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1.
In its simplest form, the longest common substring problem is to find a longest substring common to two or multiple strings. Using (generalized) suffix trees, this problem can be solved in linear time and space. A first generalization is the k -common substring problem: Given m strings of total length n, for all k with 2≤km simultaneously find a longest substring common to at least k of the strings. It is known that the k-common substring problem can also be solved in O(n) time (Hui in Proc. 3rd Annual Symposium on Combinatorial Pattern Matching, volume 644 of Lecture Notes in Computer Science, pp. 230–243, Springer, Berlin, 1992). A further generalization is the k -common repeated substring problem: Given m strings T (1),T (2),…,T (m) of total length n and m positive integers x 1,…,x m , for all k with 1≤km simultaneously find a longest string ω for which there are at least k strings \(T^{(i_{1})},T^{(i_{2})},\ldots,T^{(i_{k})}\) (1≤i 1<i 2<???<i k m) such that ω occurs at least \(x_{i_{j}}\) times in \(T^{(i_{j})}\) for each j with 1≤jk. (For x 1=???=x m =1, we have the k-common substring problem.) In this paper, we present the first O(n) time algorithm for the k-common repeated substring problem. Our solution is based on a new linear time algorithm for the k-common substring problem.  相似文献   

2.
We introduce a construction of a set of code sequences {Cn(m) : n ≥ 1, m ≥ 1} with memory order m and code length N(n). {Cn(m)} is a generalization of polar codes presented by Ar?kan in [1], where the encoder mapping with length N(n) is obtained recursively from the encoder mappings with lengths N(n ? 1) and N(n ? m), and {Cn(m)} coincides with the original polar codes when m = 1. We show that {Cn(m)} achieves the symmetric capacity I(W) of an arbitrary binary-input, discrete-output memoryless channel W for any fixed m. We also obtain an upper bound on the probability of block-decoding error Pe of {Cn(m)} and show that \({P_e} = O({2^{ - {N^\beta }}})\) is achievable for β < 1/[1+m(? ? 1)], where ? ∈ (1, 2] is the largest real root of the polynomial F(m, ρ) = ρm ? ρm ? 1 ? 1. The encoding and decoding complexities of {Cn(m)} decrease with increasing m, which proves the existence of new polar coding schemes that have lower complexity than Ar?kan’s construction.  相似文献   

3.
Let Z/(pe) be the integer residue ring modulo pe with p an odd prime and e ≥ 2. We consider the suniform property of compressing sequences derived from primitive sequences over Z/(pe). We give necessary and sufficient conditions for two compressing sequences to be s-uniform with α provided that the compressing map is of the form ?(x0, x1,...,xe?1) = g(xe?1) + η(x0, x1,..., xe?2), where g(xe?1) is a permutation polynomial over Z/(p) and η is an (e ? 1)-variable polynomial over Z/(p).  相似文献   

4.
This paper proposes a strengthening of the author’s core-accessibility theorem for balanced TU-cooperative games. The obtained strengthening relaxes the influence of the nontransitivity of classical domination αv on the quality of the sequential improvement of dominated imputations in a game v. More specifically, we establish the k-accessibility of the core C v ) of any balanced TU-cooperative game v for all natural numbers k: for each dominated imputation x, there exists a converging sequence of imputations x0, x1,..., such that x0 = x, lim x r C v ) and xr?m is dominated by any successive imputation x r with m ∈ [1, k] and rm. For showing that the TU-property is essential to provide the k-accessibility of the core, we give an example of an NTU-cooperative game G with a ”black hole” representing a nonempty closed subset B ? G(N) of dominated imputations that contains all the α G -monotonic sequential improvement trajectories originating at any point xB.  相似文献   

5.
We study the quantity p(n, k, t1, t2) equal to the maximum number of edges in a k-uniform hypergraph having the property that all cardinalities of pairwise intersections of edges lie in the interval [t1, t2]. We present previously known upper and lower bounds on this quantity and analyze their interrelations. We obtain new bounds on p(n, k, t1, t2) and consider their possible applications in combinatorial geometry problems. For some values of the parameters we explicitly evaluate the quantity in question. We also give a new bound on the size of a constant-weight error-correcting code.  相似文献   

