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Abstract. Social networks are navigable small worlds, in which two arbitrary people are likely connected by a short path of intermediate friends that can be found by a “decentralized ” routing algorithm using only local information. We develop a model of social networks based on an arbitrary metric space of points, with population density varying across the points. We consider rank-based friendships, where the probability that person u befriends person v is inversely proportional to the number of people who are closer to u than v is. Our main result is that greedy routing can find a short path (of expected polylogarithmic length) from an arbitrary source to a randomly chosen target, independent of the population densities, as long as the doubling dimension of the metric space of locations is low. We also show that greedy routing finds short paths with good probability in tree-based metrics with varying population distributions. 1

A new family of proximity graphs, called region counting graphs (RCG) is presented. The RCG for a finite set of points in the plane uses the notion of region counting distance introduced by Demaine et al. to characterize the proximity between two points p and q: the edge pq is in the RCG if and only if there is less than or exactly k vertices in a given geometric neighborhood defined by a region. These graphs generalize many common proximity graphs, such as k-nearest neighbor graphs, β-skeletons or Θ-graphs. This paper concentrates on RCGs that are invariant under translations, rotations and uniform scaling. For k = 0, we give conditions on regions R that define an RCG to ensure a number of properties including planarity, connectivity, triangle freeness, cycle freeness, bipartiteness, and bounded degree. These conditions take form of what we call tight regions: maximal or minimal regions that a region R must contain or be contained in to satisfy a given monotone property.

We present the first spatially adaptive data structure that answers approximate nearest neighbor (ANN) queries to points that reside in a geometric space of any constant dimension d. The Lt-norm approximation ratio is O(d 1+1/t), and the running time for a query q is O(d 2 lg δ(p, q)), where p is the result of the preceding query and δ(p, q) is the number of input points in a suitably-sized box containing p and q. Our data structure has O(dn) size and requires O(d 2 n lg n) preprocessing time, where n is the number of points in the data structure. The size of the bounding box for δ depends on d, and our results rely on the Random Access Machine (RAM) model with word size Θ(lg n). 1

We propose a definition of locality for properties of geometric graphs. We measure the local density of graphs using region-counting distances [8] between pairs of vertices, and we use this density to define local properties of classes of graphs. We illustrate locality by introducing the local diameter of geometric graphs: we define it as the upper bound on the size of the shortest path between any pair of vertices, expressed as a function of the density of the graph around these vertices. We determine the local diameter of several well-studied graphs such as Θ-graphs, Ordered Θ-graphs and Skip List Spanners. We also show that various operations, such as path and point queries using geometric graphs as data structures, have complexities which can be expressed as local properties. 1

We present a new priority queue data structure, the queap, that executes insertion in O(1) amortized time and extract-min in O(log(k + 2)) amortized time if there are k items that have been in the heap longer than the item to be extracted. Thus if the operations on the queap are rst-in rst-out, as on a queue, each operation will execute in constant time. This idea of trying to make operations on the least recently accessed items fast, which we call the queueish property, is a natural complement to the working set property of certain data structures, such as splay trees and pairing heaps, where operations on the most recently accessed data execute quickly. However, we show that the queueish property is in some sense more dicult than the working set property by demonstrating that it is impossible to create a queueish binary search tree, but that many search data structures can be made almost queueish with a O(log log n) amortized extra cost per operation.

views and conclusions contained in this document are those of the author and should not be interpreted as representing the official policies, either expressed or implied, of any sponsoring institution, the U.S. government or any other entity. Keywords: binary search trees, adaptive algorithms, splay trees, Unified Bound, dynamic A ubiquitous problem in the field of algorithms and data structures is that of searching for an element from an ordered universe. The simple yet powerful binary search tree (BST) model provides a rich family of solutions to this problem. Although BSTs require Ω(lg n) time per operation in the worst case, various adaptive BST algorithms are capable of exploiting patterns in the sequence of queries to achieve tighter, input-sensitive, bounds that can be o(lg n) in many cases. This thesis furthers our understanding of what is achievable in the BST model along two directions. First, we make progress in improving instance-specific lower bounds in the BST model. In particular, we introduce a framework for generating lower bounds on the cost that any BST algorithm must pay to execute a query sequence,

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