Stable sets in k-colorable -free graphs

We show that, for fixed k, there is a polynomial-time algorithm that finds a maximum (or maximum-weight) stable set in any graph that belongs to the class of k-colorable P₅-free graphs, or, more generally, to the class of P₅-free graphs that contain no clique of size k+1. This is based on the following structural result: every connected k-colorable P₅-free graph has a vertex whose non-neighbors induce a (k−1)-colorable subgraph.     

On the k-ary hypercube

In this paper, we propose and analyze a new interconnection network, the k-ary hypercube. This new architecture captures the advantages of the mesh network and those of the binary hypercube. We show that the hamiltoniacity of this network and its capability of efficiently simulating other topologies. It has a smaller degree than that of its equivalent binary hypercube (the one with at least as many nodes) and has a smaller diameter than its equivalent mesh of processors.     

On the partition dimension of a class of circulant graphs

For a vertex v   of a connected graph G(V,E)$G(V,E)$ and a subset S of V, the distance between a vertex v and S   is defined by d(v,S)=min{d(v,x):x∈S}$d(v,S)=\min \{d(v,x):x\in S\}$. For an ordered k  -partition π={_S1,_S2…_Sk}$\pi =\{S_1,S_2\dots S_k\}$ of V, the partition representation of v with respect to π is the k  -vector r(v|π)=(d(v,_S1),d(v,_S2)…d(v,_Sk))$r(v|\pi )=(d(v,S_1),d(v,S_2)\dots d(v,S_k))$. The k-partition π is a resolving partition if the k  -vectors r(v|π)$r(v|\pi )$, v∈V(G)$v\in V(G)$ are distinct. The minimum k for which there is a resolving k-partition of V is the partition dimension of G. Salman et al. [1] in their paper which appeared in Acta Mathematica Sinica, English Series   proved that partition dimension of a class of circulant graph G(n,±{1,2})$G(n,\pm \{1,2\})$, for all even n?6$n?6$ is four. In this paper we prove that it is three.     

Embedding k(n−k) edge-disjoint spanning trees in arrangement graphs

The arrangement graphs are a class of generalized star graphs. In this paper we construct a graph that consists of the maximum number of directed edge-disjoint spanning trees in an arrangement graph. The paths that connect the common root node to any given node through different spanning trees are node-disjoint, and the lengths of these paths differ from the shortest possible lengths by a small additive constant. This graph can be used to derive fault-tolerant algorithms for broadcasting and scattering problems without prior knowledge of the faulty elements of the network.     

On some questions regarding k -regular and k -context-free sequences

We answer two questions of Allouche and Shallit regarding k-regular sequences and k-context-free sequences. One of these has been solved independently in another way by Bell. We also provide a partial solution to another question of Allouche and Shallit regarding the subword complexity of k-context-free sequences.     

Answering range-based reverse kNN and why-not reverse kNN queries

1 Introduction Given a dataset D and a point q,a reverse nearest neighbor(RNN)query retrieves all the points p∈D that have q as their nearest neighbor.Although the RNN problem was first pro-posed in[1],it still has received considerable attention due to its importance in several applications involving decision support,resource allocation,profile-based marketing,etc.     

Thoughts on k-anonymization

k-Anonymity is a method for providing privacy protection by ensuring that data cannot be traced to an individual. In a k-anonymous dataset, any identifying information occurs in at least k tuples. To achieve optimal and practical k-anonymity, recently, many different kinds of algorithms with various assumptions and restrictions have been proposed with different metrics to measure quality. This paper evaluates a family of clustering-based algorithms that are more flexible and even attempts to improve precision by ignoring the restrictions of user-defined Domain Generalization Hierarchies. The evaluation of the new approaches with respect to cost metrics shows that metrics may behave differently with different algorithms and may not correlate with some applications’ accuracy on output data.     

Ray representation for k-trees

k-trees have established themselves as useful data structures in pattern recognition. A fundamental operation regarding k-trees is the construction of a k-tree. We present a method to store an object as a set of rays and an algorithm to convert such a set into a k-tree. The algorithm is conceptually simple, works for any k, and builds a k-tree from the rays very fast. It produces a minimal k-tree and does not lead to intermediate storage swell.     

Adaptive soft k-nearest-neighbour classifiers

A novel classifier is introduced to overcome the limitations of the k-NN classification systems. It estimates the posterior class probabilities using a local Parzen window estimation with the k-nearest-neighbour prototypes (in the Euclidean sense) to the pattern to classify. A learning algorithm is also presented to reduce the number of data points to store. Experimental results in two hand-written classification problems demonstrate the potential of the proposed classification system.     

支持隐私保护的k近邻分类器

k近邻（k-nearest neighbor,简称kNN）分类器在生物信息学、股票预测、网页分类以及鸢尾花分类预测等方面都有着广泛的应用.随着用户隐私保护意识的日益提高,kNN分类器也需要对密文数据提供分类支持,进而保证用户数据的隐私性,即设计一种支持隐私保护的k近邻分类器（privacy-preserving k-nearest neighbor classifier,简称PP-kNN）.首先,对kNN分类器的操作进行分析,从中提取出一些基本操作,包括加法、乘法、比较、内积等.然后,选择两种同态加密方案和一种全同态加密方案对数据进行加密.在此基础上设计了针对基本操作的安全协议,其输出结果与在明文数据上执行同一方法的输出结果一致,且证明该协议在半诚实模型下是安全的.最后,通过将基本操作的安全协议进行模块化顺序组合的方式实现kNN分类器对密文数据处理的支持.通过实验,对所设计的PP-kNN分类器进行测试.结果表明,该分类器能够以较高效率实现对密文数据的分类,同时为用户数据提供隐私性保护.     

