Bounding the locality of distributed routing algorithms

We examine bounds on the locality of routing. A local routing algorithm makes a sequence of distributed forwarding decisions, each of which is made using only local information. Specifically, in addition to knowing the node for which a message is destined, an intermediate node might also know (1) its local neighbourhood (the subgraph corresponding to all network nodes within $k$ hops of itself, for some fixed $k$ ), (2) the node from which the message originated, and (3) the incoming port (which of its neighbours last forwarded the message). Our objective is to determine, as $k$ varies, which of these parameters are necessary and/or sufficient to permit local routing on a network modelled by a connected undirected graph. In particular, we establish tight bounds on $k$ for the feasibility of deterministic $k$ -local routing for various combinations of these parameters, as well as corresponding bounds on dilation (the worst-case ratio of actual route length to shortest path length).     

Finding large k-clubs in undirected graphs

Finding cohesive subgroups is an important issue in studying social networks. Many models exist for defining cohesive subgraphs in social networks, such as clique, $k$ -clique, and $k$ -clan. The concept of $k$ -club is one of them. A $k$ -club of a graph is a maximal subset of the vertex set which induces a subgraph of diameter $k$ . It is a relaxation of a clique, which induces a subgraph of diameter $1$ . We conducted algorithmic studies on finding a $k$ -club of size as large as possible. In this paper, we show that one can find a $k$ -club of maximum size in $O^{*}(1.62^n)$ time where $n$ is the number of vertices of the input graph. We implemented a combinatorial branch-and-bound algorithm that finds a $k$ -club of maximum size and a new heuristic algorithm called IDROP given in this paper. To speed up the programs, we introduce a dynamic data structure called $k$ -DN which, under deletion of vertices from a graph, maintains for a given vertex $v$ the set of vertices at distances at most $k$ . From the experimental results that we obtained, we concluded that a $k$ -club of maximum size can be easily found in sparse graphs and dense graphs. Our heuristic algorithm finds, within reasonable time, $k$ -clubs of maximum size in most of experimental instances. The gap between the size of a $k$ -club of maximum size and a $k$ -club found by IDROP is a constant for the number of vertices that we are able to test.     

Hubness-aware shared neighbor distances for high-dimensional k-nearest neighbor classification

Learning from high-dimensional data is usually quite challenging, as captured by the well-known phrase curse of dimensionality. Data analysis often involves measuring the similarity between different examples. This sometimes becomes a problem, as many widely used metrics tend to concentrate in high-dimensional feature spaces. The reduced contrast makes it more difficult to distinguish between close and distant points, which renders many traditional distance-based learning methods ineffective. Secondary distances based on shared neighbor similarities have recently been proposed as one possible solution to this problem. However, these initial metrics failed to take hubness into account. Hubness is a recently described aspect of the dimensionality curse, and it affects all sorts of $k$ -nearest neighbor learning methods in severely negative ways. This paper is the first to discuss the impact of hubs on forming the shared neighbor similarity scores. We propose a novel, hubness-aware secondary similarity measure $simhub_s$ and an extensive experimental evaluation shows it to be much more appropriate for high-dimensional data classification than the standard $simcos_s$ measure. The proposed similarity changes the underlying $k$ NN graph in such a way that it reduces the overall frequency of label mismatches in $k$ -neighbor sets and increases the purity of occurrence profiles, which improves classifier performance. It is a hybrid measure, which takes into account both the supervised and the unsupervised hubness information. The analysis shows that both components are useful in their own ways and that the measure is therefore properly defined. This new similarity does not increase the overall computational cost, and the improvement is essentially ‘free’.     

