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A family of random trees with random edge lengths

1999
*
Random structures & algorithms (Print)
*

We i n troduce

doi:10.1002/(sici)1098-2418(199909)15:2<176::aid-rsa4>3.0.co;2-4
fatcat:yju3chr5tfgv7jl44jk63luyzi
*a**family**of*probability distributions on the space*of**trees**with*I labeled vertices and possibly extra unlabeled vertices*of*degree 3, whose*edges*have positive real*lengths*. ... An interpretation is given in terms*of*sampling from the inhomogeneous continuum*random**tree**of*Aldous and Pitman 1998. ... The purpose*of*this paper is to describe*a*new model*of**random**trees**with**edge**lengths*. ...##
###
The Continuum Random Tree III

1993
*
Annals of Probability
*

Then under an extra condition, this

doi:10.1214/aop/1176989404
fatcat:4oyqnmkw5nfbvivyfuipdchaie
*family*determines*a**random*continuum*tree*?/, which it is convenient to represent as*a**random*subset*of*11. ... Let (W(k), k 2 1) be*random**trees**with*k leaves, satisfying*a*consistency condition: Removing*a**random*leaf from R(k) gives R(k -1). ... Formally,*a**tree**with*n vertices (and hence n -1*edges*) could be represented as*a*point (1) t = (F; x1,. ., xn-1) E Tn x Rn-l where ti is the corresponding*tree*without*edge**lengths*, and t*edge**lengths*...##
###
Unidentifiable divergence times in rates-across-sites models

2004
*
IEEE/ACM Transactions on Computational Biology & Bioinformatics
*

We consider an example

doi:10.1109/tcbb.2004.34
pmid:17048388
fatcat:3pjbnutje5hjbiy473xwefbiyi
*of**a*clock-like*tree**with*three taxa, one unknown*edge**length*,*a*known root state, and*a*parametric*family**of*scale factor distributions that contain the gamma*family*. ... This model has the property that, for*a*generic choice*of*unknown*edge**length*and scale factor distribution, there is another*edge**length*and scale factor distribution which generates data*with*exactly ... Acknowledgments The authors thank the anonymous referees for*a*number*of*very helpful suggestions and observations. ...##
###
Percolation-like scaling exponents for minimal paths and trees in the stochastic mean field model

2005
*
Proceedings of the Royal Society A
*

*A*parallel study

*with*

*trees*instead

*of*paths gives scaling exponent β = 2. ... For 0 < ℓ < ∞ and in the n →∞ limit, let δ(ℓ) = 1/n × (maximum number

*of*steps in

*a*path whose average step-

*length*is ≤ℓ). ... For any

*tree*t in the graph,

*with*

*edges*e 1 , . . . , e m , write size(t) for the number

*of*

*edges*

*of*t and

*A*(t) for the average

*edge*-

*length*: size(t) = m

*A*(t) = m −1 e∈t d(e). ...

##
###
Recovering a tree from the lengths of subtrees spanned by a randomly chosen sequence of leaves

2018
*
Advances in Applied Mathematics
*

We show that if T is known

doi:10.1016/j.aam.2018.01.001
pmid:30220760
pmcid:PMC6135540
fatcat:c5x5ymk5z5bspcwpnhyzjckfvm
*a*priori to belong to one*of*various*families**of**edge*-weighted*trees*, then the answer is, "Yes." ... Given an*edge*-weighted*tree*T*with*n leaves, sample the leaves uniformly at*random*without replacement and let W k , 2 ≤ k ≤ n, be the*length**of*the subtree spanned by the first k leaves. ... The last*family**of**edge*-weighted*trees**with*general*edge*-weights whose elements we can identify up to isomorphism from the joint probability distributions*of*their*random**length*sequences is the class ...##
###
Unidentifiable divergence times in rates-across-sites models
[article]

2004
*
arXiv
*
pre-print

We consider an example

arXiv:q-bio/0408011v2
fatcat:azedgv6ykjfelehsgckxatqbka
*of**a*clock-like*tree**with*three taxa, one unknown*edge**length*, and*a*parametric*family**of*scale factor distributions that contain the gamma*family*. ... This model has the property that, for*a*generic choice*of*unknown*edge**length*and scale factor distribution, there is another*edge**length*and scale factor distribution which generates data*with*exactly ... Acknowledgments The authors thank the anonymous referees for*a*number*of*very helpful suggestions and observations. ...##
###
First Passage Percolation on Random Geometric Graphs and an Application to Shortest-Path Trees

2015
*
Advances in Applied Probability
*

*length*

*of*the longest branch in the shortest-path

*tree*extracted from

*a*typical segment system if the intensity

*of*network stations converges to 0. ... We consider Euclidean first passage percolation on

*a*large

*family*

*of*connected

*random*geometric graphs in the d-dimensional Euclidean space encompassing various well-known models from stochastic geometry ... above by total

*edge*

*length*inside certain bad clusters G =

*random*geometric graph in R 2 as above G * = Palm version

*of*G informally: shifting o to

*random*location on the

*edge*set

*of*G Description

*of*...

