₁,b₁],[a₂,b₂],…,[a_k,b_k]}, the Interval-Merging Problem is to find a minimum-cardinality set of intervals M(I)={[x₁,y₁],[x₂,y₂],…,[x_j,y_j]}, j ? k, such that the real numbers represented by equal those represented by . In this paper, we show the problem can be solved in O(d log d) sequential time, and in O(log d) parallel time using O(d) processors on an EREW PRAM, where d is the number of the endpoints of I. Moreover, if the input is given as a set of sorted endpoints, then the problem can be solved in O(d) sequential time, and in O(log d) parallel time using O(d/log d) processors on an EREW PRAM.     

A note on alternating one-pebble Turing machines with sublogarithmic space

Inoue et al. [A. Inoue, A. Ito, K. Inoue, T. Okazaki, Some properties of one-pebble Turing machines with sublogarithmic space, Theoret. Comput. Sci. 341 (2005) 138-149] conjectured that the language {ww|w∈⁺{a,b}} is not accepted by any alternating one-pebble Turing machine with sublogarithmic space. In this paper we disprove the conjecture by showing that there exists an alternating one-pebble Turing machine accepting the language in loglogn space.     

Impact of sea ice on the retrieval of water-leaving reflectance, chlorophyll a concentration and inherent optical properties from satellite ocean color data

Two physical phenomena by which satellite remotely sensed ocean color data are contaminated by sea ice at high latitudes are described through simulations and observations: (1) the adjacency effect that occurs along sea ice margins and (2) the sub-pixel contamination by a small amount of sea ice within an ocean pixel. The signal at the top of the atmosphere (TOA) was simulated using the 6S radiative transfer code that allows modeling of the adjacency effect for various types of sea ice surrounding an open water area. In situ sea ice reflectance spectra used in the simulations were measured prior to and during the melt period as part of the 2004 Canadian Arctic Shelf Exchange Study (CASES). For sub-pixel contamination, the TOA signal was simulated for various surface reflectances obtained by linear mixture of both sea ice and water-leaving reflectances (ρ_w). The standard atmospheric correction algorithm was then applied to the simulated TOA spectra to retrieve ρ_w spectra from which chlorophyll a concentrations (CHL) and inherent optical properties (IOPs) were derived. The adjacency effect was associated with large errors (> 0.002) in the retrieval of ρ_w as far as 24 km from an ice edge in the blue part of the spectrum (443 nm). Therefore, for moderate to high CHL (> 0.5 mg m^− 3), any pixel located within a distance of ∼ 10-20 km from the ice edge were unreliable. It was also found necessary to consider the adjacency effect when the total absorption coefficient (a_t) was to be retrieved using a semi-analytical algorithm. a_t(443) was underestimated by more than 35% at a distance of 20 km from an ice edge for CHL > 0.5 mg m^− 3. The effect on the retrieval of the particle backscattering coefficient (b_bp) was important only for clear waters (CHL ∼ 0.05 mg m^− 3). In contrast, sub-pixel contamination by a small amount of sea ice produced systematic underestimation of ρ_w in the blue because of incorrect interpretation of enhanced reflectance in the near infrared that is attributed to higher concentrations of atmospheric aerosols. In general, sub-pixel contamination was found to result in overestimations of CHL and a_t, and underestimations of b_bp. A simple method was proposed to flag pixels contaminated by adjacency effect.     

On ∗-λ-semirings

A ∗-λ-semiring is an ordered semiring S equipped with a star operation such that for any a, b ∈ S, a^∗b is the least fixed point of the linear mapping x ? ax + b over S. The notion of ∗-λ-semirings is a generalization of several important concepts such as continuous semirings, (weak) inductive ∗-semirings and the Kleene algebras of Conway and Kozen. We investigate several basic properties of ∗-λ-semirings and obtain results on ∗-λ-semirings in relation to preorders, duality and formal power series. Some of these results can be seen as generalizations of relevant results on inductive ∗-semirings.     

The spanning connectivity of folded hypercubes

A k-container of a graph G is a set of k internally disjoint paths between u and v. A k-container of G is a k∗-container if it contains all vertices of G. A graph G is k∗-connected if there exists a k∗-container between any two distinct vertices, and a bipartite graph G is k∗-laceable if there exists a k∗-container between any two vertices u and v from different partite sets of G for a given k. A k-connected graph (respectively, bipartite graph) G is f-edge fault-tolerant spanning connected (respectively, laceable) if G − F is w∗-connected for any w with 1 ? w ? k − f and for any set F of f faulty edges in G. This paper shows that the folded hypercube FQ_n is f-edge fault-tolerant spanning laceable if n(?3) is odd and f ? n − 1, and f-edge fault-tolerant spanning connected if n (?2) is even and f ? n − 2.     

