Fast Algorithms for the Density Finding Problem

We study the problem of finding a specific density subsequence of a sequence arising from the analysis of biomolecular sequences. Given a sequence A=(a ₁,w ₁),(a ₂,w ₂),…,(a _n ,w _n ) of n ordered pairs (a _i ,w _i ) of numbers a _i and width w _i >0 for each 1≤i≤n, two nonnegative numbers ℓ, u with ℓ≤u and a number δ, the Density Finding Problem is to find the consecutive subsequence A(i ^*,j ^*) over all O(n ²) consecutive subsequences A(i,j) with width constraint satisfying ℓ≤w(i,j)=∑ _r=i ^j w _r ≤u such that its density is closest to δ. The extensively studied Maximum-Density Segment Problem is a special case of the Density Finding Problem with δ=∞. We show that the Density Finding Problem has a lower bound Ω(nlog n) in the algebraic decision tree model of computation. We give an algorithm for the Density Finding Problem that runs in optimal O(nlog n) time and O(nlog n) space for the case when there is no upper bound on the width of the sequence, i.e., u=w(1,n). For the general case, we give an algorithm that runs in O(nlog ² m) time and O(n+mlog m) space, where and w _min=min  _r=1 ⁿ w _r . As a byproduct, we give another O(n) time and space algorithm for the Maximum-Density Segment Problem. Grants NSC95-2221-E-001-016-MY3, NSC-94-2422-H-001-0001, and NSC-95-2752-E-002-005-PAE, and by the Taiwan Information Security Center (TWISC) under the Grants NSC NSC95-2218-E-001-001, NSC95-3114-P-001-002-Y, NSC94-3114-P-001-003-Y and NSC 94-3114-P-011-001.     

From liveness to promptness

Liveness temporal properties state that something “good” eventually happens, e.g., every request is eventually granted. In Linear Temporal Logic (LTL), there is no a priori bound on the “wait time” for an eventuality to be fulfilled. That is, F θ asserts that θ holds eventually, but there is no bound on the time when θ will hold. This is troubling, as designers tend to interpret an eventuality F θ as an abstraction of a bounded eventuality F ^≤k θ, for an unknown k, and satisfaction of a liveness property is often not acceptable unless we can bound its wait time. We introduce here prompt-LTL, an extension of LTL with the prompt-eventually operator F _p . A system S satisfies a prompt-LTL formula φ if there is some bound k on the wait time for all prompt-eventually subformulas of φ in all computations of S. We study various problems related to prompt-LTL, including realizability, model checking, and assume-guarantee model checking, and show that they can be solved by techniques that are quite close to the standard techniques for LTL.     

Solving satisfiability problems with preferences

Propositional satisfiability (SAT) is a success story in Computer Science and Artificial Intelligence: SAT solvers are currently used to solve problems in many different application domains, including planning and formal verification. The main reason for this success is that modern SAT solvers can successfully deal with problems having millions of variables. All these solvers are based on the Davis–Logemann–Loveland procedure (dll). In its original version, dll is a decision procedure, but it can be very easily modified in order to return one or all assignments satisfying the input set of clauses, assuming at least one exists. However, in many cases it is not enough to compute assignments satisfying all the input clauses: Indeed, the returned assignments have also to be “optimal” in some sense, e.g., they have to satisfy as many other constraints—expressed as preferences—as possible. In this paper we start with qualitative preferences on literals, defined as a partially ordered set (poset) of literals. Such a poset induces a poset on total assignments and leads to the definition of optimal model for a formula ψ as a minimal element of the poset on the models of ψ. We show (i) how dll can be extended in order to return one or all optimal models of ψ (once converted in clauses and assuming ψ is satisfiable), and (ii) how the same procedures can be used to compute optimal models wrt a qualitative preference on formulas and/or wrt a quantitative preference on literals or formulas. We implemented our ideas and we tested the resulting system on a variety of very challenging structured benchmarks. The results indicate that our implementation has comparable performances with other state-of-the-art systems, tailored for the specific problems we consider.     

