The Role of Edges and Line-Ends in Illusory Contour Formation

Illusory contours can be induced along directions approximately collinear to edges or approximately perpendicular to the ends of lines. Using a rating scale procedure we explored the relation between the two types of inducers by systematically varying the thickness of inducing elements to result; in varying amounts of "edge-like" or "line-like" induction. Inducers for om illusory figures consisted of concentric rings with arcs missing. Observers judged the clarity and brightness of illusory figures as the number of arcs, their thicknesses, and spacings were parametrically varied. Degree of clarity and amount of induced brightness were both found to be inverted-U functions of the number of arcs. These results mandate that any valid model of illusory contour formation must account for interference effects between parallel lines or between those neural units responsible for completion of boundary signals in directions perpendicular to the ends of thin lines. Line width was found to have an effect on both clarity and brightness, a finding inconsistent with those models which employ only completion perpendicular to inducer orientation.

The Role of Edges and Line-Ends in Illusory Contour Formation

Illusory contours can be induced along direction approximately collinear to edges or approximately perpendicular to the ends of lines. Using a rating scale procedure we explored the relation between the two types of inducers by systematically varying the thickness of inducing elements to result in varying amounts of "edge-like" or "line-like" induction. Inducers for our illusory figures consisted of concentric rings with arcs missing. Observers judged the clarity and brightness of illusory figures as the number of arcs, their thicknesses, and spacings were parametrically varied. Degree of clarity and amount of induced brightness were both found to be inverted-U functions of the number of arcs. These results mandate that any valid model of illusory contour formation must account for interference effects between parallel lines or between those neural units responsible for completion of boundary signals in directions perpendicular to the ends of thin lines. Line width was found to have an efFect on both clarity and brightness, a finding inconsistent with those models which employ only completion perpendicular to inducer orientation.

Learning Unions of Rectangles with Queries

We investigate the efficient learnability of unions of k rectangles in the discrete plane (1,...,n)[2] with equivalence and membership queries. We exhibit a learning algorithm that learns any union of k rectangles with O(k^3log n) queries, while the time complexity of this algorithm is bounded by O(k^5log n). We design our learning algorithm by finding "corners" and "edges" for rectangles contained in the target concept and then constructing the target concept from those "corners" and "edges". Our result provides a first approach to on-line learning of nontrivial subclasses of unions of intersections of halfspaces with equivalence and membership queries.

On trip planning queries in spatial databases

In this paper we discuss a new type of query in Spatial Databases, called Trip Planning Query (TPQ). Given a set of points P in space, where each point belongs to a category, and given two points s and e, TPQ asks for the best trip that starts at s, passes through exactly one point from each category, and ends at e. An example of a TPQ is when a user wants to visit a set of different places and at the same time minimize the total travelling cost, e.g. what is the shortest travelling plan for me to visit an automobile shop, a CVS pharmacy outlet, and a Best Buy shop along my trip from A to B? The trip planning query is an extension of the well-known TSP problem and therefore is NP-hard. The difficulty of this query lies in the existence of multiple choices for each category. In this paper, we first study fast approximation algorithms for the trip planning query in a metric space, assuming that the data set fits in main memory, and give the theory analysis of their approximation bounds. Then, the trip planning query is examined for data sets that do not fit in main memory and must be stored on disk. For the disk-resident data, we consider two cases. In one case, we assume that the points are located in Euclidean space and indexed with an Rtree. In the other case, we consider the problem of points that lie on the edges of a spatial network (e.g. road network) and the distance between two points is defined using the shortest distance over the network. Finally, we give an experimental evaluation of the proposed algorithms using synthetic data sets generated on real road networks.

Inference and Labeling of Metric-Induced Network Topologies

The development and deployment of distributed network-aware applications and services over the Internet require the ability to compile and maintain a model of the underlying network resources with respect to (one or more) characteristic properties of interest. To be manageable, such models must be compact, and must enable a representation of properties along temporal, spatial, and measurement resolution dimensions. In this paper, we propose a general framework for the construction of such metric-induced models using end-to-end measurements. We instantiate our approach using one such property, packet loss rates, and present an analytical framework for the characterization of Internet loss topologies. From the perspective of a server the loss topology is a logical tree rooted at the server with clients at its leaves, in which edges represent lossy paths between a pair of internal network nodes. We show how end-to-end unicast packet probing techniques could b e used to (1) infer a loss topology and (2) identify the loss rates of links in an existing loss topology. Correct, efficient inference of loss topology information enables new techniques for aggregate congestion control, QoS admission control, connection scheduling and mirror site selection. We report on simulation, implementation, and Internet deployment results that show the effectiveness of our approach and its robustness in terms of its accuracy and convergence over a wide range of network conditions.

Sampling Biases in IP Topology Measurements

Considerable attention has been focused on the properties of graphs derived from Internet measurements. Router-level topologies collected via traceroute studies have led some authors to conclude that the router graph of the Internet is a scale-free graph, or more generally a power-law random graph. In such a graph, the degree distribution of nodes follows a distribution with a power-law tail. In this paper we argue that the evidence to date for this conclusion is at best insufficient. We show that graphs appearing to have power-law degree distributions can arise surprisingly easily, when sampling graphs whose true degree distribution is not at all like a power-law. For example, given a classical Erdös-Rényi sparse, random graph, the subgraph formed by a collection of shortest paths from a small set of random sources to a larger set of random destinations can easily appear to show a degree distribution remarkably like a power-law. We explore the reasons for how this effect arises, and show that in such a setting, edges are sampled in a highly biased manner. This insight allows us to distinguish measurements taken from the Erdös-Rényi graphs from those taken from power-law random graphs. When we apply this distinction to a number of well-known datasets, we find that the evidence for sampling bias in these datasets is strong.

