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.     

A Tight Lower Bound for Computing the Diameter of a 3D Convex Polytope

The diameter of a set P of n points in ℝ ^d is the maximum Euclidean distance between any two points in P. If P is the vertex set of a 3-dimensional convex polytope, and if the combinatorial structure of this polytope is given, we prove that, in the worst case, deciding whether the diameter of P is smaller than 1 requires Ω(nlog n) time in the algebraic computation tree model. It shows that the O(nlog n) time algorithm of Ramos for computing the diameter of a point set in ℝ³ is optimal for computing the diameter of a 3-polytope. We also give a linear time reduction from Hopcroft’s problem of finding an incidence between points and lines in ℝ² to the diameter problem for a point set in ℝ⁷.     

Inferring (Biological) Signal Transduction Networks via Transitive Reductions of Directed Graphs

In this paper we consider the p-ary transitive reduction (TR _p ) problem where p>0 is an integer; for p=2 this problem arises in inferring a sparsest possible (biological) signal transduction network consistent with a set of experimental observations with a goal to minimize false positive inferences even if risking false negatives. Special cases of TR _p have been investigated before in different contexts; the best previous results are as follows:

Mixed Covering of Trees and the Augmentation Problem with Odd Diameter Constraints

In this paper we present a polynomial time algorithm for solving the problem of partial covering of trees with n₁ balls of radius R₁ and n₂ balls of radius R₂ (R₁ < R₂) to maximize the total number of covered vertices. The solutions provided by this algorithm in the particular case R₁ = R – 1, R₂ = R can be used to obtain for any integer δ > 0 a factor (2+1/δ) approximation algorithm for solving the following augmentation problem with odd diameter constraints D = 2R + 1: Given a tree T, add a minimum number of new edges such that the augmented graph has diameter ≤ D. The previous approximation algorithm of Ishii, Yamamoto, and Nagamochi (2003) has factor 8.     

Shaping the drivers’ interaction: how the new vehicle systems match the technological requirements and the human needs

Although it is not always possible to derive from statistics the causes of an accident, data clearly show that they are mostly due to a lack of driver’s perception, with high direct and indirect social costs for society (Hutchinson in Road accidents statistics, Rumsby Scientific Publishing, 1987; Kawai in Convergence 1994, Conference Proceedings, 1994; Boussuge in Bilan et perspectives, Revue Gen. des Routes et des Aerodromes, N. 726, 1995). Therefore, a strong integration among all the actors involved (i.e., the drivers, the vehicle or technical system in general and the driving environment) is quite necessary. From the beginning of 1980s, there has been a shift in system concept design, moving from a technological approach towards a human-centred design approach (Norman and Draper 1986; Rouse 1991). Firstly this approach was applied to the human–computer interaction domain, and has also been extended towards complex and automated technological systems (Sheridan in Telerobotics, automation, and human supervisory control, MIT Press, 1992; Scerbo in Human performance in automated systems: theory and applications, Lawrence Erlbaum Associates, 1996) in several domains as, among the others, automotive (Michon in Generic intelligent driver support, Taylor & Francis, 1994) and industrial plants (Cacciabue in Giuseppe mantovani ergonomia, II Mulino, 2000). According to these perspectives, the role of user’s needs in the design process of a certain system becomes crucial; nevertheless, the technological requirements, i.e., the so-called “machine needs” should maintain a role of which the designer should be totally aware. Otherwise, the risk is to fail the design process making a system incoherent and ineffective. Thus, this paper aims at presenting such types of needs, the mutual interaction between machine and users, as well as the interaction of both with the surrounding environment. The work is based on several experiences conducted in the automotive domain.

Improved on-line broadcast scheduling with deadlines

We study an on-line broadcast scheduling problem in which requests have deadlines, and the objective is to maximize the weighted throughput, i.e., the weighted total length of the satisfied requests. For the case where all requested pages have the same length, we present an online deterministic algorithm named BAR and prove that it is 4.56-competitive. This improves the previous algorithm of (Kim, J.-H., Chwa, K.-Y. in Theor. Comput. Sci. 325(3):479–488, 2004) which is shown to be 5-competitive by (Chan, W.-T., et al. in Lecture Notes in Computer Science, vol. 3106, pp. 210–218, 2004). In the case that pages may have different lengths, we give a ( )-competitive algorithm where Δ is the ratio of maximum to minimum page lengths. This improves the (4Δ+3)-competitive algorithm of (Chan, W.-T., et al. in Lecture Notes in Computer Science, vol. 3106, pp. 210–218, 2004). We also prove an almost matching lower bound of Ω(Δ/log Δ). Furthermore, for small values of Δ we give better lower bounds. The work described in this paper was fully supported by grants from the Research Grants Council of the Hong Kong SAR, China [CityU 1198/03E, HKU 7142/03E, HKU 5172/03E], an NSF Grant of China [No. 10371094], and a Nuffield Foundation Grant of UK [NAL/01004/G].     

