Connections among nonlinearity,avalanche and correlation immunity

Nonlinear Boolean functions play an important role in the design of block ciphers, stream ciphers and one-way hash functions. Over the years researchers have identified a number of indicators that forecast nonlinear properties of these functions. Studying the relationships among these indicators has been an area that has received extensive research. The focus of this paper is on the interplay of three notable nonlinear indicators, namely nonlinearity, avalanche and correlation immunity. We establish, for the first time, an explicit and simple lower bound on the nonlinearity N_f of a Boolean function f of n variables satisfying the avalanche criterion of degree p, namely, N_f⩾2ⁿ⁻¹−2^{n−1−(1/2)p}. We also identify all the functions whose nonlinearity attains the lower bound. As a further contribution of this paper, we prove that except for very few cases, the sum of the degree of avalanche and the order of correlation immunity of a Boolean function of n variables is at most n−2. The new results obtained in this work further highlight the significance of the fact that while avalanche property is in harmony with nonlinearity, both go against correlation immunity.     

Construction of high degree resilient S-boxes with improved nonlinearity

We present a simple method to use an [n−d−1,m,t+1] code to construct an n-input, m-output, t-resilient function with degree d>m and nonlinearity ²n−1−²n−⌈(d+1)/2⌉−(m+1)²n−d−1. For any fixed values of parameters n,m,t and d, with d>m, the nonlinearity obtained by our construction is higher than the nonlinearity obtained by Cheon in Crypto 2001.     

An improved algorithm for the maximum agreement subtree problem

In this paper, we solve the maximum agreement subtree problem for a set T of k rooted leaf-labeled evolutionary trees on n leaves where T contains a binary tree. We show that the O(kn³)-time dynamic-programming algorithm proposed by Bryant [Building trees, hunting for trees, and comparing trees: theory and methods in phylogenetic analysis, Ph.D. thesis, Dept. Math., University of Canterbury, 1997, pp. 174-182] can be implemented in O(kn²+n²logk−2nloglogn) and O(kn3−1/(k−1)) time by using multidimensional range search related data structures proposed by Gabow et al. [Scaling and related techniques for geometry problems, in: Proc. 16th Annual ACM Symp. on Theory of Computing, 1984, pp. 135-143] and Bentley [Multidimensional binary search trees in database applications, IEEE Trans. Softw. Eng. SE-5 (4) (1979) 333-340], respectively. When k<2+(logn−logloglogn)/(loglogn), the first implementation will be significantly faster than Bryant's algorithm. For k=3, it yields the best known algorithm which runs in O(n²lognloglogn)-time.     

Embedding a long fault-free cycle in a crossed cube with more faulty nodes

The crossed cube is an important variant of the most popular hypercube network for parallel computing. In this paper, we consider the problem of embedding a long fault-free cycle in a crossed cube with more faulty nodes. We prove that for n?5 and f?2n−7, a fault-free cycle of length at least n²−f−(n−5) can be embedded in an n-dimensional crossed cube with f faulty nodes. Our work extends some previously known results in the sense of the maximum number of faulty nodes tolerable in a crossed cube.     

Edge-fault-tolerant edge-bipancyclicity of hypercubes

In this paper, we consider the problem embedding a cycle into the hypercube Qn with existence of faulty edges and show that for any edge subset F of Qn with |F|?n−1 every edge of Qn−F lies on a cycle of every even length from 6 to n² inclusive provided n?4 and all edges in F are not incident with the same vertex. This result improves some known results.     

Shorter strings containing all k-element permutations

We consider the problem of finding short strings that contain all permutations of order k over an alphabet of size n, with k?n. We show constructively that k(n−2)+3 is an upper bound on the length of shortest such strings, for n?k?10. Consequently, for n?10, the shortest strings that contain all permutations of order n have length at most n²−2n+3. These two new upper bounds improve with one unit the previous known upper bounds.     

A note on maximum independent sets in rectangle intersection graphs

Finding the maximum independent set in the intersection graph of n axis-parallel rectangles is NP-hard. We re-examine two known approximation results for this problem. For the case of rectangles of unit height, Agarwal, van Kreveld and Suri [Comput. Geom. Theory Appl. 11 (1998) 209-218] gave a (1+1/k)-factor algorithm with an O(nlogn+n^2k−1) time bound for any integer constant k?1; we describe a similar algorithm running in only O(nlogn+nΔ^k−1) time, where Δ?n denotes the maximum number of rectangles a point can be in. For the general case, Berman, DasGupta, Muthukrishnan and Ramaswami [J. Algorithms 41 (2001) 443-470] gave a ⌈log_kn⌉-factor algorithm with an O(n^k+1) time bound for any integer constant k?2; we describe similar algorithms running in O(nlogn+nΔ^k−2) and n^O(k/logk) time.     