Quadratic Lower Bound for Permanent Vs. Determinant in any Characteristic

In Valiant’s theory of arithmetic complexity, the classes VP and VNP are analogs of P and NP. A fundamental problem concerning these classes is the Permanent and Determinant Problem: Given a field \mathbbF{\mathbb{F}} of characteristic ≠ 2, and an integer n, what is the minimum m such that the permanent of an n × n matrix X = (x_ij) can be expressed as a determinant of an m × m matrix, where the entries of the determinant matrix are affine linear functions of x_ij ’s, and the equality is in \mathbbF[X]{\mathbb{F}}[{\bf X}]. Mignon and Ressayre (2004) proved a quadratic lower bound m = W(n²)m = \Omega(n^{2}) for fields of characteristic 0. We extend the Mignon–Ressayre quadratic lower bound to all fields of characteristic ≠ 2.     

A Primal-Dual Approximation Algorithm for Partial Vertex Cover: Making Educated Guesses

We study the partial vertex cover problem. Given a graph G=(V,E), a weight function w:V→R ⁺, and an integer s, our goal is to cover all but s edges, by picking a set of vertices with minimum weight. The problem is clearly NP-hard as it generalizes the well-known vertex cover problem. We provide a primal-dual 2-approximation algorithm which runs in O(nlog n+m) time. This represents an improvement in running time from the previously known fastest algorithm. Our technique can also be used to get a 2-approximation for a more general version of the problem. In the partial capacitated vertex cover problem each vertex u comes with a capacity k _u . A solution consists of a function x:V→ℕ₀ and an orientation of all but s edges, such that the number of edges oriented toward vertex u is at most x _u k _u . Our objective is to find a cover that minimizes ∑ _v∈V x _v w _v . This is the first 2-approximation for the problem and also runs in O(nlog n+m) time. Research supported by NSF Awards CCR 0113192 and CCF 0430650, and the University of Maryland Dean’s Dissertation Fellowship.     

Fault-Tolerant Embedding of Pairwise Independent Hamiltonian Paths on a Faulty Hypercube with Edge Faults

A Hamiltonian path in G is a path which contains every vertex of G exactly once. Two Hamiltonian paths P ₁=〈u ₁,u ₂,…,u _n 〉 and P ₂=〈v ₁,v ₂,…,v _n 〉 of G are said to be independent if u ₁=v ₁, u _n =v _n , and u _i ≠v _i for all 1<i<n; and both are full-independent if u _i ≠v _i for all 1≤i≤n. Moreover, P ₁ and P ₂ are independent starting at u ₁, if u ₁=v ₁ and u _i ≠v _i for all 1<i≤n. A set of Hamiltonian paths {P ₁,P ₂,…,P _k } of G are pairwise independent (respectively, pairwise full-independent, pairwise independent starting at u ₁) if any two different Hamiltonian paths in the set are independent (respectively, full-independent, independent starting at u ₁). A bipartite graph G is Hamiltonian-laceable if there exists a Hamiltonian path between any two vertices from different partite sets. It is well known that an n-dimensional hypercube Q _n is bipartite with two partite sets of equal size. Let F be the set of faulty edges of Q _n . In this paper, we show the following results:

Depth estimation for ranking query optimization

A relational ranking query uses a scoring function to limit the results of a conventional query to a small number of the most relevant answers. The increasing popularity of this query paradigm has led to the introduction of specialized rank join operators that integrate the selection of top tuples with join processing. These operators access just “enough” of the input in order to generate just “enough” output and can offer significant speed-ups for query evaluation. The number of input tuples that an operator accesses is called the input depth of the operator, and this is the driving cost factor in rank join processing. This introduces the important problem of depth estimation, which is crucial for the costing of rank join operators during query compilation and thus for their integration in optimized physical plans. We introduce an estimation methodology, termed deep, for approximating the input depths of rank join operators in a physical execution plan. At the core of deep lies a general, principled framework that formalizes depth computation in terms of the joint distribution of scores in the base tables. This framework results in a systematic estimation methodology that takes the characteristics of the data directly into account and thus enables more accurate estimates. We develop novel estimation algorithms that provide an efficient realization of the formal deep framework, and describe their integration on top of the statistics module of an existing query optimizer. We validate the performance of deep with an extensive experimental study on data sets of varying characteristics. The results verify the effectiveness of deep as an estimation method and demonstrate its advantages over previously proposed techniques.     