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.     

Analysis of an adaptive algorithm to find the two nearest neighbors

Given a setS ofN distinct elements in random order and a pivotx ∈S, we study the problem of simultaneously finding the left and the right neighbors ofx, i.e.,L=max{u|u<x} andR=min{v|v>x}. We analyze an adaptive algorithm that solves this problem by scanning the setS while maintaining current values for the neighborsL andR. Each new element inspected is compared first against the neighbor in the most populous side, then (if necessary) against the neighbor in the other side, and finally (if necessary), against the pivot. This algorithm may require 3N comparisons in the worst case, but it performs well on the average. If the pivot has rankαN, where α is fixed and <1/2, the algorithm does (1+α)N+Θ(logN) comparisons on the average, with a variance of 3 lnN+Θ(1). However, in the case where the pivot is the median, the average becomes 3/2;N+Θ(√N), while the variance grows to (1/2−π/8)N+Θ(logN). We also prove that, in the αN case, the limit distribution is Gaussian. This work has been supported in part by Grant FONDECYT(Chile) 1950622 and 1981029. Online publication October 6, 2000.     

The individual ergodic theorem on the IF-events with product

The ergodic theory and particularly the individual ergodic theorem were studied in many structures. Recently the individual ergodic theorem has been proved for MV-algebras of fuzzy sets (Riečan in Czech Math J 50(125):673–680, 2000; Riečan and Neubrunn in Integral, measure, and ordering. Kluwer, Dordrecht, 1997) and even in general MV-algebras (Jurečková in Int J Theor Phys 39:757–764, 2000). The notion of almost everywhere equality of observables was introduced by Riečan and Jurečková (Int J Theor Phys 44:1587–1597, 2005). They proved that the limit of Cesaro means is an invariant observable for P-observables. In Lendelová (Int J Theor Phys 45(5):915–923, 2006c) showed that the assumption of P-observable can be omitted. In this paper we prove the individual ergodic theorem on family of IF-events and show that each P {\mathcal{P}} -preserving transformation in this family can be expressed by two corresponding P^\flat,P^\sharp {\mathcal{P}}^\flat,{\mathcal{P}}^\sharp -preserving transformations in tribe T. {\mathcal{T}}.     

A generalisation of Jury’s conjecture to arbitrary dimensions and its proof

In 1986 E. I. Jury conjectured by analogy to the theory of digital filters that a two-dimensional analog filter is BIBO stable if its transfer function is of the form H = 1/P, where P is a very strict Hurwitz polynomial (VSHP). In this article we prove a generalisation of Jury’s conjecture to r-dimensional analog filters (r ≥ 2) with proper transfer function H = Q/P, where the denominator P is a robustly stable polynomial, i.e., a strict Hurwitz polynomial which retains this property under small variations of its coefficients. In the bivariate case these polynomials are the VSHPs. Financial support by the Austrian FWF via project 18974 is appreciated.     

Graph Spanners in the Streaming Model: An Experimental Study

This article reports the results of an extensive experimental analysis of efficient algorithms for computing graph spanners in the data streaming model, where an (α,β)-spanner of a graph G is a subgraph S⊆G such that for each pair of vertices the distance in S is at most α times the distance in G plus β. To the best of our knowledge, this is the first computational study of graph spanner algorithms in a streaming setting. We compare experimentally the randomized algorithms proposed by Baswana () and by Elkin (In: Proceedings of the 34th International Colloquium on Automata, Languages and Programming (ICALP 2007), Wroclaw, Poland, pp. 716–727, 9–13 July 2007) for general stretch factors with the deterministic algorithm presented by Ausiello et al. (In: Proceedings of the 15th Annual European Symposium on Algorithms (ESA 2007), Engineering and Applications Track, Eilat, Israel, 8–10 October 2007. LNCS, vol. 4698, pp. 605–617, 2007), designed for building small stretch spanners. All the algorithms we implemented work in a data streaming model where the input graph is given as a stream of edges in arbitrary order, and all of them need a single pass over the data. Differently from the algorithm in Ausiello et al., the algorithms in Baswana () and Elkin (In: Proceedings of the 34th International Colloquium on Automata, Languages and Programming (ICALP 2007), Wroclaw, Poland, pp. 716–727, 9–13 July 2007) need to know in advance the number of vertices in the graph. The results of our experimental investigation on several input families confirm that all these algorithms are very efficient in practice, finding spanners with stretch and size much smaller than the theoretical bounds and comparable to those obtainable by off-line algorithms. Moreover, our experimental findings confirm that small values of the stretch factor are the case of interest in practice, and that the algorithm by Ausiello et al. tends to produce spanners of better quality than the algorithms by Baswana and Elkin, while still using a comparable amount of time and space resources. Work partially supported by the Italian Ministry of University and Research under Project MAINSTREAM “Algorithms for Massive Information Structures and Data Streams”. A preliminary version of this paper was presented at the 15th Annual European Symposium on Algorithms (ESA 2007) 5.     