Learning potential functions and their representations for multi-task reinforcement learning

In multi-task learning, there are roughly two approaches to discovering representations. The first is to discover task relevant representations, i.e., those that compactly represent solutions to particular tasks. The second is to discover domain relevant representations, i.e., those that compactly represent knowledge that remains invariant across many tasks. In this article, we propose a new approach to multi-task learning that captures domain-relevant knowledge by learning potential-based shaping functions, which augment a task’s reward function with artificial rewards. We address two key issues that arise when deriving potential functions. The first is what kind of target function the potential function should approximate; we propose three such targets and show empirically that which one is best depends critically on the domain and learning parameters. The second issue is the representation for the potential function. This article introduces the notion of $k$ -relevance, the expected relevance of a representation on a sample sequence of $k$ tasks, and argues that this is a unifying definition of relevance of which both task and domain relevance are special cases. We prove formally that, under certain assumptions, $k$ -relevance converges monotonically to a fixed point as $k$ increases, and use this property to derive Feature Selection Through Extrapolation of k-relevance (FS-TEK), a novel feature-selection algorithm. We demonstrate empirically the benefit of FS-TEK on artificial domains.     

Efficient processing of k-hop reachability queries

We study the problem of answering k -hop reachability queries in a directed graph, i.e., whether there exists a directed path of length $k$ , from a source query vertex to a target query vertex in the input graph. The problem of $k$ -hop reachability is a general problem of the classic reachability (where $k=\infty $ ). Existing indexes for processing classic reachability queries, as well as for processing shortest path distance queries, are not applicable or not efficient for processing $k$ -hop reachability queries. We propose an efficient index for processing $k$ -hop reachability queries. Our experimental results on a wide range of real datasets show that our method is efficient and scalable in terms of both index construction and query processing.     

Order optimal information spreading using algebraic gossip

In this paper we study gossip based information spreading with bounded message sizes. We use algebraic gossip to disseminate $k$ distinct messages to all $n$ nodes in a network. For arbitrary networks we provide a new upper bound for uniform algebraic gossip of $O((k+\log n + D)\varDelta )$ rounds with high probability, where $D$ and $\varDelta $ are the diameter and the maximum degree in the network, respectively. For many topologies and selections of $k$ this bound improves previous results, in particular, for graphs with a constant maximum degree it implies that uniform gossip is order optimal and the stopping time is $\varTheta (k + D)$ . To eliminate the factor of $\varDelta $ from the upper bound we propose a non-uniform gossip protocol, TAG, which is based on algebraic gossip and an arbitrary spanning tree protocol $\mathcal{S } $ . The stopping time of TAG is $O(k+\log n +d(\mathcal{S })+t(\mathcal{S }))$ , where $t(\mathcal{S })$ is the stopping time of the spanning tree protocol, and $d(\mathcal{S })$ is the diameter of the spanning tree. We provide two general cases in which this bound leads to an order optimal protocol. The first is for $k=\varOmega (n)$ , where, using a simple gossip broadcast protocol that creates a spanning tree in at most linear time, we show that TAG finishes after $\varTheta (n)$ rounds for any graph. The second uses a sophisticated, recent gossip protocol to build a fast spanning tree on graphs with large weak conductance. In turn, this leads to the optimally of TAG on these graphs for $k=\varOmega (\text{ polylog }(n))$ . The technique used in our proofs relies on queuing theory, which is an interesting approach that can be useful in future gossip analysis.     

Non-unique cluster numbers determination methods based on stability in spectral clustering

Recently, a large amount of work has been devoted to the study of spectral clustering—a simple yet powerful method for finding structure in a data set using spectral properties of an associated pairwise similarity matrix. Most of the existing spectral clustering algorithms estimate only one cluster number or estimate non-unique cluster numbers based on eigengap criterion. However, the number of clusters not always exists one, and eigengap criterion lacks theoretical justification. In this paper, we propose non-unique cluster numbers determination methods based on stability in spectral clustering (NCNDBS). We first utilize the multiway normalized cut spectral clustering algorithm to cluster data set for a candidate cluster number $k$ . Then the ratio value of the multiway normalized cut criterion of the obtained clusters and the sum of the leading eigenvalues (descending sort) of the stochastic transition matrix is chosen as a standard to decide whether the $k$ is a reasonable cluster number. At last, by varying the scaling parameter in the Gaussian function, we judge whether the reasonable cluster number $k$ is also a stability one. By three stages, we can determine non-unique cluster numbers of a data set. The Lumpability theorem concluded by Meil $\breve{a}$ and Xu provides a theoretical base for our methods. NCNDBS can estimate non-unique cluster numbers of the data set successfully by illustrative experiments.     

