All-Pairs Shortest Paths with Real Weights in <Emphasis Type="Italic">O</Emphasis>(<Emphasis Type="Italic">n</Emphasis><Superscript>3</Superscript>/log <Emphasis Type="Italic">n</Emphasis>) Time

We describe an O(n ³/log n)-time algorithm for the all-pairs-shortest-paths problem for a real-weighted directed graph with n vertices. This slightly improves a series of previous, slightly subcubic algorithms by Fredman (SIAM J. Comput. 5:49–60, 1976), Takaoka (Inform. Process. Lett. 43:195–199, 1992), Dobosiewicz (Int. J. Comput. Math. 32:49–60, 1990), Han (Inform. Process. Lett. 91:245–250, 2004), Takaoka (Proc. 10th Int. Conf. Comput. Comb., Lect. Notes Comput. Sci., vol. 3106, pp. 278–289, Springer, 2004), and Zwick (Proc. 15th Int. Sympos. Algorithms and Computation, Lect. Notes Comput. Sci., vol. 3341, pp. 921–932, Springer, 2004). The new algorithm is surprisingly simple and different from previous ones. A preliminary version of this paper appeared in Proc. 9th Workshop Algorithms Data Struct. (WADS), Lect. Notes Comput. Sci., vol. 3608, pp. 318–324, Springer, 2005.     

The 1-Versus-2 Queries Problem Revisited

The 1-versus-2 queries problem, which has been extensively studied in computational complexity theory, asks in its generality whether every efficient algorithm that makes at most 2 queries to a Σ _k ^p -complete language L _k has an efficient simulation that makes at most 1 query to L _k . We obtain solutions to this problem for hypotheses weaker than previously considered. We prove that:

Planar Polyline Drawings via Graph Transformations

We study the problem of transforming plane triangulations into irreducible triangulations, which are plane graphs with a quadrangular exterior face, triangular interior faces and no separating triangles. Our linear time transformation reveals important relations between the minimum Schnyder’s realizers of plane triangulations (Bonichon et al., Proceedings of the 20th Annual Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Computer Science, vol. 2607, pp. 499–510, Springer, Berlin, 2003; Research Report RR-1279-02, LaBRI, University of Bordeaux, France; Brehm, Diploma thesis, FB Mathematik und Informatik, Freie Universität Berlin, 2000) and the transversal structures of irreducible triangulations (Fusy, Proceedings of 13th International Symposium on Graph Drawing, Lecture Notes in Computer Science, vol. 3843, pp. 177–188, Springer, Berlin, 2005; He, SIAM J. Comput. 22:1218–1226, 1993). The transformation morphs a 3-connected plane graph into an internally 4-connected plane graph. Therefore some of the graph algorithms designed specifically for 4-connected plane graphs can be applied to 3-connected plane graphs indirectly. As an example of such applications, we present a linear time algorithm that produces a planar polyline drawing for a plane graph with n vertices in a grid of size bounded by W×H, where $W\leq\lfloor\frac{2n-2}{3}\rfloorWe study the problem of transforming plane triangulations into irreducible triangulations, which are plane graphs with a quadrangular exterior face, triangular interior faces and no separating triangles. Our linear time transformation reveals important relations between the minimum Schnyder’s realizers of plane triangulations (Bonichon et al., Proceedings of the 20th Annual Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Computer Science, vol. 2607, pp. 499–510, Springer, Berlin, 2003; Research Report RR-1279-02, LaBRI, University of Bordeaux, France; Brehm, Diploma thesis, FB Mathematik und Informatik, Freie Universit?t Berlin, 2000) and the transversal structures of irreducible triangulations (Fusy, Proceedings of 13th International Symposium on Graph Drawing, Lecture Notes in Computer Science, vol. 3843, pp. 177–188, Springer, Berlin, 2005; He, SIAM J. Comput. 22:1218–1226, 1993). The transformation morphs a 3-connected plane graph into an internally 4-connected plane graph. Therefore some of the graph algorithms designed specifically for 4-connected plane graphs can be applied to 3-connected plane graphs indirectly. As an example of such applications, we present a linear time algorithm that produces a planar polyline drawing for a plane graph with n vertices in a grid of size bounded by W×H, where W ￡ ?\frac2n-23?W\leq\lfloor\frac{2n-2}{3}\rfloor , and W+H ￡ ?\frac4n-43?W+H\leq\lfloor \frac{4n-4}{3}\rfloor . It uses at most ?\frac2n-53?\lfloor\frac{2n-5}{3}\rfloor bends, and each edge uses at most one bend. Our algorithm is area optimal. Compared with the existing area optimal polyline drawing algorithm proposed in Bonichon et al. (Proceedings of the 28th International Workshop on Graph-Theoretic Concepts in Computer Science, Lecture Notes in Computer Science, vol. 2573, pp. 35–46, Springer, Berlin, 2002), our algorithm uses a smaller number of bends. Their bend bound is (n−2).     

