In this paper we study the time required for a @math -biased ( @math ) walk to visit all nodes of a supercritical Galton-Watson tree up to generation @math . Inspired by the extremal landscape approach in [ 2018] for simple random walk on binary trees, we establish the near-independent nature of extremal points for the @math -biased walk, and deduce the scaling limit of the cover time.
Particularly in @cite_6 , the authors noticed a clustering extremal landscape (Theorem 5.1, @cite_6 ): at a suitable time, if two nodes with low local time share the same ancestor in a generation of order @math , then (with high probability) they have the same ancestors all the way until a generation of order @math . This inspired us to look for properties of similar style in the biased case, leading to the key observation in our proof (Lemma ) that non-visited nodes (up to a suitable time, with high probability) never share ancestors after generations of order @math .
@cite_6
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Bootstrap percolation on an arbitrary graph has a random initial configuration, where each vertex is occupied with probability @math , independently of each other, and a deterministic spreading rule with a fixed parameter @math : if a vacant site has at least @math occupied neighbours at a certain time step, then it becomes occupied in the next step. This process is well studied on @math ; here we investigate it on regular and general infinite trees and on non-amenable Cayley graphs. The critical probability is the infimum of those values of @math for which the process achieves complete occupation with positive probability. On trees we find the following discontinuity: if the branching number of a tree is strictly smaller than @math , then the critical probability is 1, while it is @math on the @math -ary tree. A related result is that in any rooted tree @math there is a way of erasing @math children of the root, together with all their descendants, and repeating this for all remaining children, and so on, such that the remaining tree @math has branching number @math . We also prove that on any @math -regular non-amenable graph, the critical probability for the @math -rule is strictly positive. The nearestor near-neighbor query problems arise in a large variety of database applications, usually in the context of similarity searching. Of late, there has been increasing interest in building search index structures for performing similarity search over high-dimensional data, e.g., image databases, document collections, time-series databases, and genome databases. Unfortunately, all known techniques for solving this problem fall prey to the of dimensionality." That is, the data structures scale poorly with data dimensionality; in fact, if the number of dimensions exceeds 10 to 20, searching in k-d trees and related structures involves the inspection of a large fraction of the database, thereby doing no better than brute-force linear search. It has been suggested that since the selection of features and the choice of a distance metric in typical applications is rather heuristic, determining an approximate nearest neighbor should su ce for most practical purposes. In this paper, we examine a novel scheme for approximate similarity search based on hashing. The basic idea is to hash the points Supported by NAVY N00014-96-1-1221 grant and NSF Grant IIS-9811904. Supported by Stanford Graduate Fellowship and NSF NYI Award CCR-9357849. Supported by ARO MURI Grant DAAH04-96-1-0007, NSF Grant IIS-9811904, and NSF Young Investigator Award CCR9357849, with matching funds from IBM, Mitsubishi, Schlumberger Foundation, Shell Foundation, and Xerox Corporation. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the VLDB copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Very Large Data Base Endowment. To copy otherwise, or to republish, requires a fee and or special permission from the Endowment. Proceedings of the 25th VLDB Conference, Edinburgh, Scotland, 1999. from the database so as to ensure that the probability of collision is much higher for objects that are close to each other than for those that are far apart. We provide experimental evidence that our method gives signi cant improvement in running time over other methods for searching in highdimensional spaces based on hierarchical tree decomposition. Experimental results also indicate that our scheme scales well even for a relatively large number of dimensions (more than 50). Consider a uniform random rooted tree on vertices labelled by @math , with edges directed towards the root. We imagine that each node of the tree has space for a single car to park. A number @math of cars arrive one by one, each at a node chosen independently and uniformly at random. If a car arrives at a space which is already occupied, it follows the unique path oriented towards the root until it encounters an empty space, in which case it parks there; if there is no empty space, it leaves the tree. Consider @math and let @math denote the event that all @math cars find spaces in the tree. Lackner and Panholzer proved (via analytic combinatorics methods) that there is a phase transition in this model. Then if @math , we have @math , whereas if @math we have @math . We give a probabilistic explanation for this phenomenon, and an alternative proof via the objective method. Along the way, we are led to consider the following variant of the problem: take the tree to be the family tree of a Galton-Watson branching process with Poisson(1) offspring distribution, and let an independent Poisson( @math ) number of cars arrive at each vertex. Let @math be the number of cars which visit the root of the tree. Then for @math , we have @math , whereas for @math , we have @math . This discontinuous phase transition turns out to be a generic phenomenon in settings with an arbitrary offspring distribution of mean at least 1 for the tree and arbitrary arrival distribution. We start a systematic study of data structures for the nearest colored node problem on trees. Given a tree with colored nodes and weighted edges, we want to answer queries (v,c) asking for the nearest node to node v that has color c. This is a natural generalization of the well-known nearest marked ancestor problem. We give an O(n)-space O(log log n)-query solution and show that this is optimal. We also consider the dynamic case where updates can change a node's color and show that in O(n) space we can support both updates and queries in O(log n) time. We complement this by showing that O(polylog n) update time implies Omega(log n log log n) query time. Finally, we consider the case where updates can change the edges of the tree (link-cut operations). There is a known (top-tree based) solution that requires update time that is roughly linear in the number of colors. We show that this solution is probably optimal by showing that a strictly sublinear update time implies a strictly subcubic time algorithm for the classical all pairs shortest paths problem on a general graph. We also consider versions where the tree is rooted, and the query asks for the nearest ancestor descendant of node v that has color c, and present efficient data structures for both variants in the static and the dynamic setting.