A hybrid clustering algorithm based on PSO with dynamic crossover

In order to overcome the premature convergence in particle swarm optimization (PSO), we introduce dynamical crossover, a crossover operator with variable lengths and positions, to PSO, which is briefly denoted as CPSO. To get rid of the drawbacks of only finding the convex clusters and being sensitive to the initial points in $k$ -means algorithm, a hybrid clustering algorithm based on CPSO is proposed. The difference between the work and the existing ones lies in that CPSO is firstly introduced into $k$ -means. Experimental results performing on several data sets illustrate that the proposed clustering algorithm can get completely rid of the shortcomings of $k$ -means algorithms, and acquire correct clustering results. The application in image segmentation illustrates that the proposed algorithm gains good performance.     

Auto-tuning Similarity Search Algorithms on Multi-core Architectures

In recent times, large high-dimensional datasets have become ubiquitous. Video and image repositories, financial, and sensor data are just a few examples of such datasets in practice. Many applications that use such datasets require the retrieval of data items similar to a given query item, or the nearest neighbors (NN or $k$ -NN) of a given item. Another common query is the retrieval of multiple sets of nearest neighbors, i.e., multi $k$ -NN, for different query items on the same data. With commodity multi-core CPUs becoming more and more widespread at lower costs, developing parallel algorithms for these search problems has become increasingly important. While the core nearest neighbor search problem is relatively easy to parallelize, it is challenging to tune it for optimality. This is due to the fact that the various performance-specific algorithmic parameters, or “tuning knobs”, are inter-related and also depend on the data and query workloads. In this paper, we present (1) a detailed study of the various tuning knobs and their contributions on increasing the query throughput for parallelized versions of the two most common classes of high-dimensional multi-NN search algorithms: linear scan and tree traversal, and (2) an offline auto-tuner for setting these knobs by iteratively measuring actual query execution times for a given workload and dataset. We show experimentally that our auto-tuner reaches near-optimal performance and significantly outperforms un-tuned versions of parallel multi-NN algorithms for real video repository data on a variety of multi-core platforms.     

Computational complexity and approximability of social welfare optimization in multiagent resource allocation

A central task in multiagent resource allocation, which provides mechanisms to allocate (bundles of) resources to agents, is to maximize social welfare. We assume resources to be indivisible and nonshareable and agents to express their utilities over bundles of resources, where utilities can be represented in the bundle form, the $k$ -additive form, and as straight-line programs. We study the computational complexity of social welfare optimization in multiagent resource allocation, where we consider utilitarian and egalitarian social welfare and social welfare by the Nash product. Solving some of the open problems raised by Chevaleyre et al. (2006) and confirming their conjectures, we prove that egalitarian social welfare optimization is $\mathrm{NP}$ -complete for the bundle form, and both exact utilitarian and exact egalitarian social welfare optimization are $\mathrm{DP}$ -complete, each for both the bundle and the $2$ -additive form, where $\mathrm{DP}$ is the second level of the boolean hierarchy over  $\mathrm{NP}$ . In addition, we prove that social welfare optimization by the Nash product is $\mathrm{NP}$ -complete for both the bundle and the $1$ -additive form, and that the exact variants are $\mathrm{DP}$ -complete for the bundle and the $3$ -additive form. For utility functions represented as straight-line programs, we show $\mathrm{NP}$ -completeness for egalitarian social welfare optimization and social welfare optimization by the Nash product. Finally, we show that social welfare optimization by the Nash product in the $1$ -additive form is hard to approximate, yet we also give fully polynomial-time approximation schemes for egalitarian and Nash product social welfare optimization in the $1$ -additive form with a fixed number of agents.     

Nearly exact mining of frequent trees in large networks

Mining frequent patterns in a single network (graph) poses a number of challenges. Already only to match one path pattern to a network under subgraph isomorphism is NP-complete. Classical matching algorithms become intractable even for reasonably small patterns, on networks which are large or have a high average degree. Based on recent advances in parameterized complexity theory, we propose a novel miner for rooted trees in networks. The miner, for a fixed parameter $k$ k (maximal pattern size), can mine all rooted trees with delay linear in the size of the network and only mildly exponential in the fixed parameter $k$ k . This allows us to mine tractably, rooted trees, in large networks such as the WWW or social networks. We establish the practical applicability of our miner, by presenting an experimental evaluation on both synthetic and real-world data.     