Indexing Distances in Large Graphs and Applications in Search Tasks

This thesis elaborates on the problem of preprocessing a large graph so that single-pair shortest-path queries can be answered quickly at runtime. Computing shortest paths is a well studied problem, but exact algorithms do not scale well to real-world huge graphs in applications that require very short response time. The focus is on approximate methods for distance estimation, in particular in landmarks-based distance indexing. This approach involves choosing some nodes as landmarks and computing (offline), for each node in the graph its embedding, i.e., the vector of its distances from all the landmarks. At runtime, when the distance between a pair of nodes is queried, it can be quickly estimated by combining the embeddings of the two nodes. Choosing optimal landmarks is shown to be hard and thus heuristic solutions are employed. Given a budget of memory for the index, which translates directly into a budget of landmarks, different landmark selection strategies can yield dramatically different results in terms of accuracy. A number of simple methods that scale well to large graphs are therefore developed and experimentally compared. The simplest methods choose central nodes of the graph, while the more elaborate ones select central nodes that are also far away from one another. The efficiency of the techniques presented in this thesis is tested experimentally using five different real world graphs with millions of edges; for a given accuracy, they require as much as 250 times less space than the current approach which considers selecting landmarks at random. Finally, they are applied in two important problems arising naturally in large-scale graphs, namely social search and community detection.

Fast Shortest Path Distance Estimation in Large Networks

We study the problem of preprocessing a large graph so that point-to-point shortest-path queries can be answered very fast. Computing shortest paths is a well studied problem, but exact algorithms do not scale to huge graphs encountered on the web, social networks, and other applications. In this paper we focus on approximate methods for distance estimation, in particular using landmark-based distance indexing. This approach involves selecting a subset of nodes as landmarks and computing (offline) the distances from each node in the graph to those landmarks. At runtime, when the distance between a pair of nodes is needed, we can estimate it quickly by combining the precomputed distances of the two nodes to the landmarks. We prove that selecting the optimal set of landmarks is an NP-hard problem, and thus heuristic solutions need to be employed. Given a budget of memory for the index, which translates directly into a budget of landmarks, different landmark selection strategies can yield dramatically different results in terms of accuracy. A number of simple methods that scale well to large graphs are therefore developed and experimentally compared. The simplest methods choose central nodes of the graph, while the more elaborate ones select central nodes that are also far away from one another. The efficiency of the suggested techniques is tested experimentally using five different real world graphs with millions of edges; for a given accuracy, they require as much as 250 times less space than the current approach in the literature which considers selecting landmarks at random. Finally, we study applications of our method in two problems arising naturally in large-scale networks, namely, social search and community detection.

Nearest-neighbor Queries in Probabilistic Graphs

Large probabilistic graphs arise in various domains spanning from social networks to biological and communication networks. An important query in these graphs is the k nearest-neighbor query, which involves finding and reporting the k closest nodes to a specific node. This query assumes the existence of a measure of the "proximity" or the "distance" between any two nodes in the graph. To that end, we propose various novel distance functions that extend well known notions of classical graph theory, such as shortest paths and random walks. We argue that many meaningful distance functions are computationally intractable to compute exactly. Thus, in order to process nearest-neighbor queries, we resort to Monte Carlo sampling and exploit novel graph-transformation ideas and pruning opportunities. In our extensive experimental analysis, we explore the trade-offs of our approximation algorithms and demonstrate that they scale well on real-world probabilistic graphs with tens of millions of edges.

Unicast-based Characterization of Network Loss Topologies

Current Internet transport protocols make end-to-end measurements and maintain per-connection state to regulate the use of shared network resources. When a number of such connections share a common endpoint, that endpoint has the opportunity to correlate these end-to-end measurements to better diagnose and control the use of shared resources. A valuable characterization of such shared resources is the "loss topology". From the perspective of a server with concurrent connections to multiple clients, the loss topology is a logical tree rooted at the server in which edges represent lossy paths between a pair of internal network nodes. We develop an end-to-end unicast packet probing technique and an associated analytical framework to: (1) infer loss topologies, (2) identify loss rates of links in an existing loss topology, and (3) augment a topology to incorporate the arrival of a new connection. Correct, efficient inference of loss topology information enables new techniques for aggregate congestion control, QoS admission control, connection scheduling and mirror site selection. Our extensive simulation results demonstrate that our approach is robust in terms of its accuracy and convergence over a wide range of network conditions.

Implications of Selfish Neighbor Selection in Overlay Networks

In a typical overlay network for routing or content sharing, each node must select a fixed number of immediate overlay neighbors for routing traffic or content queries. A selfish node entering such a network would select neighbors so as to minimize the weighted sum of expected access costs to all its destinations. Previous work on selfish neighbor selection has built intuition with simple models where edges are undirected, access costs are modeled by hop-counts, and nodes have potentially unbounded degrees. However, in practice, important constraints not captured by these models lead to richer games with substantively and fundamentally different outcomes. Our work models neighbor selection as a game involving directed links, constraints on the number of allowed neighbors, and costs reflecting both network latency and node preference. We express a node's "best response" wiring strategy as a k-median problem on asymmetric distance, and use this formulation to obtain pure Nash equilibria. We experimentally examine the properties of such stable wirings on synthetic topologies, as well as on real topologies and maps constructed from PlanetLab and AS-level Internet measurements. Our results indicate that selfish nodes can reap substantial performance benefits when connecting to overlay networks composed of non-selfish nodes. On the other hand, in overlays that are dominated by selfish nodes, the resulting stable wirings are optimized to such great extent that even non-selfish newcomers can extract near-optimal performance through naive wiring strategies.