Model Checking Weighted Integer Reset Timed Automata

Weighted timed automata (WTA), introduced in Alur et al. (Proceedings of HSCC’01, LNCS, vol. 2034, pp. 49–62, Springer, Berlin, 2001), Behrmann et al. (Proceedings of HSCC’01, LNCS, vol. 2034, pp. 147–161, Springer, Berlin, 2001) are an extension of Alur and Dill (Theor. Comput. Sci. 126(2):183–235, 1994) timed automata, a widely accepted formalism for the modelling and verification of real time systems. Weighted timed automata extend timed automata by allowing costs on the locations and edges. There has been a lot of interest Bouyer et al. (Inf. Process. Lett. 98(5):188–194, 2006), Bouyer et al. (Log. Methods Comput. Sci. 4(2):9, 2008), Brihaye et al. (Proceedings of FORMATS/FTRTFT’04, LNCS, vol. 3253, pp. 277–292, Springer, Berlin, 2004), Brihaye et al. (Inf. Comput. 204(3):408–433, 2006) in studying the model checking problem of weighted timed automata. The properties of interest are written using logic weighted CTL (WCTL), an extension of CTL with costs. It has been shown Bouyer et al. (Log. Methods Comput. Sci. 4(2):9, 2008) that the problem of model checking WTAs with a single clock using WCTL with no external cost variables is decidable, while 3 clocks render the problem undecidable Bouyer et al. (Inf. Process. Lett. 98(5):188–194, 2006). The question of 2 clocks is open. In this paper, we introduce a subclass of weighted timed automata called weighted integer reset timed automata (WIRTA) and study the model checking problem. We give a clock reduction technique for WIRTA. Given a WIRTA A\mathcal{A} with n≥1 clocks, we show that a single clock WIRTA A￠\mathcal{A}' preserving the paths and costs of A\mathcal{A} can be obtained. This gives us the decidability of model checking WIRTA with n≥1 clocks and m≥1 costs using WCTL with no external cost variables. We then show that for a restricted version of WCTL with external cost variables, the model checking problem is undecidable for WIRTA with 3 stopwatch costs and 1 clock. Finally, we show that model checking WTA with 2 clocks and 1 stopwatch cost against WCTL with no external cost variables is undecidable, thereby answering a question that has remained long open.     

The Baire Partial Quasi-Metric Space: A?Mathematical Tool for Asymptotic Complexity Analysis in Computer Science

In 1994, S.G. Matthews introduced the notion of partial metric space in order to obtain a suitable mathematical tool for program verification (Ann. N.Y. Acad. Sci. 728:183–197, 1994). He gave an application of this new structure to parallel computing by means of a partial metric version of the celebrated Banach fixed point theorem (Theor. Comput. Sci. 151:195–205, 1995). Later on, M.P. Schellekens introduced the theory of complexity (quasi-metric) spaces as a part of the development of a topological foundation for the asymptotic complexity analysis of programs and algorithms (Electron. Notes Theor. Comput. Sci. 1:211–232, 1995). The applicability of this theory to the asymptotic complexity analysis of Divide and Conquer algorithms was also illustrated by Schellekens. In particular, he gave a new proof, based on the use of the aforenamed Banach fixed point theorem, of the well-known fact that Mergesort algorithm has optimal asymptotic average running time of computing. In this paper, motivated by the utility of partial metrics in Computer Science, we discuss whether the Matthews fixed point theorem is a suitable tool to analyze the asymptotic complexity of algorithms in the spirit of Schellekens. Specifically, we show that a slight modification of the well-known Baire partial metric on the set of all words over an alphabet constitutes an appropriate tool to carry out the asymptotic complexity analysis of algorithms via fixed point methods without the need for assuming the convergence condition inherent to the definition of the complexity space in the Schellekens framework. Finally, in order to illustrate and to validate the developed theory we apply our results to analyze the asymptotic complexity of Quicksort, Mergesort and Largesort algorithms. Concretely we retrieve through our new approach the well-known facts that the running time of computing of Quicksort (worst case behaviour), Mergesort and Largesort (average case behaviour) are in the complexity classes O(n²)\mathcal{O}(n^{2}), O(nlog₂(n))\mathcal{O}(n\log_{2}(n)) and O(2(n-1)-log₂(n))\mathcal{O}(2(n-1)-\log_{2}(n)), respectively.     

