7 research outputs found
Long and short paths in uniform random recursive dags
In a uniform random recursive k-dag, there is a root, 0, and each node in
turn, from 1 to n, chooses k uniform random parents from among the nodes of
smaller index. If S_n is the shortest path distance from node n to the root,
then we determine the constant \sigma such that S_n/log(n) tends to \sigma in
probability as n tends to infinity. We also show that max_{1 \le i \le n}
S_i/log(n) tends to \sigma in probability.Comment: 16 page
On martingale tail sums in affine two-color urn models with multiple drawings
In two recent works, Kuba and Mahmoud (arXiv:1503.090691 and
arXiv:1509.09053) introduced the family of two-color affine balanced Polya urn
schemes with multiple drawings. We show that, in large-index urns (urn index
between and ) and triangular urns, the martingale tail sum for the
number of balls of a given color admits both a Gaussian central limit theorem
as well as a law of the iterated logarithm. The laws of the iterated logarithm
are new even in the standard model when only one ball is drawn from the urn in
each step (except for the classical Polya urn model). Finally, we prove that
the martingale limits exhibit densities (bounded under suitable assumptions)
and exponentially decaying tails. Applications are given in the context of node
degrees in random linear recursive trees and random circuits.Comment: 17 page
Longest path distance in random circuits
We study distance properties of a general class of random directed acyclic
graphs (DAGs). In a DAG, many natural notions of distance are possible, for
there exists multiple paths between pairs of nodes. The distance of interest
for circuits is the maximum length of a path between two nodes. We give laws of
large numbers for the typical depth (distance to the root) and the minimum
depth in a random DAG. This completes the study of natural distances in random
DAGs initiated (in the uniform case) by Devroye and Janson (2009+). We also
obtain large deviation bounds for the minimum of a branching random walk with
constant branching, which can be seen as a simplified version of our main
result.Comment: 21 pages, 2 figure
Random structures for partially ordered sets
This thesis is presented in two parts. In the first part, we study a family of models
of random partial orders, called classical sequential growth models, introduced by
Rideout and Sorkin as possible models of discrete space-time. We analyse a particular
model, called a random binary growth model, and show that the random partial
order produced by this model almost surely has infinite dimension. We also give
estimates on the size of the largest vertex incomparable to a particular element of
the partial order. We show that there is some positive probability that the random
partial order does not contain a particular subposet. This contrasts with other existing
models of partial orders. We also study "continuum limits" of sequences of
classical sequential growth models. We prove results on the structure of these limits
when they exist, highlighting a deficiency of these models as models of space-time.
In the second part of the thesis, we prove some correlation inequalities for mappings
of rooted trees into complete trees. For T a rooted tree we can define the proportion
of the total number of embeddings of T into a complete binary tree that map the
root of T to the root of the complete binary tree. A theorem of Kubicki, Lehel and
Morayne states that, for two binary trees with one a subposet of the other, this
proportion is larger for the larger tree. They conjecture that the same is true for
two arbitrary trees with one a subposet of the other. We disprove this conjecture
by analysing the asymptotics of this proportion for large complete binary trees.
We show that the theorem of Kubicki, Lehel and Morayne can be thought of as a
correlation inequality which enables us to generalise their result in other directions
On the Depth of Randomly Generated Circuits
This research is motivated by the Circuit Value Problem; this problem is well known to be inherently sequential. We consider Boolean circuits with descriptions length d that consist of gates with a fixed fan-in f and a constant number of inputs. Assuming uniform distribution of descriptions, we show that such a circuit has expected depth O(log d). This improves on the best known result. More precisely, we prove for circuits of size n their depth is asymptotically ef ln n with extremely high probability. Our proof uses the coupling technique to bound circuit depth from above and below by those of two alternative discrete-time processes. We are able to establish the result by embedding the processes in suitable continuous-time branching processes. As a simple consequence of our result we obtain that monotone CVP is in the class average NC. Key Words: random circuits, depth, recursive trees, domination by coupling, continuous Poisson process. 1 The Problem and Motivation A circuit is a ..
On the depth of randomly generated circuits
This research is motivated by the Circuit Value Problem; this problem is well known to be inherently sequential. We consider Boolean Circuits with descriptions of length d that consist of gates with a fixed fan-in f and a constant number of inputs. Assuming uniform distribution of descriptions, we show that such a circuit has expected depth O(log d). This improves on the best known result. More precisely, we prove for circuits of size n their depth is asymptotically ef ln n with extremely high probability. Our proof uses the coupling technique to bound circuit depth from above and below by those of two alternative discrete time processes. We are able to establish the result by embedding the processes in suitable continuous time branching processes. As a simple consequence of our result we obtain that monotone CVP is in the class average NCPostprint (author's final draft
On the depth of randomly generated circuits
This research is motivated by the Circuit Value Problem; this problem is well known to be inherently sequential. We consider Boolean Circuits with descriptions of length d that consist of gates with a fixed fan-in f and a constant number of inputs. Assuming uniform distribution of descriptions, we show that such a circuit has expected depth O(log d). This improves on the best known result. More precisely, we prove for circuits of size n their depth is asymptotically ef ln n with extremely high probability. Our proof uses the coupling technique to bound circuit depth from above and below by those of two alternative discrete time processes. We are able to establish the result by embedding the processes in suitable continuous time branching processes. As a simple consequence of our result we obtain that monotone CVP is in the class average N