Parameterized Complexity of Induced Graph Matching on Claw-Free Graphs

The Induced Graph Matching problem asks to find \(k\) disjoint induced subgraphs isomorphic to a given graph  \(H\) in a given graph \(G\) such that there are no edges between vertices of different subgraphs. This problem generalizes the classical Independent Set and Induced Matching problems, among several other problems. We show that Induced Graph Matching is fixed-parameter tractable in \(k\) on claw-free graphs when \(H\) is a fixed connected graph, and even admits a polynomial kernel when  \(H\) is a complete graph. Both results rely on a new, strong, and generic algorithmic structure theorem for claw-free graphs. Complementing the above positive results, we prove \(\mathsf {W}[1]\) -hardness of Induced Graph Matching on graphs excluding \(K_{1,4}\) as an induced subgraph, for any fixed complete graph \(H\) . In particular, we show that Independent Set is \(\mathsf {W}[1]\) -hard on \(K_{1,4}\) -free graphs. Finally, we consider the complexity of Induced Graph Matching on a large subclass of claw-free graphs, namely on proper circular-arc graphs. We show that the problem is either polynomial-time solvable or \(\mathsf {NP}\) -complete, depending on the connectivity of \(H\) and the structure of \(G\) .     

Time versus space trade-offs for rendezvous in trees

Two identical (anonymous) mobile agents start from arbitrary nodes of an unknown tree and have to meet at some node. Agents move in synchronous rounds: in each round an agent can either stay at the current node or move to one of its neighbors. We consider deterministic algorithms for this rendezvous task. The main result of this paper is a tight trade-off between the optimal time of completing rendezvous and the size of memory of the agents. For agents with $k$ memory bits, we show that optimal rendezvous time is $\Theta (n+n^2/k)$ in $n$ -node trees. More precisely, if $k \ge c\log n$ , for some constant $c$ , we design agents accomplishing rendezvous in arbitrary trees of size $n$ (unknown to the agents) in time $O(n+n^2/k)$ , starting with arbitrary delay. We also show that no pair of agents can accomplish rendezvous in time $o(n+n^2/k)$ , even in the class of lines of known length and even with simultaneous start. Finally, we prove that at least logarithmic memory is necessary for rendezvous, even for agents starting simultaneously in a $n$ -node line.     

Efficient query processing for XML keyword queries based on the IDList index

Keyword search over XML data has attracted a lot of research efforts in the last decade, where one of the fundamental research problems is how to efficiently answer a given keyword query w.r.t. a certain query semantics. We found that the key factor resulting in the inefficiency for existing methods is that they all heavily suffer from the common-ancestor-repetition problem. In this paper, we propose a novel form of inverted list, namely the IDList; the IDList for keyword $k$ consists of ordered nodes that directly or indirectly contain $k$ . We then show that finding keyword query results based on the smallest lowest common ancestor and exclusive lowest common ancestor semantics can be reduced to ordered set intersection problem, which has been heavily optimized due to its application in areas such as information retrieval and database systems. We propose several algorithms that exploit set intersection in different directions and with or without using additional indexes. We further propose several algorithms that are based on hash search to simplify the operation of finding common nodes from all involved IDLists. We have conducted an extensive set of experiments using many state-of-the-art algorithms and several large-scale datasets. The results demonstrate that our proposed methods outperform existing methods by up to two orders of magnitude in many cases.     

Aspects of insertion in random trees

A method formulated by Yao and used by Brown has yielded bounds on the fraction of nodes with specified properties in trees bult by a sequence of random internal nodes in a random tree built by binary search and insertion, and show that in such a tree about bounds better than those now known. We then apply these methods to weight-balanced trees and to a type of “weakly balanced” trees. We determine the distribution of the weight-balance factors of the internal nodes in a random tree built by binary search and insertion and show that in such a tree about 72% of all internal nodes have weight balance factors lying between \(1 - \sqrt 2 /2\) and \(\sqrt 2 /2\) .     