On bounded lattices satisfying Elkan’s law

In this short paper we shall prove that every bounded lattice L with the conditions: (c1) 1′ =  0 and (EL): (a · b′)′ =  b +  a′ · b′ for all a, b ∈ L is a Boolean algebra. This is a more general result than that of Renedo et al. (Proceedings NAFIPS’04, 2004), in which it is proved that every orthocomplemented lattice with (EL) is a Boolean algebra.     

Absolute approximation ratios for packing rectangles into bins

We consider the problem of packing rectangles into bins that are unit squares, where the goal is to minimize the number of bins used. All rectangles have to be packed non-overlapping and orthogonal, i.e., axis-parallel. We present an algorithm with an absolute worst-case ratio of 2 for the case where the rectangles can be rotated by 90 degrees. This is optimal provided P 1 NP\mathcal{P}\not=\mathcal{NP} . For the case where rotation is not allowed, we prove an approximation ratio of 3 for the algorithm Hybrid First Fit which was introduced by Chung et al. (SIAM J. Algebr. Discrete Methods 3(1):66–76, 1982) and whose running time is slightly better than the running time of Zhang’s previously known 3-approximation algorithm (Zhang in Oper. Res. Lett. 33(2):121–126, 2005).     

Neural Model Approach to the Basic Law of Psychophysics

Stevens’ law, which is one of the well-known psychophysical laws, suggests that the perceived intensity R of a biological system is proportional to the power of the stimulus strength I, R ∝ I ⁿ . In order to realize a self-sustainable system that adapts to changes of the environment, it is important to understand the neural mechanism behind this law. Here, we propose a new neural scheme based on the shunting short-term memory (STM) model with the physiological properties of the nervous system, and examine the relation between the neural system and Stevens’ law through computer simulations of the firing rate f with respect to the stimulus strength I. The simulations showed that the feedback-inputting connectivity plays an important role in reproducing the n > 1 and n < 1 cases of Stevens’ law.     

Combinatorial Algorithms for Data Migration to Minimize Average Completion Time

The data migration problem is to compute an efficient plan for moving data stored on devices in a network from one configuration to another. It is modeled by a transfer graph, where vertices represent the storage devices, and edges represent data transfers required between pairs of devices. Each vertex has a non-negative weight, and each edge has a processing time. A vertex completes when all the edges incident on it complete; the constraint is that two edges incident on the same vertex cannot be processed simultaneously. The objective is to minimize the sum of weighted completion times of all vertices. Kim (J. Algorithms 55, 42–57, 2005) gave an LP-rounding 3-approximation algorithm when edges have unit processing times. We give a more efficient primal-dual algorithm that achieves the same approximation guarantee. When edges have arbitrary processing times we give a primal-dual 5.83-approximation algorithm. We also study a variant of the open shop scheduling problem. This is a special case of the data migration problem in which the transfer graph is bipartite and the objective is to minimize the sum of completion times of edges. We present a simple algorithm that achieves an approximation ratio of , thus improving the 1.796-approximation given by Gandhi et al. (ACM Trans. Algorithms 2(1), 116–129, 2006). We show that the analysis of our algorithm is almost tight. A preliminary version of the paper appeared in the Proceedings of the 9th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2006. Research of R. Gandhi partially supported by Rutgers University Research Council Grant. Research of J. Mestre done at the University of Maryland; supported by NSF Awards CCR-0113192 and CCF-0430650, and the University of Maryland Dean’s Dissertation Fellowship.     