Longest fault-free paths in hypercubes with vertex faults

The hypercube is one of the most versatile and efficient interconnection networks (networks for short) so far discovered for parallel computation. Let f denote the number of faulty vertices in an n-cube. This study demonstrates that when f ? n − 2, the n-cube contains a fault-free path with length at least 2ⁿ − 2f − 1 (or 2ⁿ − 2f − 2) between two arbitrary vertices of odd (or even) distance. Since an n-cube is a bipartite graph with two partite sets of equal size, the path is longest in the worst-case. Furthermore, since the connectivity of an n-cube is n, the n-cube cannot tolerate n − 1 faulty vertices. Hence, our result is optimal.     

Algorithms for transitive closure

Let σ′(n) denote the number of all strongly connected graphs on the n-element set. We prove that σ′(n)?2ⁿ²·(1−n(n−1)/2ⁿ⁻¹). Hence the algorithm computing a transitive closure by a reduction to acyclic graphs has the expected time O(n²), under the assumption of uniform distribution of input graphs. Furthermore, we present a new algorithm constructing the transitive closure of an acyclic graph.     

Fault-tolerant hamiltonian laceability of hypercubes

It is known that every hypercube Q_n is a bipartite graph. Assume that n?2 and F is a subset of edges with |F|?n−2. We prove that there exists a hamiltonian path in Q_n−F between any two vertices of different partite sets. Moreover, there exists a path of length 2ⁿ−2 between any two vertices of the same partite set. Assume that n?3 and F is a subset of edges with |F|?n−3. We prove that there exists a hamiltonian path in Q_n−{v}−F between any two vertices in the partite set without v. Furthermore, all bounds are tight.     

Embedding meshes into locally twisted cubes

As a newly introduced interconnection network for parallel computing, the locally twisted cube possesses many desirable properties. In this paper, mesh embeddings in locally twisted cubes are studied. Let LTQ_n(V, E) denote the n-dimensional locally twisted cube. We present three major results in this paper: (1) For any integer n ? 1, a 2 × 2ⁿ⁻¹ mesh can be embedded in LTQ_n with dilation 1 and expansion 1. (2) For any integer n ? 4, two node-disjoint 4 × 2ⁿ⁻³ meshes can be embedded in LTQ_n with dilation 1 and expansion 2. (3) For any integer n ? 3, a 4  × (2ⁿ⁻² − 1) mesh can be embedded in LTQ_n with dilation 2. The first two results are optimal in the sense that the dilations of all embeddings are 1. The embedding of the 2 × 2ⁿ⁻¹ mesh is also optimal in terms of expansion. We also present the analysis of 2p × 2q mesh embedding in locally twisted cubes.     

Birth and growth of multicyclic components in random hypergraphs

Define an ?-component to be a connected b-uniform hypergraph with k edges and k(b−1)−? vertices. In this paper, we investigate the growth of size and complexity of connected components of a random hypergraph process. We prove that the expected number of creations of ?-components during a random hypergraph process tends to 1 as b is fixed and ? tends to infinity with the total number of vertices n while remaining ?=o(n^1/3). We also show that the expected number of vertices that ever belong to an ?-component is ∼12^1/3?^1/3n^2/3(b−1)^−1/3. We prove that the expected number of times hypertrees are swallowed by ?-components is ∼2^1/33^−1/3n^1/3?^−1/3(b−1)^−5/3. It follows that with high probability the largest ?-component during the process is of size of order O(?^1/3n^2/3(b−1)^−1/3). Our results give insight into the size of giant components inside the phase transition of random hypergraphs and generalize previous results about graphs.     

Embedding meshes into crossed cubes

Crossed cubes are important variants of hypercubes. In this paper, we consider embeddings of meshes in crossed cubes. The major research findings in this paper are: (1) For any integer n ? 1, a 2 × 2ⁿ⁻¹ mesh can be embedded in the n-dimensional crossed cube with dilation 1 and expansion 1. (2) For any integer n ? 4, two node-disjoint 4 × 2ⁿ⁻³ meshes can be embedded in the n-dimensional crossed cube with dilation 1 and expansion 2. The obtained results are optimal in the sense that the dilations of the embeddings are 1. The embedding of the 2 × 2ⁿ⁻¹ mesh is also optimal in terms of expansion because it has the smallest expansion 1.     