Efficient coding of labeled directed acyclic graphs

 We use an adaptation of the Prüfer code for trees to encode labeled directed acyclic graphs, which are often abbreviated to DAGs (or ADGs). In this paper, each DAG is assigned a unique DAG code, which allows an easy handling for several purposes. The set of all possible DAG codes (and therefore the set of all DAGs) for a fixed number of n vertices can be generated efficiently. Furthermore, we are able to rank DAGs, i.e., we provide an algorithm that assigns every DAG a unique number in the set {0,…,a _n −1}, where a _n is the cardinality of the set of labeled DAGs with n≥1 vertices, and we are able to unrank DAGs, which is the inverse operation. We also gain recurrence relations, which can be used to calculate a _n and a _n,q , i.e., the number of DAGs with n vertices and q edges. Finally, it is possible to generate, enumerate, rank and unrank DAGs with given number of edges and also DAGs with bounded indegree. RID="*" ID="*" This research was supported by the Austrian Science Fund (FWF), P13261-INF. I want to thank the reviewers, specially the one who suggested to add the algorithm for unranking DAG codes, for reading the paper very carefully and for the helpful comments.     

Bounding the Inefficiency of Length-Restricted Prefix Codes

Abstract. We consider an alphabet Σ= {a ₁ ,\ldots,a _n } with corresponding symbol probabilities p ₁ ,\ldots,p _n . For each prefix code associated to Σ , let l _i be the length of the codeword associated to a _i . The average code length c is defined by c=\sum _i=1 ⁿ p _i l _i . An optimal prefix code for Σ is one that minimizes c . An optimal L -restricted prefix code is a prefix code that minimizes c constrained to l _i ≤ L for i=1,\ldots,n . The value of the length restriction L is an integer no smaller than \lceil log n \rceil . Let A be the average length of an optimal prefix code for Σ . Also let A _L be the average length of an optimal L -restricted prefix code for Σ . The average code length difference ɛ is defined by ɛ=A _L -A . Let ψ be the golden ratio 1.618. In this paper we show that ɛ < 1/ψ ^{L-\lceil\log (n+\lceil\log n\rceil-L)\rceil-1} when L > \lceil log n \rceil . We also prove the sharp bound ɛ < \lceil log n \rceil -1 , when L = \lceil log n \rceil . By showing the lower bound 1/(ψ ^{L-\lceil\log n\rceil+2+\lceil\log (n/(n-L))\rceil} -1) on the maximum value of ɛ , we guarantee that our bound is asymptotically tight in the range \lceil log n \rceil < L ≤ n/2 . When L\geq \lceil log n \rceil +11 , the bound guarantees that ɛ < 0.01 . From a practical point of view, this is a negligible loss of compression efficiency. Furthermore, we present an O(n) time and space 1/ψ ^{L-\lceil\log (n+\lceil\log n\rceil-L)\rceil-1} -approximative algorithm to construct L -restricted prefix codes, assuming that the given probabilities are already sorted. The results presented in this paper suggest that one can efficiently implement length restricted prefix codes, obtaining also very effective codes.     

Batch scheduling of step deteriorating jobs

In this paper we consider the problem of scheduling n jobs on a single machine, where the jobs are processed in batches and the processing time of each job is a step function depending on its waiting time, which is the time between the start of the processing of the batch to which the job belongs and the start of the processing of the job. For job i, if its waiting time is less than a given threshold value D, then it requires a basic processing time a _i ; otherwise, it requires an extended processing time a _i +b _i . The objective is to minimize the completion time of the last job. We first show that the problem is NP-hard in the strong sense even if all b _i are equal, it is NP-hard even if b _i =a _i for all i, and it is non-approximable in polynomial time with a constant performance guarantee Δ<3/2, unless . We then present O(nlog n) and O(n ^3F−1log n/F ^F ) algorithms for the case where all a _i are equal and for the case where there are F, F≥2, distinct values of a _i , respectively. We further propose an O(n ²log n) approximation algorithm with a performance guarantee for the general problem, where m ^* is the number of batches in an optimal schedule. All the above results apply or can be easily modified for the corresponding open-end bin packing problem.     

Augmenting the Edge Connectivity of Planar Straight Line Graphs to Three

We characterize the planar straight line graphs (Pslgs) that can be augmented to 3-connected and 3-edge-connected Pslgs, respectively. We show that if a Pslg with n vertices can be augmented to a 3-edge-connected Pslg, then at most 2n−2 new edges are always sufficient and sometimes necessary for the augmentation. If the input Pslg is, in addition, already 2-edge-connected, then n−2 new edges are always sufficient and sometimes necessary for the augmentation to a 3-edge-connected Pslg.     