The accelerated k-in-a-row game

We investigate two versions of the well-known k-in-a-row game. While in the most intriguing k = 5 case the outcome of the game has been recently settled, very little is known about what happens when the rules are changed. A natural modification is that the players take more than one square of the board per move in order to speed up the game. Our main goal is to improve the quadratic bound on the error term, given by Csirmaz in Csirmaz (Discrete Math. 29 (1980) 19–23), to a logarithmic one for the accelerated k-in-a-row. The other issue is the extreme sensitivity of k-in-a-row under biased rules. Beck proposed in Beck (unpublished lecture notes) that a player may trade some of his freedom of choice for the right of taking more squares than his opponent. We prove logarithmic bounds on the error term in that case, too.     

基于k-Lin假设的同态加密方案

作为数字货币的底层核心技术之一,区块链随着数字货币的快速发展而受到了广泛关注.由于区块链具有去中心化、防篡改、可追溯等性质,如今越来越多的企业和个人用户选择利用区块链技术来实现数据的传输和记录.区块链公开透明的特性,一方面充分保证了数据的可用性;但另一方面,又给用户的隐私信息带来了严重威胁.为了同时兼顾用户数据的机密性和可用性,同态加密常常被用到区块链的安全解决方案之中.然而,现实应用对于所部署的同态加密方案的安全强度要求也很可能会随着时间推移而有所变化.考虑到区块链应用场景的复杂多样性和分布式特点,同态加密方案一旦部署下去,之后,当随着时间推移需要调整安全性强度时,相应的工作量将会非常繁重.此外,在区块链的现实应用中,考虑到监管方面的需求,很多情况下（尤其是针对某些群组成员发布和传输的数据）需要允许某可信第三方（如监管方）能够对链上的相应密文数据进行解密.若采用传统的同态加密方案对数据进行加密,可信第三方需要存储所有用户的私钥,这将给密钥管理和存储带来巨大压力.针对当前的区块链应用场景和安全需求,提出了一个基于ZN2*（N=pq）上的判定性k-Lin假设的加法同态加密方案.该方案不仅在标准模型下能够满足IND-CCA1安全性,还具有3个特殊优势：（i）可以通过对参数k的调控细粒度地调节加密方案的安全性强度;（ii）加密方案具有双解密机制：存在两种私钥,一种由用户本人持有,另一种由可信第三方持有,其中,可信第三方的私钥可用于该加密体制所有用户的密文解密;（iii）加密方案可以极为便利地退化为IND-CPA安全的公钥加密方案,退化后的方案不仅其公私钥长度和密文长度变得更短,而且同样具有加法同态性和双解密机制.     

Algorithm 28 algorithm for generating graphs of a given partition

An algorithm which produces with comparatively little redundancy all simple graphs realizing a given partition is discussed, and a FORTRAN implementation is given. The program generates all graphs of order 8 in approximately 40 seconds on an ICL 1906A computer.     

On generalization/specialization for conceptual graphs

Abstract

This paper centres on the generalization/specialization relation in the framework of conceptual graphs (this relation corresponds to logical subsumption when considering logical formulas associated with conceptual graphs). Results given here apply more generally to any model where knowledge is described by labelled graphs and reasoning is based on graph subsumption, as in semantic networks or in structural machine learning. The generalization/specialization relation, as defined by Sowa, is first precisely analysed, in particular its links with a graph morphism, called projection. Besides Sowa's specialization relation (which is a preorder), another one is actually used in some practical applications (which is an order). These are comparatively studied. The second topic of this paper is the design of efficient algorithms for computing these specialization relations. Since the associated problems are NP-hard, the form of the graphs is restricted in order to arrive at polynomial algorithms. In particular, polynomial algorithms are presented for computing a projection from a conceptual ‘tree’ to any conceptual graph, and for counting the number of such projections. The algorithms are also described in a generic way, replacing the projection by a parametrized graph morphism, and conceptual graphs by directed labelled graphs.     

Strongly Hamiltonian laceability of the even k-ary n-cube

The interconnection network considered in this paper is the k-ary n-cube that is an attractive variance of the well-known hypercube. Many interconnection networks can be viewed as the subclasses of the k-ary n-cubes include the cycle, the torus and the hypercube. A bipartite graph is Hamiltonian laceable if there exists a Hamiltonian path joining every two vertices which are in distinct partite sets. A bipartite graph G is strongly Hamiltonian laceable if it is Hamiltonian laceable and there exists a path of length N − 2 joining each pair of vertices in the same partite set, where N = |V(G)|. We prove that the k-ary n-cube is strongly Hamiltonian laceable for k is even and n  2.     

On induced-universal graphs for the class of bounded-degree graphs

For a family F of graphs, a graph U is said to be F-induced-universal if every graph of F is an induced subgraph of U. We give a construction for an induced-universal graph for the family of graphs on n vertices with degree at most k. For k even, our induced-universal graph has O(nk/2) vertices and for k odd it has O(n⌈k/2⌉−1/klog2+2/kn) vertices. This construction improves a result of Butler by a multiplicative constant factor for the even case and by almost a multiplicative n1/k factor for the odd case. We also construct induced-universal graphs for the class of oriented graphs with bounded incoming and outgoing degree, slightly improving another result of Butler.     

Separating k-separated eNCE graph languages

An eNCE graph grammar is k-separated (k1) if the distance between any two nonterminal nodes in any of its sentential forms is at least k. Let SEP_k denote the class of graph languages generated by k-separated grammars. Then, SEP₁ (SEP₂) is the class of eNCE (boundary eNCE) graph languages, and so SEP₂SEP₁. Recently, Engelfriet (1991) showed that SEP₃SEP₂ and conjectured that, in fact, SEP_k+1SEP_k for each k 1. We prove this conjecture affirmatively.