Edge-pancyclicity and path-embeddability of bijective connection graphs

An n-dimensional Bijective Connection graph (in brief BC graph) is a regular graph with 2ⁿ nodes and n2ⁿ⁻¹ edges. The n-dimensional hypercube, crossed cube, Möbius cube, etc. are some examples of the n-dimensional BC graphs. In this paper, we propose a general method to study the edge-pancyclicity and path-embeddability of the BC graphs. First, we prove that a path of length l with dist(X_n, x, y) + 2 ? l ? 2ⁿ − 1 can be embedded between x and y with dilation 1 in X_n for x, y ∈ V(X_n) with x ≠ y in X_n, where X_n (n ? 4) is a n-dimensional BC graph satisfying the three specific conditions and V(X_n) is the node set of X_n. Furthermore, by this result, we can claim that X_n is edge-pancyclic. Lastly, we show that these results can be applied to not only crossed cubes and Möbius cubes, but also other BC graphs except crossed cubes and Möbius cubes. So far, the research on edge-pancyclicity and path-embeddability has been limited in some specific interconnection architectures such as crossed cubes, Möbius cubes.     

The twin domination number in generalized de Bruijn digraphs

Let G=(V,A) be a digraph. A set T of vertices of G is a twin dominating set of G if for every vertex v∈V?T, there exist u,w∈T (possibly u=w) such that arcs (u,v),(v,w)∈A. The twin domination numberγ^∗(G) of G is the cardinality of a minimum twin dominating set of G. In this paper we investigate the twin domination number in generalized de Bruijn digraphs GB(n,d). For the digraphs GB(n,d), we first establish sharp bounds on the twin domination number. Secondly, we give the exact values of the twin domination number for several types of GB(n,d) by constructing minimum twin dominating sets in the digraphs. Finally, we present sharp upper bounds for some special generalized de Bruijn digraphs.     

On conditional diagnosability of the folded hypercubes

The n-dimensional folded hypercube FQ_n, a variation of the hypercube proposed by Ahmed et al. [A. El-Amawy, S. Latifi, Properties and performance of folded hypercubes, IEEE Transactions on Parallel and Distributed Systems 2(3) (1991) 31-42], is an (n + 1)-regular (n + 1)-connected graph. Conditional diagnosability, a new measure of diagnosability introduced by Lai et al. [Pao-Lien Lai, Jimmy J.M. Tan, Chien-Ping Chuang, Lih-Hsing Hsu, Conditional diagnosability measures for large multiprocessor systems, IEEE Transactions on Computers 54(2) (2005) 165-175] can better measure the diagnosability of regular interconnection networks. This paper determines that under PMC-model the conditional diagnosability of FQ_n (t_c(FQ_n)) is 4n − 3 when n = 5 or n ? 8; t_c(FQ₃) = 3, t_c(FQ₄) = 7.     

Periodic input estimation for linear periodic systems: Automotive engine applications

In this paper, we consider periodic linear systems driven by T₀-periodic signals that we desire to reconstruct. The systems under consideration are of the form , y=C(t)x, x∈Rⁿ, w∈R^m, y∈R^p, (m?p?n) where A(t), A₀(t), and C(t) are T₀-periodic matrices. The period T₀ is known. The T₀-periodic input signal w(t) is unknown but is assumed to admit a finite dimensional Fourier decomposition. Our contribution is a technique to estimate w from the measurements y. In both full state measurement and partial state measurement cases, we propose an efficient observer for the coefficients of the Fourier decomposition of w(t). The proposed techniques are particularly attractive for automotive engine applications where sampling time is short. In this situation, standard estimation techniques based on Kalman filters are often discarded (because of their relative high computational burden). Relevance of our approach is supported by two practical cases of application. Detailed convergence analysis is also provided. Under standard observability conditions, we prove asymptotic convergence when the tuning parameters are chosen sufficiently small.     

Hamiltonian laceability of bubble-sort graphs with edge faults

It is known that the n-dimensional bubble-sort graph B_n is bipartite, (n − 1)-regular, and has n! vertices. We first show that, for any vertex v, B_n − v has a hamiltonian path between any two vertices in the same partite set without v. Let F be a subset of edges of B_n. We next show that B_n − F has a hamiltonian path between any two vertices of different partite sets if ∣F∣ is at most n − 3. Then we also prove that B_n − F has a path of length n! − 2 between any pair of vertices in the same partite set.     

A fast algorithm for computing a longest common increasing subsequence

Let A=〈a₁,a₂,…,am〉 and B=〈b₁,b₂,…,bn〉 be two sequences, where each pair of elements in the sequences is comparable. A common increasing subsequence of A and B is a subsequence 〈ai₁=bj₁,ai₂=bj₂,…,ail=bjl〉, where i₁<i₂<?<il and j₁<j₂<?<jl, such that for all 1?k<l, we have aik<aik+1. A longest common increasing subsequence of A and B is a common increasing subsequence of the maximum length. This paper presents an algorithm for delivering a longest common increasing subsequence in O(mn) time and O(mn) space.     

