A spectral lower bound for the treewidth of a graph and its consequences

We give a lower bound for the treewidth of a graph in terms of the second smallest eigenvalue of its Laplacian matrix. We use this lower bound to show that the treewidth of a d-dimensional hypercube is at least ⌊3·2^d/(2(d+4))⌋−1. The currently known upper bound is . We generalize this result to Hamming graphs. We also observe that every graph G on n vertices, with maximum degree Δ

(1)

contains an induced cycle (chordless cycle) of length at least 1+log_Δ(μn/8) (provided G is not acyclic),

(2)

has a clique minor K_h for some ,

where μ is the second smallest eigenvalue of the Laplacian matrix of G.     

Known-plaintext cryptanalysis of the Domingo-Ferrer algebraic privacy homomorphism scheme

We propose cryptanalysis of the First Domingo-Ferrer's algebraic privacy homomorphism where n=pq. We show that the scheme can be broken by (d+1) known plaintexts in O(d³log²n) time. Even when the modulus n is kept secret, it can be broken by 2(d+1) known plaintexts in O(d⁴logdn+d³log²n+?(m)) time with overwhelming probability.     

Efficient approximation algorithms for shortest cycles in undirected graphs

We describe a simple combinatorial approximation algorithm for finding a shortest (simple) cycle in an undirected graph. Given an adjacency-list representation of an undirected graph G with n vertices and unknown girth k, our algorithm returns with high probability a cycle of length at most 2k for even k and 2k+2 for odd k, in time . Thus, in general, it yields a approximation. For a weighted, undirected graph, with non-negative edge weights in the range {1,2,…,M}, we present a simple combinatorial 2-approximation algorithm for a minimum weight (simple) cycle that runs in time O(n²logn(logn+logM)).     

Alphabetic coding with exponential costs

This note considers an alphabetic binary tree formulation in a family of nonlinear problems. An application of this family occurs when a random outcome needs to be determined via alphabetically ordered search within a stochastic time window. Rather than finding a decision tree minimizing , this variant involves minimizing for a given a∈(0,1). Herein a dynamic programming algorithm finds the optimal solution in O(n³) time and O(n²) space; methods traditionally used to improve the speed of optimizations in related problems, such as the Hu-Tucker procedure, fail for this problem. This note thus also introduces two algorithms which can find a suboptimal solution in linear time (for one) or O(nlogn) time (for the other), with associated redundancy bounds guaranteeing their coding efficiency.     

New instability results for high-dimensional nearest neighbor search

Consider a dataset of n(d) points generated independently from Rd according to a common p.d.f. fd with support(fd)=d[0,1] and sup{fd(Rd)} growing sub-exponentially in d. We prove that: (i) if n(d) grows sub-exponentially in d, then, for any query point and any ?>0, the ratio of the distance between any two dataset points and is less that 1+? with probability →1 as d→∞; (ii) if n(d)>d[4(1+?)] for large d, then for all (except a small subset) and any ?>0, the distance ratio is less than 1+? with limiting probability strictly bounded away from one. Moreover, we provide preliminary results along the lines of (i) when .     

The complexity of on-line simulations between multidimensional turing machines and random access machines

To study different implementations of arrays, we present four results on the time complexities of on-line simulations between multidimensional Turing machines and random access machines (RAMs). First, everyd-dimensional Turing machine of time complexityt can be simulated by a log-cost RAM running inO(t(logt)^1–(1/d)(log logt)^1/d) time. Second, everyd-dimensional Turing machine of time complexityt can be simulated by a unit-cost RAM running inO(t/(logt)^1/d) time, provided that the input length iso(t/(logt)^1/d). Third, there is a log-cost RAMR of time complexityO(n), wheren is the input length, such that, for anyd-dimensional Turing machineM that simulatesR on-line,M requires (n ^{1 + (1/d)})/(logn(log logn)^{1 + (1/d)})) time. Fourth, every unit-cost RAM of time complexityt can be simulated by ad-dimensional Turing machine inO(t ²(logt)^1/2) time ifd = 2, and inO(t ²) time ifd 3. This result uses the weight-balanced trees of Nievergelt and Reingold.This paper was prepared while M. C. Loui was visiting the National Science Foundation in Washington, DC, and the Institute for Advanced Computer Studies, University of Maryland, College Park, MD. The views, opinions, and conclusions in this paper are those of the authors and should not be construed as an official position of the National Science Foundation, Department of Defense, U.S. Air Force, or any other U.S. government agency. The research of M. C. Loui was supported by the National Science Foundation under Grant CCR-8922008.     