Improved Approximation Algorithms for Maximum Resource Bin Packing and Lazy Bin Covering Problems

In this paper, we study two variants of the bin packing and covering problems called Maximum Resource Bin Packing (MRBP) and Lazy Bin Covering (LBC) problems, and present new approximation algorithms for them. For the offline MRBP problem, the previous best known approximation ratio is \frac65\frac{6}{5} (=1.2) achieved by the classical First-Fit-Increasing (FFI) algorithm (Boyar et al. in Theor. Comput. Sci. 362(1–3):127–139, 2006). In this paper, we give a new FFI-type algorithm with an approximation ratio of \frac8071\frac{80}{71} (≈1.12676). For the offline LBC problem, it has been shown in Lin et al. (COCOON, pp. 340–349, 2006) that the classical First-Fit-Decreasing (FFD) algorithm achieves an approximation ratio of \frac7160\frac{71}{60} (≈1.18333). In this paper, we present a new FFD-type algorithm with an approximation ratio of \frac1715\frac{17}{15} (≈1.13333). Our algorithms are based on a pattern-based technique and a number of other observations. They run in near linear time (i.e., O(nlog n)), and therefore are practical.     

On the performance of Dijkstra’s third self-stabilizing algorithm for mutual exclusion and related algorithms

In Dijkstra (Commun ACM 17(11):643–644, 1974) introduced the notion of self-stabilizing algorithms and presented three such algorithms for the problem of mutual exclusion on a ring of n processors. The third algorithm is the most interesting of these three but is rather non intuitive. In Dijkstra (Distrib Comput 1:5–6, 1986) a proof of its correctness was presented, but the question of determining its worst case complexity—that is, providing an upper bound on the number of moves of this algorithm until it stabilizes—remained open. In this paper we solve this question and prove an upper bound of 3\frac1318 n² + O(n){3\frac{13}{18} n^2 + O(n)} for the complexity of this algorithm. We also show a lower bound of 1\frac56 n² - O(n){1\frac{5}{6} n^2 - O(n)} for the worst case complexity. For computing the upper bound, we use two techniques: potential functions and amortized analysis. We also present a new-three state self-stabilizing algorithm for mutual exclusion and show a tight bound of \frac56 n² + O(n){\frac{5}{6} n^2 + O(n)} for the worst case complexity of this algorithm. In Beauquier and Debas (Proceedings of the second workshop on self-stabilizing systems, pp 17.1–17.13, 1995) presented a similar three-state algorithm, with an upper bound of 5\frac34n²+O(n){5\frac{3}{4}n^2+O(n)} and a lower bound of \frac18n²-O(n){\frac{1}{8}n^2-O(n)} for its stabilization time. For this algorithm we prove an upper bound of 1\frac12n² + O(n){1\frac{1}{2}n^2 + O(n)} and show a lower bound of n ²−O(n). As far as the worst case performance is considered, the algorithm in Beauquier and Debas (Proceedings of the second workshop on self-stabilizing systems, pp 17.1–17.13, 1995) is better than the one in Dijkstra (Commun ACM 17(11):643–644, 1974) and our algorithm is better than both.     