6.
Let Ω = AN be a space of right-sided infinite sequences drawn from a finite alphabet A = {0,1}, N = {1,2,…}. Let ρ(x, yk=1|x k ? y k |2?k be a metric on Ω = AN, and μ the Bernoulli measure on Ω with probabilities p0, p1 > 0, p0 + p1 = 1. Denote by B(x,ω) an open ball of radius r centered at ω. The main result of this paper \(\mu (B(\omega ,r))r + \sum\nolimits_{n = 0}^\infty {\sum\nolimits_{j = 0}^{{2^n} - 1} {{\mu _{n,j}}} } (\omega )\tau ({2^n}r - j)\), where τ(x) = 2min {x,1 ? x}, 0 ≤ x ≤ 1, (τ(x) = 0, if x < 0 or x > 1 ), \({\mu _{n,j}}(\omega ) = (1 - {p_{{\omega _{n + 1}}}})\prod _{k = 1}^n{p_{{\omega _k}}} \oplus {j_k}\), \(j = {j_1}{2^{n - 1}} + {j_2}{2^{n - 2}} + ... + {j_n}\). The family of functions 1, x, τ(2 n r ? j), j = 0,1,…, 2 n ? 1, n = 0,1,…, is the Faber–Schauder system for the space C([0,1]) of continuous functions on [0, 1]. We also obtain the Faber–Schauder expansion for Lebesgue’s singular function, Cezaro curves, and Koch–Peano curves. Article is published in the author’s wording.  相似文献   

7.
Tracking frequent items (also called heavy hitters) is one of the most fundamental queries in real-time data due to its wide applications, such as logistics monitoring, association rule based analysis, etc. Recently, with the growing popularity of Internet of Things (IoT) and pervasive computing, a large amount of real-time data is usually collected from multiple sources in a distributed environment. Unfortunately, data collected from each source is often uncertain due to various factors: imprecise reading, data integration from multiple sources (or versions), transmission errors, etc. In addition, due to network delay and limited by the economic budget associated with large-scale data communication over a distributed network, an essential problem is to track the global frequent items from all distributed uncertain data sites with the minimum communication cost. In this paper, we focus on the problem of tracking distributed probabilistic frequent items (TDPF). Specifically, given k distributed sites S = {S 1, … , S k }, each of which is associated with an uncertain database \(\mathcal {D}_{i}\) of size n i , a centralized server (or called a coordinator) H, a minimum support ratio r, and a probabilistic threshold t, we are required to find a set of items with minimum communication cost, each item X of which satisfies P r(s u p(X) ≥ r × N) > t, where s u p(X) is a random variable to describe the support of X and \(N={\sum }_{i=1}^{k}n_{i}\). In order to reduce the communication cost, we propose a local threshold-based deterministic algorithm and a sketch-based sampling approximate algorithm, respectively. The effectiveness and efficiency of the proposed algorithms are verified with extensive experiments on both real and synthetic uncertain datasets.  相似文献   

8.
The Doob graph D(m, n), where m > 0, is a Cartesian product of m copies of the Shrikhande graph and n copies of the complete graph K 4 on four vertices. The Doob graph D(m, n) is a distance-regular graph with the same parameters as the Hamming graph H(2m + n, 4). We give a characterization of MDS codes in Doob graphs D(m, n) with code distance at least 3. Up to equivalence, there are m 3/36+7m 2/24+11m/12+1?(m mod 2)/8?(m mod 3)/9 MDS codes with code distance 2m + n in D(m, n), two codes with distance 3 in each of D(2, 0) and D(2, 1) and with distance 4 in D(2, 1), and one code with distance 3 in each of D(1, 2) and D(1, 3) and with distance 4 in each of D(1, 3) and D(2, 2).  相似文献   

9.
The (s + t + 1)-dimensional exchanged crossed cube, denoted as ECQ(s, t), combines the strong points of the exchanged hypercube and the crossed cube. It has been proven that ECQ(s, t) has more attractive properties than other variations of the fundamental hypercube in terms of fewer edges, lower cost factor and smaller diameter. In this paper, we study the embedding of paths of distinct lengths between any two different vertices in ECQ(s, t). We prove the result in ECQ(s, t): if s ≥ 3, t ≥ 3, for any two different vertices, all paths whose lengths are between \( \max \left\{9,\left\lceil \frac{s+1}{2}\right\rceil +\left\lceil \frac{t+1}{2}\right\rceil +4\right\} \) and 2 s+t+1 ? 1 can be embedded between the two vertices with dilation 1. Note that the diameter of ECQ(s, t) is \( \left\lceil \frac{s+1}{2}\right\rceil +\left\lceil \frac{t+1}{2}\right\rceil +2 \). The obtained result is optimal in the sense that the dilations of path embeddings are all 1. The result reveals the fact that ECQ(s, t) preserves the path embedding capability to a large extent, while it only has about one half edges of CQ n .  相似文献   