A classification of generic families of control-affine systems and their bifurcations

We study control-affine systems with n − 1 inputs evolving on an n-dimensional manifold for which the distribution spanned by the control vector fields is involutive and of constant rank (equivalently, they may be considered as 1-dimensional systems with n − 1 inputs entering nonlinearly). We provide a complete classification of such generic systems and their one-parameter families. We show that a generic family for n > 2 is equivalent (with respect to feedback or orbital feedback transformations) to one of nine canonical forms which differ from those for n = 2 by quadratic terms only. We also describe all generic bifurcations of 1-parameter families of systems of the above form.     

Minimum Weakly Fundamental Cycle Bases Are Hard To Find

In the last years, new variants of the minimum cycle basis (MCB) problem and new classes of cycle bases have been introduced, as motivated by several applications from disparate areas of scientific and technological inquiry. At present, the complexity status of the MCB problem is settled only for undirected, directed, and strictly fundamental cycle bases (SFCB’s). Weakly fundamental cycle bases (WFCB’s) form a natural superclass of SFCB’s. A cycle basis of a graph G is a WFCB iff ν=0 or there exists an edge e of G and a circuit C _i in such that is a WFCB of G∖e. WFCB’s still possess several of the nice properties offered by SFCB’s. At the same time, several classes of graphs enjoying WFCB’s of cost asymptotically inferior to the cost of the cheapest SFCB’s have been found and exhibited in the literature. Considered also the computational difficulty of finding cheap SFCB’s, these works advocated an in-depth study of WFCB’s. In this paper, we settle the complexity status of the MCB problem for WFCB’s (the MWFCB problem). The problem turns out to be -hard. However, in this paper, we also offer a simple and practical 2⌈log ₂ n⌉-approximation algorithm for the MWFCB problem. In O(n ν) time, this algorithm actually returns a WFCB whose cost is at most 2⌈log ₂ n⌉∑ _e∈E(G) w _e , thus allowing a fast 2⌈log ₂ n⌉-approximation also for the MCB problem. With this algorithm, we provide tight bounds on the cost of any MCB and MWFCB.     

Testing Periodicity

We study the string-property of being periodic and having periodicity smaller than a given bound. Let Σ be a fixed alphabet and let p,n be integers such that p ￡ \fracn2p\leq \frac{n}{2} . A length-n string over Σ, α=(α ₁,…,α _n ), has the property Period(p) if for every i,j∈{1,…,n}, α _i =α _j whenever i≡j (mod p). For an integer parameter g ￡ \fracn2,g\leq \frac{n}{2}, the property Period(≤g) is the property of all strings that are in Period(p) for some p≤g. The property Period( ￡ \fracn2)\mathit{Period}(\leq \frac{n}{2}) is also called Periodicity.     

Incremental Medians via Online Bidding

In the k-median problem we are given sets of facilities and customers, and distances between them. For a given set F of facilities, the cost of serving a customer u is the minimum distance between u and a facility in F. The goal is to find a set F of k facilities that minimizes the sum, over all customers, of their service costs. Following the work of Mettu and Plaxton, we study the incremental medians problem, where k is not known in advance. An incremental algorithm produces a nested sequence of facility sets F ₁⊆F ₂⊆⋅⋅⋅⊆F _n , where |F _k |=k for each k. Such an algorithm is called c -cost-competitive if the cost of each F _k is at most c times the optimum k-median cost. We give improved incremental algorithms for the metric version of this problem: an 8-cost-competitive deterministic algorithm, a 2e≈5.44-cost-competitive randomized algorithm, a (24+ε)-cost-competitive, polynomial-time deterministic algorithm, and a 6e+ε≈16.31-cost-competitive, polynomial-time randomized algorithm. We also consider the competitive ratio with respect to size. An algorithm is s -size-competitive if the cost of each F _k is at most the minimum cost of any set of k facilities, while the size of F _k is at most sk. We show that the optimal size-competitive ratios for this problem, in the deterministic and randomized cases, are 4 and e. For polynomial-time algorithms, we present the first polynomial-time O(log m)-size-approximation algorithm for the offline problem, as well as a polynomial-time O(log m)-size-competitive algorithm for the incremental problem. Our upper bound proofs reduce the incremental medians problem to the following online bidding problem: faced with some unknown threshold T∈ℝ⁺, an algorithm must submit “bids” b∈ℝ⁺ until it submits a bid b≥T, paying the sum of all its bids. We present folklore algorithms for online bidding and prove that they are optimally competitive. We extend some of the above results for incremental medians to approximately metric distance functions and to incremental fractional medians. Finally, we consider a restricted version of the incremental medians problem where k is restricted to one of two given values, for which we give a deterministic algorithm with a nearly optimal cost-competitive ratio. The conference version of this paper appeared in (Chrobak, M., et al. in Lecture Notes in Computer Science, vol. 3887, pp. 311–322, 2006). Research of M. Chrobak supported by NSF Grant CCR-0208856.     