The sudden death of entanglement in constructed Yang–Baxter systems

Some Hamiltonians are constructed from the unitary \checkR_i,i+1(q, j){\check{R}_{i,i+1}(\theta, \varphi)}-matrices, where θ and j{\varphi} are time-independent parameters. We show that the entanglement sudden death (ESD) can happen in these closed Yang–Baxter systems. It is found that the ESD is not only sensitive to the initial condition, but also has a great connection with different Yang–Baxter systems. Especially, we find that the meaningful parameter j{\varphi} has a great influence on ESD.     

Generalized fuzzy interior ideals of semigroups

The concept of $(\overline{\in},\overline{\in} \vee \overline{q})The concept of ([`( ? )],[`( ? )] ú[`(q)])(\overline{\in},\overline{\in} \vee \overline{q})-fuzzy interior ideals of semigroups is introduced and some related properties are investigated. In particular, we describe the relationships among ordinary fuzzy interior ideals, (∈, ∈ ∨ q)-fuzzy interior ideals and ([`( ? )],[`( ? )] ú[`(q)])(\overline{\in},\overline{\in} \vee \overline{q})-fuzzy interior ideals of semigroups. Finally, we give some characterization of [F] _t by means of (∈, ∈ ∨ q)-fuzzy interior ideals.     

Comparing Nontriviality for E and EXP

A set A is nontrivial for the linear-exponential-time class E=DTIME(2 ^lin ) if for any k≥1 there is a set B _k ∈E such that B _k is (p-m-)reducible to A and B_k ? DTIME(2^k·n)B_{k} \not\in \mathrm{DTIME}(2^{k\cdot n}). I.e., intuitively, A is nontrivial for E if there are arbitrarily complex sets in E which can be reduced to A. Similarly, a set A is nontrivial for the polynomial-exponential-time class EXP=DTIME(2 ^poly ) if for any k≥1 there is a set [^(B)]_k ? EXP\hat{B}_{k} \in \mathrm {EXP} such that [^(B)]_k\hat{B}_{k} is reducible to A and [^(B)]_k ? DTIME(2^nk)\hat{B}_{k} \not\in \mathrm{DTIME}(2^{n^{k}}). We show that these notions are independent, namely, there are sets A ₁ and A ₂ in E such that A ₁ is nontrivial for E but trivial for EXP and A ₂ is nontrivial for EXP but trivial for E. In fact, the latter can be strengthened to show that there is a set A∈E which is weakly EXP-hard in the sense of Lutz (SIAM J. Comput. 24:1170–1189, 11) but E-trivial.     

The Deutsch–Jozsa problem: de-quantisation and entanglement

The Deutsch–Jozsa problem is one of the most basic ways to demonstrate the power of quantum computation. Consider a Boolean function f : {0, 1} ⁿ → {0, 1} and suppose we have a black-box to compute f. The Deutsch–Jozsa problem is to determine if f is constant (i.e. f(x) = const, "x ? {0,1}ⁿf(x) = \hbox {const, } \forall x \in \{0,1\}^n) or if f is balanced (i.e. f(x) = 0 for exactly half the possible input strings x ? {0,1}ⁿx \in \{0,1\}^n) using as few calls to the black-box computing f as is possible, assuming f is guaranteed to be constant or balanced. Classically it appears that this requires at least 2 ⁿ⁻¹ + 1 black-box calls in the worst case, but the well known quantum solution solves the problem with probability one in exactly one black-box call. It has been found that in some cases the algorithm can be de-quantised into an equivalent classical, deterministic solution. We explore the ability to extend this de-quantisation to further cases, and examine with more detail when de-quantisation is possible, both with respect to the Deutsch–Jozsa problem, as well as in more general cases.     

Communication Complexity Under Product and Nonproduct Distributions

We solve an open problem in communication complexity posed by Kushilevitz and Nisan (1997). Let R_∈(f) and $D^\mu_\in (f)$D^\mu_\in (f) denote the randomized and μ-distributional communication complexities of f, respectively (∈ a small constant). Yao’s well-known minimax principle states that $R_{\in}(f) = max_\mu \{D^\mu_\in(f)\}$R_{\in}(f) = max_\mu \{D^\mu_\in(f)\}. Kushilevitz and Nisan (1997) ask whether this equality is approximately preserved if the maximum is taken over product distributions only, rather than all distributions μ. We give a strong negative answer to this question. Specifically, we prove the existence of a function f : {0, 1}ⁿ ×{0, 1}ⁿ ? {0, 1}f : \{0, 1\}^n \times \{0, 1\}^n \rightarrow \{0, 1\} for which max_{μ product} {D^m _? (f)} = Q(1)  but R _? (f) = Q(n)\{D^\mu_\in (f)\} = \Theta(1) \,{\textrm but}\, R_{\in} (f) = \Theta(n). We also obtain an exponential separation between the statistical query dimension and signrank, solving a problem previously posed by the author (2007).     