10.
Representative skyline computation is a fundamental issue in database area, which has attracted much attention in recent years. A notable definition of representative skyline is the distance-based representative skyline (DBRS). Given an integer k, a DBRS includes k representative skyline points that aims at minimizing the maximal distance between a non-representative skyline point and its nearest representative. In the 2D space, the state-of-the-art algorithm to compute the DBRS is based on dynamic programming (DP) which takes O(k m 2) time complexity, where m is the number of skyline points. Clearly, such a DP-based algorithm cannot be used for handling large scale datasets due to the quadratic time cost. To overcome this problem, in this paper, we propose a new approximate algorithm called ARS, and a new exact algorithm named PSRS, based on a carefully-designed parametric search technique. We show that the ARS algorithm can guarantee a solution that is at most ?? larger than the optimal solution. The proposed ARS and PSRS algorithms run in O(klog2mlog(T/??)) and O(k 2 log3m) time respectively, where T is no more than the maximal distance between any two skyline points. We also propose an improved exact algorithm, called PSRS+, based on an effective lower and upper bounding technique. We conduct extensive experimental studies over both synthetic and real-world datasets, and the results demonstrate the efficiency and effectiveness of the proposed algorithms.  相似文献   

11.
Consider a random k-conjunctive normal form Fk(n, rn) with n variables and rn clauses. We prove that if the probability that the formula Fk(n, rn) is satisfiable tends to 0 as n→∞, then r ? 2.83, 8.09, 18.91, 40.81, and 84.87, for k = 3, 4, 5, 6, and 7, respectively.  相似文献   

12.
We consider a class of graphs G(n, r, s) = (V (n, r),E(n, r, s)) defined as follows:
$$V(n,r) = \{ x = ({x_{1,}},{x_2}...{x_n}):{x_i} \in \{ 0,1\} ,{x_{1,}} + {x_2} + ... + {x_n} = r\} ,E(n,r,s) = \{ \{ x,y\} :(x,y) = s\} $$
where (x, y) is the Euclidean scalar product. We study random subgraphs G(G(n, r, s), p) with edges independently chosen from the set E(n, r, s) with probability p each. We find nontrivial lower and upper bounds on the clique number of such graphs.
  相似文献   

13.
In the framework of parameterized complexity, exploring how one parameter affects the complexity of a different parameterized (or unparameterized problem) is of general interest. A well-developed example is the investigation of how the parameter treewidth influences the complexity of (other) graph problems. The reason why such investigations are of general interest is that real-world input distributions for computational problems often inherit structure from the natural computational processes that produce the problem instances (not necessarily in obvious, or well-understood ways). The max leaf number ml(G) of a connected graph G is the maximum number of leaves in a spanning tree for G. Exploring questions analogous to the well-studied case of treewidth, we can ask: how hard is it to solve 3-Coloring, Hamilton Path, Minimum Dominating Set, Minimum Bandwidth or many other problems, for graphs of bounded max leaf number? What optimization problems are W[1]-hard under this parameterization? We do two things:
  1. (1)
    We describe much improved FPT algorithms for a large number of graph problems, for input graphs G for which ml(G)≤k, based on the polynomial-time extremal structure theory canonically associated to this parameter. We consider improved algorithms both from the point of view of kernelization bounds, and in terms of improved fixed-parameter tractable (FPT) runtimes O *(f(k)).
     
  2. (2)
    The way that we obtain these concrete algorithmic results is general and systematic. We describe the approach, and raise programmatic questions.
     
  相似文献   

14.
We analyze the asymptotic behavior of the j-independence number of a random k-uniform hypergraph H(n, k, p) in the binomial model. We prove that in the strongly sparse case, i.e., where \(p = c/\left( \begin{gathered} n - 1 \hfill \\ k - 1 \hfill \\ \end{gathered} \right)\) for a positive constant 0 < c ≤ 1/(k ? 1), there exists a constant γ(k, j, c) > 0 such that the j-independence number α j (H(n, k, p)) obeys the law of large numbers \(\frac{{{\alpha _j}\left( {H\left( {n,k,p} \right)} \right)}}{n}\xrightarrow{P}\gamma \left( {k,j,c} \right)asn \to + \infty \) Moreover, we explicitly present γ(k, j, c) as a function of a solution of some transcendental equation.  相似文献   