Models of Greedy Algorithms for Graph Problems

Borodin et al. (Algorithmica 37(4):295–326, 2003) gave a model of greedy-like algorithms for scheduling problems and Angelopoulos and Borodin (Algorithmica 40(4):271–291, 2004) extended their work to facility location and set cover problems. We generalize their model to include other optimization problems, and apply the generalized framework to graph problems. Our goal is to define an abstract model that captures the intrinsic power and limitations of greedy algorithms for various graph optimization problems, as Borodin et al. (Algorithmica 37(4):295–326, 2003) did for scheduling. We prove bounds on the approximation ratio achievable by such algorithms for basic graph problems such as shortest path, weighted vertex cover, Steiner tree, and independent set. For example, we show that, for the shortest path problem, no algorithm in the FIXED priority model can achieve any approximation ratio (even one dependent on the graph size), but the well-known Dijkstra’s algorithm is an optimal ADAPTIVE priority algorithm. We also prove that the approximation ratio for weighted vertex cover achievable by ADAPTIVE priority algorithms is exactly 2. Here, a new lower bound matches the known upper bounds (Johnson in J. Comput. Syst. Sci. 9(3):256–278, 1974). We give a number of other lower bounds for priority algorithms, as well as a new approximation algorithm for minimum Steiner tree problem with weights in the interval [1,2]. S. Davis’ research supported by NSF grants CCR-0098197, CCR-0313241, and CCR-0515332. Views expressed are not endorsed by the NSF. R. Impagliazzo’s research supported by NSF grant CCR-0098197, CCR-0313241, and CCR-0515332. Views expressed are not endorsed by the NSF. Some work done while at the Institute for Advanced Study, supported by the State of New Jersey.     

On Total Variation Minimization and Surface Evolution Using Parametric Maximum Flows

In a recent paper Boykov et al. (LNCS, Vol. 3953, pp. 409–422, 2006) propose an approach for computing curve and surface evolution using a variational approach and the geo-cuts method of Boykov and Kolmogorov (International conference on computer vision, pp. 26–33, 2003). We recall in this paper how this is related to well-known approaches for mean curvature motion, introduced by Almgren et al. (SIAM Journal on Control and Optimization 31(2):387–438, 1993) and Luckhaus and Sturzenhecker (Calculus of Variations and Partial Differential Equations 3(2):253–271, 1995), and show how the corresponding problems can be solved with sub-pixel accuracy using Parametric Maximum Flow techniques. This provides interesting algorithms for computing crystalline curvature motion, possibly with a forcing term. A. Chambolle’s research supported by ANR project “MICA”, grant ANR-08-BLAN-0082. J. Darbon’s research supported by ONR grant N000140710810.     

Improved multi-processor scheduling for?flow time and?energy

Energy usage has been an important concern in recent research on online scheduling. In this paper, we study the tradeoff between flow time and energy (Albers and Fujiwara in ACM Trans. Algorithms 3(4), 2007; Bansal et al. in Proceedings of ACM-SIAM Symposium on Discrete Algorithms, pp. 805–813, 2007b, Bansal et al. in Proceedings of International Colloquium on Automata, Languages and Programming, pp. 409–420, 2008; Lam et al. in Proceedings of European Symposium on Algorithms, pp. 647–659, 2008b) in the multi-processor setting. Our main result is an enhanced analysis of a simple non-migratory online algorithm called CRR (classified round robin) on m≥2 processors, showing that its flow time plus energy is within O(1) times of the optimal non-migratory offline algorithm, when the maximum allowable speed is slightly relaxed. The result still holds even if the comparison is made against the optimal migratory offline algorithm. This improves previous analysis that CRR is O(log P)-competitive where P is the ratio of the maximum job size to the minimum job size.     

A permuted factors approach for the linearization of polynomial matrices

In Antoniou and Vologiannidis (Electron J Linear Algebra 11:78–87, 2004; 15:107–114, 2006), a new family of companion forms associated with a regular polynomial matrix T (s) has been presented, using products of permutations of n elementary matrices, generalizing similar results presented in Fiedler (Linear Algebra Its Appl 371:325–331, 2003) where the scalar case was considered. In this paper, extending this “permuted factors” approach, we present a broader family of companion-like linearizations, using products of up to n(n−1)/2 elementary matrices, where n is the degree of the polynomial matrix. Under given conditions, the proposed linearizations can be shown to consist of block entries where the coefficients of the polynomial matrix appear intact. Additionally, we provide a criterion for those linearizations to be block symmetric. We also illustrate several new block symmetric linearizations of the original polynomial matrix T (s), where in some of them the constraint of nonsingularity of the constant term and the coefficient of maximum degree are not a prerequisite.