Following the long-held belief that the Internet is hierarchical, the network topology generators most widely used by the Internet research community, Transit-Stub and Tiers, create networks with a deliberately hierarchical structure. However, in 1999 a seminal paper by Faloutsos et al. revealed that the Internet's degree distribution is a power-law. Because the degree distributions produced by the Transit-Stub and Tiers generators are not power-laws, the research community has largely dismissed them as inadequate and proposed new network generators that attempt to generate graphs with power-law degree distributions.

As panoramic photography becomes increasingly popular, there is a greater need for high-quality software to automatically create panoramic images. Existing algorithms either produce a rough "stitch " that cannot deal with common artifacts, or require user input. This paper presents methods for dealing with two artifacts that often occur in practice. Our first contribution is a method for dealing with objects that move between different views of a dynamic scene. If such moving objects are left in, they will appear blurry and "ghosted". Treating such regions as nodes in a graph, we use a vertex cover algorithm to selectively remove all but one instance of each object. Our second contribution is a method for continuously adjusting exposure across multiple images in order to eliminate visible shifts in brightness or hue. We compute exposure corrections on a block-by block basis, then smoothly interpolate the parameters using a spline to get spatially continuous exposure adjustment. Our enhancements, combined with previously published techniques for automatic image stitching, result in a high-quality automated stitcher that exhibits far fewer artifacts than existing software. 1.

We address the problem of finding parallel plans for SQL queries using the two-phase approach of join ordering followed by parallelization. We focus on the parallelization phase and develop algorithms for exploiting pipelined parallelism. We formulate parallelization as scheduling a weighted operator tree to minimize response time. Our model of response time captures the fundamental tradeoff between parallel execution and its communication overhead. We assess the quality of an optimization algorithm by its performance ratio which is the ratio of the response time of the generated schedule to that of the optimal. We develop fast algorithms that produce near-optimal schedules -- the performance ratio is extremely close to 1 on the average and has a worst case bound of about 2 for many cases. 1 Introduction We address the problem of parallel query optimization, which is to find optimal parallel plans for executing SQL queries. Following Hong and Stonebraker [HS91], we break the optimi...

We survey the problem of comparing labeled trees based on simple local operations of deleting, inserting and relabeling nodes. These operations lead to the tree edit distance, alignment distance and inclusion problem. For each problem we review the results available and present, in detail, one or more of the central algorithms for solving the problem.

We introduce a class of novel multiprocessor scheduling problems that arise in the optimization of SQL queries for parallel machines. These consist of scheduling a tree of interdependent communicating operators while exploiting both inter-operator and intra-operator parallelism. We develop algorithms for the specific problem of scheduling a Pipelined Operator Tree in which all operators run in parallel using inter-operator parallelism. Weights associated with nodes and edges represent respectively the cost of operators and communication. Communication cost is incurred only if adjacent operators are assigned different processors. The optimization problem is to assign operators to processors so as to minimize the maximum processor load. We develop two approximation algorithms for this NP-hard problem. The faster algorithm has a performance ratio of 3.56 while the slower algorithm has a ratio of 2.87. 1 Introduction Exploiting parallel execution [DG92, Val93] to speed up database querie...

The tree and tour cover problems on an edge-weighted graph are to compute a minimum weight tree and closed walk, respectively, whose vertices form a vertex cover. Both problems are NP-hard. In this note we give strongly polynomial time, constant factor approximation algorithms for both problems. An interesting feature of our algorithms is how they combine approximations of other problems, namely the weighted vertex cover, traveling salesman, and Steiner tree problems. Our starting point is the weighted vertex cover problem: Given a graph G = (V, E) with weights on the vertices find a minimum weight set of vertices that “cover ” all edges, i.e., a set of vertices V ′ ⊆ V such that for each edge {u, v} ∈ E at least one of u and v belongs to V ′. This problem is NP-hard [GJ], however it can be approximated by a cover whose weight is at most double the optimal weight [Ho, BE] (see [Mo] for

This paper reports work investigating various evolutionary approaches to vertex cover (VC), a well-known NP-Hard optimization problem. Central to each of the algorithms is a novel encoding scheme for VC and related problems that treats each chromosome as a binary decision diagram. As a result, the encoding allows only a (guaranteed optimal) subset of feasible solutions. The encoding also incorporates features of a powerful traditional heuristic for VC that allow initial EA populations to be seeded in known promising regions of the search space. The resulting evolutionary algorithms have displayed exceptionally strong empirical performance on various vertex cover, independent set, and maximum clique problem classes.

This paper compares three different evolutionary algorithms for solving the node covering problem: EA-I relies on the definition of the problem only without using any domain knowledge, while EA-II and EA-III employ extra heuristic knowledge. In theory, it is proven that all three algorithms can find an optimal solution in finite generations and find a feasible solution efficiently; but none of them can find the optimal solution efficiently for all instances of the problem. Through experiments, it is observed that all three algorithms can find a feasible solution efficiently, and the algorithms with extra heuristic knowledge can find better approximation solutions; but none of them can find the optimal solution to the first instance efficiently. This paper shows that heuristic knowledge is helpful for evolutionary algorithms to find good approximation solutions, but it contributes little to search the optimal solution for some instances.

Introduction This course will cover some basic topics in the design and analysis of approximation algorithms. The study of approximation algorithms has developed from the seeming intractability of a number of widely-applicable NP-hard optimization problems. These optimization problems are unlikely to admit efficient (polynomial-time computable) optimal solutions. Consequently, a number of techniques have been designed to provide approximate (near-optimal) solutions that can be obtained efficiently. Of course, we would like to sacrifice as little optimality as possible, while gaining as much as possible in efficiency. Trading off optimality in favor of efficiency is the paradigm of approximation algorithms. In this lecture, we provide a brief overview of notions of complexity, approximations, and polynomialtime transformations among problems. Much of the material covered in this lecture is taken from [CLR90] and an excellent set of lecture notes prepared by Raj

Introduction This lecture looks at some known algorithms for the vertex cover, which is known to be NP-complete and, compares their approximation of the optimal. Definition 1 (Vertex Cover) A vertex cover in an undirected graph G = (V; E) is the set of vertices C such that each edge of G has at least one endpoint in C. ffl Instance: G = (V; E). ffl Solutions: Set of vertices that cover every edge. ffl Value: j V 0 j, the number of vertices in the vertex cover V 0 and, the goal is to minimize it. 2 Approximating Vertex Cover 2.1 Greedy Algorithm One The most natural heuristic is a greedy algorithm which repeatedly picks an edge that has not yet bee