15.
The algebraic immunity of a Boolean function is a parameter that characterizes the possibility to bound this function from above or below by a nonconstant Boolean function of a low algebraic degree. We obtain lower bounds on the algebraic immunity for a class of functions expressed through the inversion operation in the field GF(2 n ), as well as for larger classes of functions defined by their trace forms. In particular, for n ≥ 5, the algebraic immunity of the function Tr n (x ?1) has a lower bound ?2√n + 4? ? 4, which is close enough to the previously obtained upper bound ?√n? + ?n/?√n?? ? 2. We obtain a polynomial algorithm which, give a trace form of a Boolean function f, computes generating sets of functions of degree ≤ d for the following pair of spaces. Each function of the first (linear) space bounds f from below, and each function of the second (affine) space bounds f from above. Moreover, at the output of the algorithm, each function of a generating set is represented both as its trace form and as a polynomial of Boolean variables.  相似文献   

16.
Two new constructions of Steiner quadruple systems S(v, 4, 3) are given. Both preserve resolvability of the original Steiner system and make it possible to control the rank of the resulting system. It is proved that any Steiner system S(v = 2 m , 4, 3) of rank rv ? m + 1 over F2 is resolvable and that all systems of this rank can be constructed in this way. Thus, we find the number of all different Steiner systems of rank r = v ? m + 1.  相似文献   

17.
This article proposes a method to study M / E s / 1 / m, E r E s /1 / m, and E r / M / n / m queuing systems including the case when m = ∞. Recurrence relations are obtained to compute the stationary distribution of the number of customers in a system and its steady-state characteristics. The developed algorithms are tested on examples using simulation models constructed with the help of the GPSS World tools.  相似文献   

18.
Recall that Lebesgue’s singular function L(t) is defined as the unique solution to the equation L(t) = qL(2t) + pL(2t ? 1), where p, q > 0, q = 1 ? p, pq. The variables M n = ∫01t n dL(t), n = 0,1,… are called the moments of the function The principal result of this work is \({M_n} = {n^{{{\log }_2}p}}{e^{ - \tau (n)}}(1 + O({n^{ - 0.99}}))\), where the function τ(x) is periodic in log2x with the period 1 and is given as \(\tau (x) = \frac{1}{2}1np + \Gamma '(1)lo{g_2}p + \frac{1}{{1n2}}\frac{\partial }{{\partial z}}L{i_z}( - \frac{q}{p}){|_{z = 1}} + \frac{1}{{1n2}}\sum\nolimits_{k \ne 0} {\Gamma ({z_k})L{i_{{z_k} + 1}}( - \frac{q}{p})} {x^{ - {z_k}}}\), \({z_k} = \frac{{2\pi ik}}{{1n2}}\), k ≠ 0. The proof is based on poissonization and the Mellin transform.  相似文献   

19.
An outer-connected dominating set in a graph G = (V, E) is a set of vertices D ? V satisfying the condition that, for each vertex v ? D, vertex v is adjacent to some vertex in D and the subgraph induced by V?D is connected. The outer-connected dominating set problem is to find an outer-connected dominating set with the minimum number of vertices which is denoted by \(\tilde {\gamma }_{c}(G)\). In this paper, we determine \(\tilde {\gamma }_{c}(S(n,k))\), \(\tilde {\gamma }_{c}(S^{+}(n,k))\), \(\tilde {\gamma }_{c}(S^{++}(n,k))\), and \(\tilde {\gamma }_{c}(S_{n})\), where S(n, k), S +(n, k), S ++(n, k), and S n are Sierpi\(\acute {\mathrm {n}}\)ski-like graphs.  相似文献   

20.
The ambiguity of a nondeterministic finite automaton (NFA) N for input size n is the maximal number of accepting computations of N for inputs of size n. For every natural number k we construct a family \((L_{r}^{k}\;|\;r\in \mathbb{N})\) of languages which can be recognized by NFA’s with size k?poly(r) and ambiguity O(n k ), but \(L_{r}^{k}\) has only NFA’s with size exponential in r, if ambiguity o(n k ) is required. In particular, a hierarchy for polynomial ambiguity is obtained, solving a long standing open problem (Ravikumar and Ibarra, SIAM J. Comput. 19:1263–1282, 1989, Leung, SIAM J. Comput. 27:1073–1082, 1998).  相似文献   

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