Optimal per-edge processing times in the semi-streaming model

We present semi-streaming algorithms for basic graph problems that have optimal per-edge processing times and therefore surpass all previous semi-streaming algorithms for these tasks. The semi-streaming model, which is appropriate when dealing with massive graphs, forbids random access to the input and restricts the memory to bits.Particularly, the formerly best per-edge processing times for finding the connected components and a bipartition are O(α(n)), for determining k-vertex and k-edge connectivity O(k²n) and O(n⋅logn) respectively for any constant k and for computing a minimum spanning forest O(logn). All these time bounds we reduce to O(1).Every presented algorithm determines a solution asymptotically as fast as the best corresponding algorithm up to date in the classical RAM model, which therefore cannot convert the advantage of unlimited memory and random access into superior computing times for these problems.     

On generalized middle-level problem

Let be the subgraph of the hypercube Q_n induced by levels between k and n-k, where n?2k+1 is odd. The well-known middle-level conjecture asserts that is Hamiltonian for all k?1. We study this problem in for fixed k. It is known that and are Hamiltonian for all odd n?3. In this paper we prove that also is Hamiltonian for all odd n?5, and we conjecture that is Hamiltonian for every k?0 and every odd n?2k+1.     

Edge-fault-tolerant bipanconnectivity of hypercubes

This paper shows that for any two distinct vertices u and v with distance d in the hypercube Q_n (n?3) with at most faulty edges and each vertex incident with least two fault-free edges, there exist fault-free uv-paths of length ? in Q_n for every ? with d+4???2ⁿ-1 and . This result improves some known results on edge-fault bipanconnectivity of hypercubes. The proof is based on the recursive structure of Q_n.     

Improved generalized Atkin algorithm for computing square roots in finite fields

Recently, S. Müller developed a generalized Atkin algorithm for computing square roots, which requires two exponentiations in finite fields GF(q) when . In this paper, we present a simple improvement to it and the improved algorithm requires only one exponentiation for half of squares in finite fields GF(q) when . Furthermore, in finite fields GF(pm), where and m is odd, we reduce the complexity of the algorithm from O(m³log³p) to O(m²log²p(logm+logp)) using the Frobenius map and normal basis representation.     

Improved approximation bounds for the minimum rainbow subgraph problem

In this paper we consider the Minimum Rainbow Subgraph problem (MRS): Given a graph G with n vertices whose edges are coloured with p colours, find a subgraph F⊆G of minimum order and with p edges such that F contains each colour exactly once.We present a polynomial time -approximation algorithm for the MRS problem for an arbitrary small positive ?. This improves the previously best known approximation ratio of . We also prove the MRS problem to be NP-hard and APX-hard for graphs with maximum degree 2. Finally we present an algorithm to find an optimal solution in running time O(²(p+2plog₂Δ)nO(1)).     

The lower bounds on the second order nonlinearity of three classes of Boolean functions with high nonlinearity

The rth order nonlinearity of Boolean functions is an important cryptographic criterion associated with some attacks on stream and block ciphers. It is also very useful in coding theory, since it is related to the covering radii of Reed-Muller codes. This paper tightens the lower bounds of the second order nonlinearity of three classes of Boolean functions in the form f(x)=tr(x^d) in n variables, where (1) d=2^m+1+3 and n=2m, or (2) , n=2m and m is odd, or (3) d=2^2r+2^r+1+1 and n=4r.     

How much precision is needed to compare two sums of square roots of integers?

In this paper, we investigate the problem of the minimum nonzero difference between two sums of square roots of integers. Let r(n,k) be the minimum positive value of where ai and bi are integers not larger than integer n. We prove by an explicit construction that r(n,k)=O(n−2k+3/2) for fixed k and any n. Our result implies that in order to compare two sums of k square roots of integers with at most d digits per integer, one might need precision of as many as digits. We also prove that this bound is optimal for a wide range of integers, i.e., r(n,k)=Θ(n−2k+3/2) for fixed k and for those integers in the form of and , where n^′ is any integer satisfied the form and i is any integer in [0,k−1]. We finally show that for k=2 and any n, this bound is also optimal, i.e., r(n,2)=Θ(n−7/2).     

Optimal search for rationals

We present a Θ(log₂M)-time algorithm that determines an unknown rational number x in by asking at most 2log₂M+O(1) queries of the form “Is x?y?”.     

On the longest increasing subsequence of a circular list

The longest increasing circular subsequence (LICS) of a list is considered. A Monte Carlo algorithm to compute it is given which has worst case execution time O(n3/2logn) and storage requirement O(n). It is proved that the expected length μ(n) of the LICS satisfies . Numerical experiments with the algorithm suggest that .