Entanglement and Berry phase in a 9 × 9 Yang–Baxter system

A M-matrix which satisfies the Hecke algebraic relations is presented. Via the Yang–Baxterization approach, we obtain a unitary solution \breveR(q,j₁,j₂){\breve{R}(\theta,\varphi_{1},\varphi_{2})} of Yang–Baxter equation. It is shown that any pure two-qutrit entangled states can be generated via the universal \breveR{\breve{R}}-matrix assisted by local unitary transformations. A Hamiltonian is constructed from the \breveR{\breve{R}}-matrix, and Berry phase of the Yang–Baxter system is investigated. Specifically, for j₁ = j₂{\varphi_{1}\,{=}\,\varphi_{2}}, the Hamiltonian can be represented based on three sets of SU(2) operators, and three oscillator Hamiltonians can be obtained. Under this framework, the Berry phase can be interpreted.     

Approximation Algorithms for Reconstructing the Duplication History of Tandem Repeats

Computing the duplication history of a tandem repeated region is an important problem in computational biology (Fitch in Genetics 86:623–644, 1977; Jaitly et al. in J. Comput. Syst. Sci. 65:494–507, 2002; Tang et al. in J. Comput. Biol. 9:429–446, 2002). In this paper, we design a polynomial-time approximation scheme (PTAS) for the case where the size of the duplication block is 1. Our PTAS is faster than the previously best PTAS in Jaitly et al. (J. Comput. Syst. Sci. 65:494–507, 2002). For example, to achieve a ratio of 1.5, our PTAS takes O(n ⁵) time while the PTAS in Jaitly et al. (J. Comput. Syst. Sci. 65:494–507, 2002) takes O(n ¹¹) time. We also design a ratio-6 polynomial-time approximation algorithm for the case where the size of each duplication block is at most 2. This is the first polynomial-time approximation algorithm with a guaranteed ratio for this case. Part of work was done during a Z.-Z. Chen visit at City University of Hong Kong.     

Distributed discovery of large near-cliques

Given an undirected graph and 0 ￡ e ￡ 1{0\le\epsilon\le1}, a set of nodes is called an e{\epsilon}-near clique if all but an e{\epsilon} fraction of the pairs of nodes in the set have a link between them. In this paper we present a fast synchronous network algorithm that uses small messages and finds a near-clique. Specifically, we present a constant-time algorithm that finds, with constant probability of success, a linear size e{\epsilon}-near clique if there exists an e³{\epsilon^3}-near clique of linear size in the graph. The algorithm uses messages of O(log n) bits. The failure probability can be reduced to n ^−Ω(1) by increasing the time complexity by a logarithmic factor, and the algorithm also works if the graph contains a clique of size Ω(n/(log log n) ^α ) for some a ? (0,1){\alpha \in (0,1)}. Our approach is based on a new idea of adapting property testing algorithms to the distributed setting.     

Quantum search in a possible three-dimensional complex subspace

Suppose we are given an unsorted database with N items and N is sufficiently large. By using a simpler approximate method, we re-derive the approximate formula cos² Φ, which represents the maximum success probability of Grover’s algorithm corresponding to the case of identical rotation angles f = q{\phi=\theta} for any fixed deflection angle F ? [0,p/2){\Phi \in\left[0,\pi/2\right)}. We further show that for any fixed F ? [0,p/2){\Phi \in\left[0,\pi/2\right)}, the case of identical rotation angles f = q{\phi=\theta} is energetically favorable compared to the case |q- f| >> 0{\left|{\theta - \phi}\right|\gg 0} for enhancing the probability of measuring a unique desired state.     