Computing multiple pitchfork bifurcation points

A point (x*,λ*) is called apitchfork bifurcation point of multiplicityp≥1 of the nonlinear systemF(x, λ)=0,F:ℝⁿ×ℝ¹→ℝⁿ, if rank∂ _xF(x*, λ*)=n−1, and if the Ljapunov-Schmidt reduced equation has the normal formg(ξ, μ)=±ξ ²⁺ p±μξ=0. It is shown that such points satisfy a minimally extended systemG(y)=0,G:ℝ ⁿ⁺²→ℝⁿ⁺² the dimensionn+2 of which is independent ofp. For solving this system, a two-stage Newton-type method is proposed. Some numerical tests show the influence of the starting point and of the bordering vectors used in the definition of the extended system on the behavior of the iteration.     

Controllability of right invariant systems on real simple Lie groups of typeF 4,G 2,C n,andB n *

We deal with controllability of right-invariant systems for some real simple Lie groups ofF ₄,G ₂,C _n , andB _n types. We prove that the so-calledcontrollability rank condition is a necessary and sufficient condition for controllability for an open class of systems. In other papers, analogous results were obtained for Lie groups of the remaining types (i.e.,E ₆,E ₇,E ₈,A _n , andD _n ) using a special property of the root systems of their Lie algebras.     

Approximation Algorithms for Bounded Degree Phylogenetic Roots

The Degree- Δ Closest Phylogenetic k th Root Problem (ΔCPR _k ) is the problem of finding a (phylogenetic) tree T from a given graph G=(V,E) such that (1) the degree of each internal node in T is at least 3 and at most Δ, (2) the external nodes (i.e. leaves) of T are exactly the elements of V, and (3) the number of disagreements, i.e., |E ⊕{{u,v} : u,v are leaves of T and d _T (u,v)≤k}|, is minimized, where d _T (u,v) denotes the distance between u and v in tree T. This problem arises from theoretical studies in evolutionary biology and generalizes several important combinatorial optimization problems such as the maximum matching problem. Unfortunately, it is known to be NP-hard for all fixed constants Δ,k such that either both Δ≥3 and k≥3, or Δ>3 and k=2. This paper presents a polynomial-time 8-approximation algorithm for Δ CPR ₂ for any fixed Δ>3, a quadratic-time 12-approximation algorithm for 3CPR ₃, and a polynomial-time approximation scheme for the maximization version of Δ CPR _k for any fixed Δ and k.     

Embedding of Cycles in Twisted Cubes with Edge-Pancyclic

In this paper, we study the embedding of cycles in twisted cubes. It has been proven in the literature that, for any integer l, 4≤l≤2 ⁿ , a cycle of length l can be embedded with dilation 1 in an n-dimensional twisted cube, n≥3. We obtain a stronger result of embedding of cycles with edge-pancyclic. We prove that, for any integer l, 4≤l≤2 ⁿ , and a given edge (x,y) in an n-dimensional twisted cube, n≥3, a cycle C of length l can be embedded with dilation 1 in the n-dimensional twisted cube such that (x,y) is in C in the twisted cube. Based on the proof of the edge-pancyclicity of twisted cubes, we further provide an O(llog l+n ²+nl) algorithm to find a cycle C of length l that contains (u,v) in TQ _n for any (u,v)∈E(TQ _n ) and any integer l with 4≤l≤2 ⁿ .