Bernstein-type operators which preserve polynomials

In this paper we present the sequence of linear Bernstein-type operators defined for f∈C[0,1] by B_n(f°τ⁻¹)°τ, B_n being the classical Bernstein operators and τ being any function that is continuously differentiable ∞ times on [0,1], such that τ(0)=0, τ(1)=1 and τ^′(x)>0 for x∈[0,1]. We investigate its shape preserving and convergence properties, as well as its asymptotic behavior and saturation. Moreover, these operators and others of King type are compared with each other and with B_n. We present as an interesting byproduct sequences of positive linear operators of polynomial type with nice geometric shape preserving properties, which converge to the identity, which in a certain sense improve B_n in approximating a number of increasing functions, and which, apart from the constant functions, fix suitable polynomials of a prescribed degree. The notion of convexity with respect to τ plays an important role.     

Windows scheduling of arbitrary-length jobs on multiple machines

The generalized windows scheduling problem for n jobs on multiple machines is defined as follows: Given is a sequence, I=〈(w ₁,ℓ ₁),(w ₂,ℓ ₂),…,(w _n ,ℓ _n )〉 of n pairs of positive integers that are associated with the jobs 1,2,…,n, respectively. The processing length of job i is ℓ _i slots where a slot is the processing time of one unit of length. The goal is to repeatedly and non-preemptively schedule all the jobs on the fewest possible machines such that the gap (window) between two consecutive beginnings of executions of job i is at most w _i slots. This problem arises in push broadcast systems in which data are transmitted on multiple channels. The problem is NP-hard even for unit-length jobs and a (1+ε)-approximation algorithm is known for this case by approximating the natural lower bound W(I)=?_i=1ⁿ(1/w_i)W(I)=\sum_{i=1}^{n}(1/w_{i}). The techniques used for approximating unit-length jobs cannot be extended for arbitrary-length jobs mainly because the optimal number of machines might be arbitrarily larger than the generalized lower bound W(I)=?_i=1ⁿ(l_i/w_i)W(I)=\sum_{i=1}^{n}(\ell_{i}/w_{i}). The main result of this paper is an 8-approximation algorithm for the WS problem with arbitrary lengths using new methods, different from those used for the unit-length case. The paper also presents another algorithm that uses 2(1+ε)W(I)+logw _max machines and a greedy algorithm that is based on a new tree representation of schedules. The greedy algorithm is optimal for some special cases, and computational experiments show that it performs very well in practice.     

The edge-orientation problem and some of its variants on weighted graphs

Let G(V, E) be a connected undirected graph with n vertices and m edges, where each vertex v is associated with a cost C(v) and each edge e = (u, v) is associated with two weights, W(u → v) and W(v → u). The issue of assigning an orientation to each edge so that G becomes a directed graph is resolved in this paper. Determining a scheme to assign orientations of all edges such that max_x∈V{C(x)+∑_x→zW(x→z)} is minimized is the objective. This issue is called the edge-orientation problem (the EOP). Two variants of the EOP, the Out-Degree-EOP and the Vertex-Weighted EOP, are first proposed and then efficient algorithms for solving them on general graphs are designed. Ascertaining that the EOP is NP-hard on bipartite graphs and chordal graphs is the second result. Finally, an O(n log n)-time algorithm for the EOP on trees is designed. In general, the algorithmic results in this paper facilitate the implementation of the weighted fair queuing (WFQ) on real networks. The objective of the WFQ is to assign an effective weight for each flow to enhance link utilization. Our findings consequently can be easily extended to other classes of graphs, such as cactus graphs, block graphs, and interval graphs.     

Fast algorithms for minimum matrix norm with application in computer graphics

In this paper we consider the following problem. Given (r ₁,r ₂, ...,r _n) R ⁿ, for anyI= (I ₁,I ₂,...,I _n) Z ⁿ, letE ₁=(e _ij), wheree _ij=(r _i–r_j)–(I _i–I_j), findI Z ⁿ such that |E _I| is minimized, where |·| is a matrix norm. This problem arises from optimal curve rasterization in computer graphics, where minimum distortion of curve dynamic context is sought. Until now, there has been no polynomial-time solution to this computer graphics problem. We present a very simpleO(n lgn)-time algorithm to solve this problem under various matrix norms.This research was supported by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0046373.     

Cyclic codes over R = Fp + uFp +?+ uFp with length pn

In this paper, all cyclic codes with length p^sn, (n prime to p) over the ring R = F_p + uF_p +?+ u^k−1F_p are classified. It is first proved that Tor_j(C) is an ideal of , so that the structure of ideals over extension ring S_u^k(m,ω)=GR(u^k,m)[ω]/〈ω^ps-1〉 is determined. Then, an isomorphism between R[X]/〈X^N − 1〉 and a direct sum ⊕_h∈IS_u^k(m_h,ω) can be obtained using discrete Fourier transform. The generator polynomial representation of the corresponding ideals over F_p + uF_p +?+ u^k−1F_p is calculated via the inverse isomorphism. Moreover, torsion codes, MS polynomial and inversion formula are described.