Complexity of Hard-Core Set Proofs

We study a fundamental result of Impagliazzo (FOCS’95) known as the hard-core set lemma. Consider any function f:{0,1}ⁿ?{0,1}{f:\{0,1\}^n\to\{0,1\}} which is “mildly hard”, in the sense that any circuit of size s must disagree with f on at least a δ fraction of inputs. Then, the hard-core set lemma says that f must have a hard-core set H of density δ on which it is “extremely hard”, in the sense that any circuit of size s￠=O(s/(\frac1e²log(\frac1ed))){s'=O(s/(\frac{1}{\epsilon^2}\log(\frac{1}{\epsilon\delta})))} must disagree with f on at least (1-e)/2{(1-\epsilon)/2} fraction of inputs from H.     

New Approximation Bounds for Lpt Scheduling

We provide new bounds for the worst case approximation ratio of the classic Longest Processing Time (Lpt) heuristic for related machine scheduling (Q||C _max?). For different machine speeds, Lpt was first considered by Gonzalez et al. (SIAM J. Comput. 6(1):155–166, 1977). The best previously known bounds originate from more than 20 years back: Dobson (SIAM J. Comput. 13(4):705–716, 1984), and independently Friesen (SIAM J. Comput. 16(3):554–560, 1987) showed that the worst case ratio of Lpt is in the interval (1.512,1.583), and in (1.52,1.67), respectively. We tighten the upper bound to $1+\sqrt{3}/3\approx1.5773We provide new bounds for the worst case approximation ratio of the classic Longest Processing Time (Lpt) heuristic for related machine scheduling (Q||C _max). For different machine speeds, Lpt was first considered by Gonzalez et al. (SIAM J. Comput. 6(1):155–166, 1977). The best previously known bounds originate from more than 20 years back: Dobson (SIAM J. Comput. 13(4):705–716, 1984), and independently Friesen (SIAM J. Comput. 16(3):554–560, 1987) showed that the worst case ratio of Lpt is in the interval (1.512,1.583), and in (1.52,1.67), respectively. We tighten the upper bound to 1+?3/3 ? 1.57731+\sqrt{3}/3\approx1.5773 , and the lower bound to 1.54. Although this improvement might seem minor, we consider the structure of potential lower bound instances more systematically than former works. We present a scheme for a job-exchanging process, which, repeated any number of times, gradually increases the lower bound. For the new upper bound, this systematic method together with a new idea of introducing fractional jobs, facilitated a proof that is surprisingly simple, relative to the result. We present the upper-bound proof in parameterized terms, which leaves room for further improvements.     

Lower Bounds for Agnostic Learning via Approximate Rank

We prove that the concept class of disjunctions cannot be pointwise approximated by linear combinations of any small set of arbitrary real-valued functions. That is, suppose that there exist functions f₁, ?, f_r\phi_{1}, \ldots , \phi_{r} : {− 1, 1}ⁿ → \mathbbR{\mathbb{R}} with the property that every disjunction f on n variables has $\|f - \sum\nolimits_{i=1}^{r} \alpha_{i}\phi _{i}\|_{\infty}\leq 1/3$\|f - \sum\nolimits_{i=1}^{r} \alpha_{i}\phi _{i}\|_{\infty}\leq 1/3 for some reals a₁, ?, a_r\alpha_{1}, \ldots , \alpha_{r}. We prove that then $r \geq exp \{\Omega(\sqrt{n})\}$r \geq exp \{\Omega(\sqrt{n})\}, which is tight. We prove an incomparable lower bound for the concept class of decision lists. For the concept class of majority functions, we obtain a lower bound of W(2ⁿ/n)\Omega(2^{n}/n) , which almost meets the trivial upper bound of 2ⁿ for any concept class. These lower bounds substantially strengthen and generalize the polynomial approximation lower bounds of Paturi (1992) and show that the regression-based agnostic learning algorithm of Kalai et al. (2005) is optimal.     

SZK Proofs for Black-Box Group Problems

In this paper we classify several algorithmic problems in group theory in the complexity classes PZK and SZK (problems with perfect/statistical zero-knowledge proofs respectively). Prior to this, these problems were known to be in . As , we have a tighter upper bound for these problems. Specifically: