On sunflowers and matrix multiplication

We present several variants of the sunflower conjecture of Erd?s & Rado (J Lond Math Soc 35:85–90, 1960) and discuss the relations among them. We then show that two of these conjectures (if true) imply negative answers to the questions of Coppersmith & Winograd (J Symb Comput 9:251–280, 1990) and Cohn et al. (2005) regarding possible approaches for obtaining fast matrix-multiplication algorithms. Specifically, we show that the Erd?s–Rado sunflower conjecture (if true) implies a negative answer to the “no three disjoint equivoluminous subsets” question of Coppersmith & Winograd (J Symb Comput 9:251–280, 1990); we also formulate a “multicolored” sunflower conjecture in ${\mathbb{Z}_3^n}$ and show that (if true) it implies a negative answer to the “strong USP” conjecture of Cohn et al. (2005) (although it does not seem to impact a second conjecture in Cohn et al. (2005) or the viability of the general group-theoretic approach). A surprising consequence of our results is that the Coppersmith–Winograd conjecture actually implies the Cohn et al. conjecture. The multicolored sunflower conjecture in ${\mathbb{Z}_3^n}$ is a strengthening of the well-known (ordinary) sunflower conjecture in ${\mathbb{Z}_3^n}$ , and we show via our connection that a construction from Cohn et al. (2005) yields a lower bound of (2.51 . . .) ⁿ on the size of the largest multicolored 3-sunflower-free set, which beats the current best-known lower bound of (2.21 . . . ) ⁿ Edel (2004) on the size of the largest 3-sunflower-free set in ${\mathbb{Z}_3^n}$ .     

A Uniform Paradigm to Succinctly Encode Various Families of Trees

We propose a uniform method to encode various types of trees succinctly. These families include ordered (ordinal), k-ary (cardinal), and unordered (free) trees. We will show the approach is intrinsically suitable for obtaining entropy-based encodings of trees (such as the degree-distribution entropy). Previously-existing succinct encodings of trees use ad hoc techniques to encode each particular family of trees. Additionally, the succinct encodings obtained using the uniform approach improve upon the existing succinct encodings of each family of trees; in the case of ordered trees, it simplifies the encoding while supporting the full set of navigational operations. It also simplifies the implementation of many supported operations. The approach applied to k-ary trees yields a succinct encoding that supports both cardinal-type operations (e.g. determining the child label i) as well as the full set of ordinal-type operations (e.g. reporting the number of siblings to the left of a node). Previous work on succinct encodings of k-ary trees does not support both types of operations simultaneously (Benoit et al. in Algorithmica 43(4):275–292, 2005; Raman et al. in ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 233–242, 2002). For unordered trees, the approach achieves the first succinct encoding. The approach is based on two recursive decompositions of trees into subtrees. Recursive decomposition of a structure into substructures is a common technique in succinct encodings and has even been used to encode (ordered) trees (Geary et al. in ACM Trans. Algorithms 2(4):510–534, 2006; He et al. in ICALP, pp. 509–520, 2007) and dynamic binary trees (Munro et al. in ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 529–536, 2001; Storm in Representing dynamic binary trees succinctly, Master’s thesis, 2000). The main distinction of the approach in this paper is that a tree is decomposed into subtrees in a manner that the subtrees are maximally isolated from each other. This intermediate decomposition result is interesting in its own right and has proved useful in other applications (Farzan et al. in ICALP (1), pp. 451–462, 2009; Farzan and Munro in ICALP (1), pp. 439–450, 2009; Farzan and Kamali in ICALP, 2011).     

pth Moment Exponential Stability of Stochastic Recurrent Neural Networks with Markovian Switching

This paper investigates the problem of the pth moment exponential stability for a class of stochastic recurrent neural networks with Markovian jump parameters. With the help of Lyapunov function, stochastic analysis technique, generalized Halanay inequality and Hardy inequality, some novel sufficient conditions on the pth moment exponential stability of the considered system are derived. The results obtained in this paper are completely new and complement and improve some of the previously known results (Liao and Mao, Stoch Anal Appl, 14:165–185, 1996; Wan and Sun, Phys Lett A, 343:306–318, 2005; Hu et al., Chao Solitions Fractals, 27:1006–1010, 2006; Sun and Cao, Nonlinear Anal Real, 8:1171–1185, 2007; Huang et al., Inf Sci, 178:2194–2203, 2008; Wang et al., Phys Lett A, 356:346–352, 2006; Peng and Liu, Neural Comput Appl, 20:543–547, 2011). Moreover, a numerical example is also provided to demonstrate the effectiveness and applicability of the theoretical results.     

Functional un|unparsing

Danvy??s functional unparsing problem (Danvy in J. Funct. Program. 8(6), 621?C625, 1998) is to implement a type-safe ??printf?? function, which converts a sequence of heterogeneous arguments to a string according to a given format. The dual problem is to implement a type-safe ??scanf?? function, which extracts a sequence of heterogeneous arguments from a string by interpreting (Friedman and Wand in LFP, pp. 348?C355, 1984 and in Essentials of Programming Languages, MIT Press, 2008) the same format as an equally heterogeneous sequence of patterns that binds zero or more variables. We derive multiple solutions to both problems (Wand in J. ACM 27(1), 164?C180, 1980) from their formal specifications (Wand in Theor. Comput. Sci. 20(1), 3?C32, 1982). On one hand, our solutions show how the Hindley-Milner type system, unextended, permits accessing heterogeneous sequences with the static assurance of type safety. On the other hand, our solutions demonstrate the use of control operators (Felleisen et al. in Proceedings of the 1988 ACM Conference on Lisp and Functional Programming, pp. 52?C62, ACM Press, New York, 1988; Wand in POPL 85: Conference Record of the Annual ACM Symposium on Principles of Programming Languages, vol. 16, ACM Press, New York, 1985; Meyer and Wand in Logics of Programs, Lecture Notes in Computer Science, vol. 193, pp. 219?C224, Springer, Berlin, 1985) to communicate with formats as coroutines (Wand in Proceedings of the 1980 ACM Conference on Lisp and Functional Programming, vol. 12, pp. 285?C299, ACM Press, New York, 1980 and Haynes et al. in LFP, pp. 293?C298, 1984).     

Comments and further improvements on “pth moment stability of stochastic neural networks with Markov volatilities”

In a very recent paper, Peng and Liu (Neural Comput Appl 20:543–547, 2011) investigated the pth moment stability of the stochastic Grossberg–Hopfield neural networks with Markov volatilities by Mao et al. (Bernoulli 6:73–90, 2000, Theorem 4.1). We should point out that Mao et al. (Bernoulli 6:73–90, 2000, Theorem 4.1) investigated the pth moment exponentially stable for a class of stochastic dynamical systems with constant delay; however, this theorem cannot apply to the case of variable time delays. It is also worthy to emphasize that Peng and Liu (Neural Comput Appl 20:543–547, 2011) discussed by Mao et al. (Bernoulli 6:73–90, 2000, Theorem 4.1) the pth moment exponentially stable for the Grossberg–Hopfield neural networks with variable delays, and therefore, there are some gaps between Peng and Liu (Neural Comput Appl 20:543–547, 2011, Theorem 1) and Mao et al. (Bernoulli 6:73–90, 2000, Theorem 4.1). In this paper, we fill up this gap. Moreover, a numerical example is also provided to demonstrate the effectiveness and applicability of the theoretical results.     

A new spatial IP assignment method for IP-based wireless sensor networks

Wireless sensor networks (WSNs), one of the commercial wireless mesh networks (WMNs), are envisioned to provide an effective solution for sensor-based AmI (Ambient Intelligence) systems and applications. To enable the communications between AmI sensor networks and the most popular TCP/IP networks seamlessly, the best solution model is to run TCP/IP directly on WSNs (Mulligan et al. 2009; Hui and Culler 2008; Han and Mam 2007; Kim et al. 2007; Xiaohua et al. 2004; Dunkels et al. 2004; Dunkels et al. 2004; Dunkels 2001; Dunkels et al. 2004). In this case, an IP assignment method is required to assign each sensor node a unique IP address. SIPA (Dunkels et al. 2004) is the best known IP assignment method that uses spatial relations and locations of sensor nodes to assign their IP addresses. It has been applied in Contiki (Dunkels et al. 2004), a famous WSN operating system, to support the 6LowPAN protocol. In Chang et al. (2009), we proposed the SLIPA (Scan-Line IP Assignment) algorithm to improve the assignment success rate (ASR) obtained by SIPA. SLIPA can achieve a good ASR when sensor nodes are uniformly distributed. However, if sensor nodes are deployed by other distributions, the improvements would be limited. This paper proposes a new spatial IP assignment method, called SLIPA-Q (SLIPA with equal-quantity partition), to improve SLIPA. Experiments show that, by testing the proposed method 1,000 times with 1,000 randomly deployed sensor nodes, the average ASR obtained by SLIPA-Q is over two times of that obtained by SLIPA. Under the same 88% ASR, the average numbers of sensor nodes those can be successfully assigned by SLIPA-Q, SLIPA, and SIPA are 950, 850, and 135, respectively. Comparing to previous spatial IP assignment methods, SLIPA-Q can achieve dramatic improvements in ASR for assigning IP addresses to a large set of sensor nodes.     

An Efficient Algorithm for ℓ 0 Minimization in Wavelet Frame Based Image Restoration

Wavelet frame based models for image restoration have been extensively studied for the past decade (Chan et al. in SIAM J. Sci. Comput. 24(4):1408–1432, 2003; Cai et al. in Multiscale Model. Simul. 8(2):337–369, 2009; Elad et al. in Appl. Comput. Harmon. Anal. 19(3):340–358, 2005; Starck et al. in IEEE Trans. Image Process. 14(10):1570–1582, 2005; Shen in Proceedings of the international congress of mathematicians, vol. 4, pp. 2834–2863, 2010; Dong and Shen in IAS lecture notes series, Summer program on “The mathematics of image processing”, Park City Mathematics Institute, 2010). The success of wavelet frames in image restoration is mainly due to their capability of sparsely approximating piecewise smooth functions like images. Most of the wavelet frame based models designed in the past are based on the penalization of the ? ₁ norm of wavelet frame coefficients, which, under certain conditions, is the right choice, as supported by theories of compressed sensing (Candes et al. in Appl. Comput. Harmon. Anal., 2010; Candes et al. in IEEE Trans. Inf. Theory 52(2):489–509, 2006; Donoho in IEEE Trans. Inf. Theory 52:1289–1306, 2006). However, the assumptions of compressed sensing may not be satisfied in practice (e.g. for image deblurring and CT image reconstruction). Recently in Zhang et al. (UCLA CAM Report, vol. 11-32, 2011), the authors propose to penalize the ? ₀ “norm” of the wavelet frame coefficients instead, and they have demonstrated significant improvements of their method over some commonly used ? ₁ minimization models in terms of quality of the recovered images. In this paper, we propose a new algorithm, called the mean doubly augmented Lagrangian (MDAL) method, for ? ₀ minimizations based on the classical doubly augmented Lagrangian (DAL) method (Rockafellar in Math. Oper. Res. 97–116, 1976). Our numerical experiments show that the proposed MDAL method is not only more efficient than the method proposed by Zhang et al. (UCLA CAM Report, vol. 11-32, 2011), but can also generate recovered images with even higher quality. This study reassures the feasibility of using the ? ₀ “norm” for image restoration problems.     

Incentive-compatible interdomain routing

The routing of traffic between Internet domains, or Autonomous Systems (ASes), a task known as interdomain routing, is currently handled by the Border Gateway Protocol (BGP, Rekhter and Li in RFC 4271 of the Internet Engineering Task Force, 2006). Using BGP, ASes can apply semantically rich routing policies to choose interdomain routes in a distributed fashion. This expressiveness in routing-policy choice supports domains?? autonomy in network operations and in business decisions, but it comes at a price: The interaction of locally defined routing policies can lead to unexpected global anomalies, including route oscillations or overall protocol divergence (see, e.g., Varadhan et?al. in Comput Networks 32(1):1?C16, 2000). Networking researchers have addressed this problem by devising constraints on policies that guarantee BGP convergence without unduly limiting expressiveness and autonomy (see, e.g., Gao and Rexford in IEEE/ACM Trans Network 9(6):681?C692, 2001; Griffin et?al. in Proceedings of 9th ACM Conference on Applications, Technologies, Architectures, and Protocols for Computer Communication (SIGCOMM??03), pp. 61?C72. ACM Press, New York, 2003). In addition to taking this engineering or ??protocol- design?? approach, researchers have approached interdomain routing from an economic or ??mechanism-design?? point of view. It is known that lowest-cost-path (LCP) routing can be implemented in an incentive-compatible, BGP-compatible manner (Feigenbaum et?al. in Distribut. Comput 18(1):61?C72, 2005; Shneidman and Parkes in Proceedings of 23rd ACM Symposium on Principles of Distributed Computing (PODC??04), pp. 88?C97. ACM Press, New York, 2004) but that several other natural classes of policies cannot (Feigenbaum et?al. in Theor Comput Sci 378(2):175?C189, 2007; Feigenbaum et?al. in Distribut Comput 18(4):293?C305, 2006). In this paper, we present the first example of a class of interdomain-routing policies that is more general than LCP routing and for which BGP itself is both incentive-compatible and guaranteed to converge. We also present several steps toward a general theory of incentive-compatible, BGP-compatible interdomain routing.     

On anonymizing transactions with sensitive items

K-anonymity (Samarati and Sweeny 1998; Samarati, IEEE Trans Knowl Data Eng, 13(6):1010–1027, 2001; Sweeny, Int J Uncertain, Fuzziness Knowl-Based Syst, 10(5):557–570, 2002) and its variants, l-diversity (Machanavajjhala et al., ACM TKDD, 2007) and tcloseness (Li et al. 2007) among others are anonymization techniques for relational data and transaction data, which are used to protect privacy against re-identification attacks. A relational dataset D is k-anonymous if every record in D has at least k-1 other records with identical quasi-identifier attribute values. The combination of released data with external data will never allow the recipient to associate each released record with less than k individuals (Samarati, IEEE Trans Knowl Data Eng, 13(6):1010–1027, 2001). However, the current concept of k-anonymity on transaction data treats all items as quasi-identifiers. The anonymized data set has k identical transactions in groups and suffers from lower data utility (He and Naughton 2009; He et al. 2011; Liu and Wang 2010; Terrovitis et al., VLDB J, 20(1):83–106, 2011; Terrovitis et al. 2008). To improve the utility of anonymized transaction data, this work proposes a novel anonymity concept on transaction data that contain both quasi-identifier items (QID) and sensitive items (SI). A transaction that contains sensitive items must have at least k-1 other identical transactions (Ghinita et al. IEEE TKDE, 33(2):161–174, 2011; Xu et al. 2008). For a transaction that does not contain a sensitive item, no anonymization is required. A transaction dataset that satisfies this property is said to be sensitive k-anonymous. Three algorithms, Sensitive Transaction Neighbors (STN) Gray Sort Clustering (GSC) and Nearest Neighbors for K-anonymization (K-NN), are developed. These algorithms use adding/deleting QID items and only adding SI to achieve sensitive k-anonymity on transaction data. Additionally, a simple “privacy value” is proposed to evaluate the degree of privacy for different types of k-anonymity on transaction data. Extensive numerical simulations were carried out to demonstrate the characteristics of the proposed algorithms and also compared to other types of k-anonymity approaches. The results show that each technique possesses its own advantage under different criteria such as running time, operation, and information loss. The results obtained here can be used as a guideline of the selection of anonymization technique on different data sets and for different applications.     

Fast Phylogeny Reconstruction Through Learning of Ancestral Sequences

Given natural limitations on the length DNA sequences, designing phylogenetic reconstruction methods which are reliable under limited information is a crucial endeavor. There have been two approaches to this problem: reconstructing partial but reliable information about the tree (Mossel in IEEE Comput. Biol. Bioinform. 4:108–116, 2007; Daskalakis et al. in SIAM J. Discrete Math. 25:872–893, 2011; Daskalakis et al. in Proc. of RECOMB 2006, pp. 281–295, 2006; Gronau et al. in Proc. of the 19th Annual SODA 2008, pp. 379–388, 2008), and reaching “deeper” in the tree through reconstruction of ancestral sequences. In the latter category, Daskalakis et al. (Proc. of the 38th Annual STOC, pp. 159–168, 2006) settled an important conjecture of M. Steel (My favourite conjecture. Preprint, 2001), showing that, under the CFN model of evolution, all trees on n leaves with edge lengths bounded by the Ising model phase transition can be recovered with high probability from genomes of length O(logn) with a polynomial time algorithm. Their methods had a running time of O(n ¹⁰). Here we enhance our methods from Daskalakis et al. (Proc. of RECOMB 2006, pp. 281–295, 2006) with the learning of ancestral sequences and provide an algorithm for reconstructing a sub-forest of the tree which is reliable given available data, without requiring a-priori known bounds on the edge lengths of the tree. Our methods are based on an intuitive minimum spanning tree approach and run in O(n ³) time. For the case of full reconstruction of trees with edges under the phase transition, we maintain the same asymptotic sequence length requirements as in Daskalakis et al. (Proc. of the 38th Annual STOC, pp. 159–168, 2006), despite the considerably faster running time.     

Some types of filters in residuated lattices

In this paper, inspired by some types of $BL$ -algebra filters (deductive systems) introduced in Haveshki et al. (Soft Comput 10:657–664, 2006), Kondo and Dudek (Soft Comput 12:419–423, 2008) and Turunen (Arch Math Log 40:467–473, 2001), we defined residuated lattice versions of them and study them in connection with Van Gasse et al. (Inf Sci 180(16):3006–3020, 2010), Lianzhen and Kaitai (Inf Sci 177:5725–5738, 2007), Zhu and Xu (Inf Sci 180:3614–3632, 2010). Also we consider some relations between these filters and quotient residuated lattice that are constructed via these filters.     

Generalising Unit-Refutation Completeness and SLUR via Nested Input Resolution

The class ${\mathcal{SLUR}}$ (Single Lookahead Unit Resolution) was introduced in Schlipf et al. (Inf Process Lett 54:133–137, 1995) as an umbrella class for efficient (poly-time) SAT solving, with linear-time SAT decision, while the recognition problem was not considered. ?epek et al. (2012) and Balyo et al. (2012) extended this class in various ways to hierarchies covering all of CNF (all clause-sets). We introduce a hierarchy ${\mathcal{SLUR}}_k$ which we argue is the natural “limit” of such approaches. The second source for our investigations is the class ${\mathcal{UC}}$ of unit-refutation complete clause-sets, introduced in del Val (1994) as a target class for knowledge compilation. Via the theory of “hardness” of clause-sets as developed in Kullmann (1999), Kullmann (Ann Math Artif Intell 40(3–4):303–352, 2004) and Ansótegui et al. (2008) we obtain a natural generalisation ${\mathcal{UC}}_k$ , containing those clause-sets which are “unit-refutation complete of level k”, which is the same as having hardness at most k. Utilising the strong connections to (tree-)resolution complexity and (nested) input resolution, we develop basic methods for the determination of hardness (the level k in ${\mathcal{UC}}_k$ ). A fundamental insight now is that ${\mathcal{SLUR}}_k = {\mathcal{UC}}_k$ holds for all k. We can thus exploit both streams of intuitions and methods for the investigations of these hierarchies. As an application we can easily show that the hierarchies from ?epek et al. (2012) and Balyo et al. (2012) are strongly subsumed by ${\mathcal{SLUR}}_k$ . Finally we consider the problem of “irredundant” clause-sets in ${\mathcal{UC}}_k$ . For 2-CNF we show that strong minimisations are possible in polynomial time, while already for (very special) Horn clause-sets minimisation is NP-complete. We conclude with an extensive discussion of open problems and future directions. We envisage the concepts investigated here to be the starting point for a theory of good SAT translations, which brings together the good SAT-solving aspects from ${\mathcal{SLUR}}$ together with the knowledge-representation aspects from ${\mathcal{UC}}$ , and expands this combination via notions of “hardness”.     

Quantity-based buffer-constrained two-machine flowshop problem: active and passive prefetch models for multimedia applications

Conventional studies on buffer-constrained flowshop scheduling problems have considered applications with a limitation on the number of jobs that are allowed in the intermediate storage buffer before flowing to the next machine. The study in Lin et al. (Comput. Oper. Res. 36(4):1158?C1175, 2008a) considered a two-machine flowshop problem with ??processing time-dependent?? buffer constraints for multimedia applications. A???passive?? prefetch model (the PP-problem), in which the download process is suspended unless the buffer is sufficient for keeping an incoming media object, was applied in Lin et al. (Comput. Oper. Res. 36(4):1158?C1175, 2008a). This study further considers an ??active?? prefetch model (the AP-problem) that exploits the unoccupied buffer space by advancing the download of the incoming object by a computed maximal duration that possibly does not cause a buffer overflow. We obtain new complexity results for both problems. This study also proposes a new lower bound which improves the branch and bound algorithm presented in Lin et al. (Comput. Oper. Res. 36(4):1158?C1175, 2008a). For the PP-problem, compared to the lower bounds developed in Lin et al. (Comput. Oper. Res. 36(4):1158?C1175, 2008a), on average, the results of the simulation experiments show that the proposed new lower bound cuts about 38% of the nodes and 32% of the execution time for searching the optimal solutions.     

Comparison Between Reduced Basis and Stochastic Collocation Methods for Elliptic Problems

The stochastic collocation method (Babu?ka et al. in SIAM J Numer Anal 45(3):1005–1034, 2007; Nobile et al. in SIAM J Numer Anal 46(5):2411–2442, 2008a; SIAM J Numer Anal 46(5):2309–2345, 2008b; Xiu and Hesthaven in SIAM J Sci Comput 27(3):1118–1139, 2005) has recently been applied to stochastic problems that can be transformed into parametric systems. Meanwhile, the reduced basis method (Maday et al. in Comptes Rendus Mathematique 335(3):289–294, 2002; Patera and Rozza in Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations Version 1.0. Copyright MIT, http://augustine.mit.edu, 2007; Rozza et al. in Arch Comput Methods Eng 15(3):229–275, 2008), primarily developed for solving parametric systems, has been recently used to deal with stochastic problems (Boyaval et al. in Comput Methods Appl Mech Eng 198(41–44):3187–3206, 2009; Arch Comput Methods Eng 17:435–454, 2010). In this work, we aim at comparing the performance of the two methods when applied to the solution of linear stochastic elliptic problems. Two important comparison criteria are considered: (1), convergence results of the approximation error; (2), computational costs for both offline construction and online evaluation. Numerical experiments are performed for problems from low dimensions $O(1)$ to moderate dimensions $O(10)$ and to high dimensions $O(100)$ . The main result stemming from our comparison is that the reduced basis method converges better in theory and faster in practice than the stochastic collocation method for smooth problems, and is more suitable for large scale and high dimensional stochastic problems when considering computational costs.     

Fault Estimation for a Quad-Rotor MAV Using a Polynomial Observer

This work addresses the problem of fault detection and diagnosis (FDD) for a quad-rotor mini air vehicle (MAV). Actuator faults are considered on this paper. The basic idea behind the proposed method is to estimate the faults signals using the extended state observers theory. To estimate the faults, a polynomial observer (Aguilar et al. 2011; Mata-Machuca et al., Commun Nonlinear Sci Numer Simul 15(12):4114–4130, 2010, BioSystems 100(1):65–69, 2010) is presented by using the available measurements and know inputs of the system. In order to investigate the diagnosability properties of the system, a differential algebra approach is proposed (Cruz-Victoria et al., J Frankl Inst 345(2):102–118, 2008; and Martinez-Guerra and Diop, IEE P-Contr Theor Ap 151(1):130–135, 2004). The effectiveness of the methodology is illustrated by means of numerical simulations.     

Analytic expressions of quantum correlations in qutrit Werner states

Quantum correlations in qutrit Werner states are extensively investigated with five popular methods, namely, original quantum discord (OQD) (Ollivier and Zurek in Phys Rev Lett 88:017901, 2001), measurement-induced disturbance (MID) (Luo in Phys Rev A 77:022301, 2008), ameliorated MID (AMID) (Girolami et al. in J Phys A Math Theor 44:352002, 2011), relative entropy (RE) (Modi et al. in Phys Rev Lett 104:080501, 2010) and geometric discord (GD) (Daki? et al. in Phys Rev Lett 105:190502, 2010). Two different analytic expressions of quantum correlations are derived. Quantum correlations captured by the former four methods are same and bigger than those obtained via the GD method. Nonetheless, they all qualitatively characterize quantum correlations in the concerned states. Moreover, as same as the qubit case, there exist quantum correlations in separable qutrit Werner states, too.     

Unbounded Contention Resolution in Multiple-Access Channels

A frequent problem in settings where a unique resource must be shared among users is how to resolve the contention that arises when all of them must use it, but the resource allows only for one user each time. The application of efficient solutions for this problem spans a myriad of settings such as radio communication networks or databases. For the case where the number of users is unknown, recent work has yielded fruitful results for local area networks and radio networks, although either a (possibly loose) upper bound on the number of users needs to be known (Fernández Anta and Mosteiro in Discrete Math., Algorithms Appl. 2(4):445–456, 2010), or the solution is suboptimal (Bender et al. in ACM 17th Annual Symposium on Parallel Algorithms and Architectures, pp. 325–332, 2005), or it is only implicit (Greenberg and Leiserson in Adv. Comput. Res. 5:345–374, 1989) or embedded (Farach-Colton et al. in Theor. Comput. Sci. 472:60–80, 2013) in other problems, with bounds proved only asymptotically. In this paper, under the assumption that collision detection or information on the number of contenders is not available, we present a novel protocol for contention resolution in radio networks, and we recreate a protocol previously used for other problems (Greenberg and Leiserson in Adv. Comput. Res. 5:345–374, 1989, Farach-Colton et al. in Theor. Comput. Sci. 472:60–80, 2013), tailoring the constants for our needs. In contrast with previous work, both protocols are proved to be optimal up to a small constant factor and with high probability for big enough number of contenders. Additionally, the protocols are evaluated and contrasted with the previous work by extensive simulations. The evaluation shows that the complexity bounds obtained by the analysis are rather tight, and that both protocols proposed have small and predictable complexity for many system sizes (unlike previous protocols).     

The Parameterized Complexity of Unique Coverage and Its Variants

In this paper we study the parameterized complexity of the Unique Coverage problem, a variant of the classic Set Cover problem. This problem admits several parameterizations and we show that all, except the standard parameterization and a generalization of it, are unlikely to be fixed-parameter tractable. We use results from extremal combinatorics to obtain the best-known kernel for Unique Coverage and the well-known color-coding technique of Alon et al. (J. ACM 42(4), 844–856, 1995) to show that a weighted version of this problem is fixed-parameter tractable. Our application of color-coding uses an interesting variation of s-perfect hash families called (k,s)-hash families which were studied by Alon et al. (J. Comb. Theory Ser. A 104(1), 207–215, 2003) in the context of a class of codes called parent identifying codes (Barg et al. in SIAM J. Discrete Math. 14(3), 423–431, 2001). To the best of our knowledge, this is the first application of (k,s)-hash families outside the domain of coding theory. We prove the existence of such families of size smaller than the best-known s-perfect hash families using the probabilistic method (Alon and Spencer in The Probabilistic Method, Wiley, New York, 2000). Explicit constructions of such families of size promised by the probabilistic method is open.