The Random Projection Tree (RPTREE) structures proposed in [1] are space partitioning data structures that automatically adapt to various notions of intrinsic dimensionality of data. We prove new results for both the RPTREE-MAX and the RPTREE-MEAN data structures. Our result for RPTREE-MAX gives a nearoptimal bound on the number of levels required by this data structure to reduce the size of it...

A random suffix search tree is a binary search tree constructed for the suffixes Xi = 0.BiBi+1Bi+2 . . . of a sequence B1, B2, B3., . . . of independent identically distributed random b-ary digits Bj . Let Dn denote the depth of the node for Xn in this tree when B1 is uniform on Zb. We show that for any value of b > 1, EDn = 2 log n + O(log log n), just as for the random binary search tree. We ...

We study fragmentation of a random recursive tree into a forest by repeated removal of nodes. The initial tree consists of N nodes and it is generated by sequential addition of nodes with each new node attaching to a randomly-selected existing node. As nodes are removed from the tree, one at a time, the tree dissolves into an ensemble of separate trees, namely, a forest. We study statistical pr...

2 Binary search trees 2 2.1 Definition of a binary search tree . . . . . . . . . . . . . . . . . . 2 2.2 Profile of a binary search tree . . . . . . . . . . . . . . . . . . . . 3 2.2.1 Level polynomial. BST martingale . . . . . . . . . . . . . 3 2.2.2 Embedding in continuous time. Yule tree . . . . . . . . . . 6 2.2.3 Connection Yule tree binary search tree . . . . . . . . . . 7 2.2.4 Asymptoti...

These notes provide an elementary and self-contained introduction to branching random walks. Section 1 gives a brief overview of Galton–Watson trees, whereas Section 2 presents the classical law of large numbers for branching random walks. These two short sections are not exactly indispensable, but they introduce the idea of using size-biased trees, thus giving motivations and an avant-goût to ...

We consider unimodular random rooted trees (URTs) and invariant forests in Cayley graphs. We show that URTs of bounded degree are the same as the law of the component of the root in an invariant percolation on a regular tree. We use this to give a new proof that URTs are sofic, a result of Elek. We show that ends of invariant forests in the hyperbolic plane converge to ideal boundary points. We...

This paper describes a probabilistic algorithm that, given a connected, undirected graph G with n vertices, produces a spanning tree of G chosen uniformly at random among the spanning trees of G. The expected running time is O(n logn) per generated tree for almost all graphs, and O(n3) for the worst graphs. Previously known deterministic algorithms and much more complicated and require O(n3) ti...

Random trees planning has been used to solve connectivity problems in a variety of problem domains. In particular, these approaches are some of the most successful approaches to robotic motion planning. In this work we examine the broad array of random-tree that have been proposed for motion planning. From this survey we distill a general algorithmic framework for random-tree planning that abst...

We design a compact adaptive tree structure for random access to coherent spatial data. Many papers explore compressed trees, but nearly all involve entropy coding of a sequential traversal, thus precluding random queries required by rendering algorithms. Instead, we use fixed-rate encoding for both the tree topology and its data. Key elements include the replacement of pointers by local offset...