Lower bounds on the rotation distance of binary trees

The rotation distanced(S,T) between two binary trees S, T of n vertices is the minimum number of rotations to transform S into T. While it is known that d(S,T)?2n−6, a well-known conjecture states that there are trees for which this bound is sharp for any value of n?11. We are unable to prove the conjecture, but we give here some simple criteria for lower bound evaluation, leading for example to individuate some “regular” tree structures for which d(S,T)=3n/2−O(1), or d(S,T)=5n/3−O(1).     

Fault-free cycles in folded hypercubes with more faulty elements

Let FFv (respectively, FFe) be the set of faulty vertices (respectively, faulty edges) in an n-dimensional folded hypercube FQn. In this paper, we show that FQn−FFv−FFe contains a fault-free cycle with length at least n²−2|FFv| if |FFe|+|FFv|?2n−4 and |FFe|?n−1, where n?3. Our result improves the previously known result of [S.-Y. Hsieh, A note on cycle embedding in folded hypercubes with faulty elements, Information Processing Letters (2008), in press, doi:10.1016/j.ipl.2008.04.003] where |FFe|+|FFv|?n−1 and n?4.     

The m-pancycle-connectivity of a WK-Recursive network

In this paper, we study the m-pancycle-connectivity of a WK-Recursive network. We show that a WK-Recursive network with amplitude W and level L is strictly (5 × 2^L−1 − 2)-pancycle-connected for W ? 3. That is, each pair of vertices in a WK-recursive network with amplitude greater than or equal to 3 resides in a common cycle of every length ranging from 5 × 2^L−1 − 2 to N, where N is the size of the interconnection network; and the value 5 × 2^L−1 − 2 reaches the lower bound of the problem.     

A note on cycle embedding in hypercubes with faulty vertices

Let fv denote the number of faulty vertices in an n-dimensional hypercube. This note shows that a fault-free cycle of length of at least n²−2fv can be embedded in an n-dimensional hypercube with fv=2n−3 and n?5. This result not only enhances the previously best known result, and also answers a question in [J.-S. Fu, Fault-tolerant cycle embedding in the hypercube, Parallel Computing 29 (2003) 821-832].     

Cycles passing through a prescribed path in a hypercube with faulty edges

Let Qn denote an n-dimensional hypercube with n?2, P be a path of length h in Qn and F⊂E(Qn)\E(P). Recently, Tsai proved that if 1?h?n−1 and |F|?n−1−h, then in the graph Qn−F the path P lies on a cycle of every even length from 2h+2 to n², and P also lies on a cycle of length 2h if |F|?h−2. In this paper, we show that if 1?h?2n−3 and |F|?n−2−⌊h/2⌋, then in Qn−F the path P lies on a cycle of every even length from 2h+2 to n², and P also lies on a cycle of length 2h if P contains two edges of the same dimension or P is a shortest path and |F∩E(Qh)|?h−2, where Qh is the h-dimensional subcube containing the path P. Moreover, the upper bound 2n−3 of h is sharp and the upper bound n−2−⌊h/2⌋ of |F| is sharp for any given h with 1?h?2n−3.     

A note on the query complexity of the Condorcet winner problem

Given an unknown tournament over {1,…,n}, we show that the query complexity of the question “Is there a vertex with outdegree n−1?” (known as a Condorcet winner in social choice theory) is exactly 2n−⌊log(n)⌋−2. This stands in stark contrast to the evasiveness of this property in general digraphs.     

Primal-dual approximation algorithms for the Prize-Collecting Steiner Tree Problem

The primal-dual scheme has been used to provide approximation algorithms for many problems. Goemans and Williamson gave a (2−1/(n−1))-approximation for the Prize-Collecting Steiner Tree Problem that runs in O(n³logn) time—it applies the primal-dual scheme once for each of the n vertices of the graph. We present a primal-dual algorithm that runs in O(n²logn), as it applies this scheme only once, and achieves the slightly better ratio of (2−2/n). We also show a tight example for the analysis of the algorithm and discuss briefly a couple of other algorithms described in the literature.