Strong Conflict-Free Coloring for Intervals

We consider the \(k\) -strong conflict-free ( \(k\) -SCF) coloring of a set of points on a line with respect to a family of intervals: Each point on the line must be assigned a color so that the coloring is conflict-free in the following sense: in every interval \(I\) of the family there are at least \(k\) colors each appearing exactly once in \(I\) . We first present a polynomial-time approximation algorithm for the general problem; the algorithm has approximation ratio 2 when \(k=1\) and \(5-\frac{2}{k}\) when \(k\ge 2\) . In the special case of a family that contains all possible intervals on the given set of points, we show that a 2-approximation algorithm exists, for any \(k \ge 1\) . We also provide, in case \(k=O({{\mathrm{polylog}}}(n))\) , a quasipolynomial time algorithm to decide the existence of a \(k\) -SCF coloring that uses at most \(q\) colors.     

Simple characterization of strict residuated lattices with an involutive negation

We consider property of strict residuated lattices (SRL-algebras) with a new involutive negation $\lnot, $ called here by SRL $_{\lnot }$ -algebras, and give a simple characterization of SRL $_{\lnot }$ -algebras. We also prove a prime filter theorem of SRL $_{\lnot }$ -algebras, from which we conclude that every linearly ordered SRL $_{\lnot }$ -algebra is simple. As a corollary to this fact, we have a well-known result that every SML $_{\lnot }$ -algebra (SBL $_{\lnot }$ -algebra) is a subdirect product of linearly ordered SML $_{\lnot }$ -algebras (SBL $_{\lnot }$ -algebras).     

Exact and Parameterized Algorithms for Max Internal Spanning Tree

We consider the $\mathcal{NP}$ -hard problem of finding a spanning tree with a maximum number of internal vertices. This problem is a generalization of the famous Hamiltonian Path problem. Our dynamic-programming algorithms for general and degree-bounded graphs have running times of the form $\mathcal{O}^{*}(c^{n})$ with c≤2. For graphs with bounded degree, c<2. The main result, however, is a branching algorithm for graphs with maximum degree three. It only needs polynomial space and has a running time of $\mathcal{O}(1.8612^{n})$ when analyzed with respect to the number of vertices. We also show that its running time is $2.1364^{k} n^{\mathcal{O}(1)}$ when the goal is to find a spanning tree with at least k internal vertices. Both running time bounds are obtained via a Measure & Conquer analysis, the latter one being a novel use of this kind of analysis for parameterized algorithms.     

Multi-dimensional color image storage and retrieval for a normal arbitrary quantum superposition state

Multi-dimensional color image processing has two difficulties: One is that a large number of bits are needed to store multi-dimensional color images, such as, a three-dimensional color image of $1024 \times 1024 \times 1024$ needs $1024 \times 1024 \times 1024 \times 24$  bits. The other one is that the efficiency or accuracy of image segmentation is not high enough for some images to be used in content-based image search. In order to solve the above problems, this paper proposes a new representation for multi-dimensional color image, called a $(n\,+\,1)$ -qubit normal arbitrary quantum superposition state (NAQSS), where $n$ qubits represent colors and coordinates of ${2^n}$ pixels (e.g., represent a three-dimensional color image of $1024 \times 1024 \times 1024$ only using 30 qubits), and the remaining 1 qubit represents an image segmentation information to improve the accuracy of image segmentation. And then we design a general quantum circuit to create the NAQSS state in order to store a multi-dimensional color image in a quantum system and propose a quantum circuit simplification algorithm to reduce the number of the quantum gates of the general quantum circuit. Finally, different strategies to retrieve a whole image or the target sub-image of an image from a quantum system are studied, including Monte Carlo sampling and improved Grover’s algorithm which can search out a coordinate of a target sub-image only running in $O(\sqrt{N/r} )$ where $N$ and $r$ are the numbers of pixels of an image and a target sub-image, respectively.