##
###
Recovering a tree from the lengths of subtrees spanned by a randomly chosen sequence of leaves
[article]

2015
*
arXiv
*
pre-print

We show that if T is known

arXiv:1506.01091v1
fatcat:x7mh2kw6p5harcyikhxgucufae
*a*priori to belong to one*of*various*families**of**edge*-weighted*trees*, then the answer is, "Yes." ... Given an*edge*-weighted*tree*T*with*n leaves, sample the leaves uniformly at*random*without replacement and let W_k, 2 < k < n, be the*length**of*the subtree spanned by the first k leaves. ... The last*family**of**edge*-weighted*trees**with*general*edge*-weights whose elements we can identify up to isomorphism from the joint probability distributions*of*their*random**length*sequences is the class ...##
###
Color-coding

1994
*
Proceedings of the twenty-sixth annual ACM symposium on Theory of computing - STOC '94
*

We describe

doi:10.1145/195058.195179
dblp:conf/stoc/AlonYZ94
fatcat:7escc6ce2nfmtowkqhrauzn3ae
*a*novel*randomized*method, the method*of*color-coding for finding simple paths and cycles*of**a*specified*length*k, and other small subgraphs, within*a*given graph G = (V, E). ... The*randomized*algorithms obtained using this method can be derandomized using*families**of*perfect hash functions. ... Definition 6.2*A**tree*-decomposition*of**a*graph G = (V, E) is*a*pair (X, T ) where T = (I, F ) is*a**tree*and X = {X i : i ∈ I} is*a**family**of*subsets*of*V such that (i) ∪ i∈I X i = V ; (ii) for every*edge*...##
###
Right-angled hexagon tilings of the hyperbolic plane
[article]

2015
*
arXiv
*
pre-print

We prove that, for each measure μ in

arXiv:1503.05510v1
fatcat:or3spjzauvbjpn6varq2htb5em
*a*certain natural*family**of*measures on right-angled hexagons, there is an isometry-invariant measure on Ω whose marginal distribution on tiles is μ. ... We study isometry-invariant probability measures on the space Ω*of*tilings*of*the hyperbolic plane*with*right-angled hexagons*of*varying shapes. ... Then, for each vertex*of*the*tree*, use the three adjacent*random*variables as*lengths**of**edges*1, 3, 5*of**a*RAH. ...##
###
On the Hyperbolicity of Small-World and Tree-Like Random Graphs
[chapter]

2012
*
Lecture Notes in Computer Science
*

may easily destroy the hyperbolicity

doi:10.1007/978-3-642-35261-4_31
fatcat:v4o3zxazprd6dgc3tmmkcvyjum
*of*the graphs, except for*a*class*of**random**edges*added using an exponentially decaying probability function based on the ring distance among the leaves. ... On the other hand, for*a*class*of**tree*-like graphs called ringed*trees*that have constant hyperbolicity, adding*random*links among the leaves in*a*manner similar to the small-world graph constructions ... The second*family**of*graphs is*random*ringed*trees*.*A*ringed*tree*is*a*binary*tree**with*nodes in each level*of*the*tree*connected by*a*ring (Figure 1(d) ). ...##
###
On the Hyperbolicity of Small-World and Treelike Random Graphs

2013
*
Internet Mathematics
*

may easily destroy the hyperbolicity

doi:10.1080/15427951.2013.828336
fatcat:nulyjmpdnfcn7iu5bdqbpxpqtm
*of*the graphs, except for*a*class*of**random**edges*added using an exponentially decaying probability function based on the ring distance among the leaves. ... On the other hand, for*a*class*of**tree*-like graphs called ringed*trees*that have constant hyperbolicity, adding*random*links among the leaves in*a*manner similar to the small-world graph constructions ... The second*family**of*graphs is*random*ringed*trees*.*A*ringed*tree*is*a*binary*tree**with*nodes in each level*of*the*tree*connected by*a*ring (Figure 1(d) ). ...##
###
Color-coding

1995
*
Journal of the ACM
*

We describe

doi:10.1145/210332.210337
fatcat:xcxlixjrcbbrxkgn2lamsijyke
*a*novel*randomized*method, the method*of*color-coding for finding simple paths and cycles*of**a*specified*length*k, and other small subgraphs, within*a*given graph G = (V, E). ... The*randomized*algorithms obtained using this method can be derandomized using*families**of*perfect hash functions. ... Definition 6.2*A**tree*-decomposition*of**a*graph G = (V, E) is*a*pair (X, T ) where T = (I, F ) is*a**tree*and X = {X i : i ∈ I} is*a**family**of*subsets*of*V such that (i) ∪ i∈I X i = V ; (ii) for every*edge*...##
###
Embedding spanning trees in random graphs
[article]

2010
*
arXiv
*
pre-print

We prove that if T is

arXiv:1007.2326v2
fatcat:vwinzoc4inbuxhorazvjifuvzq
*a**tree*on n vertices wih maximum degree D and the*edge*probability p(n) satisfies: np>c*maxD*logn,n^ϵ for some constant ϵ>0, then*with*high probability the*random*graph G(n,p) contains ...*a*copy*of*T. ... Let G be*a**random*bipartite graph*with*sides*A*and B, where each pair (*a*, b),*a*∈*A*, b ∈ B, is an*edge**of*G*with*probability p, independently*of*other pairs. ...##
###
Poisson point process limits in size-biased Galton-Watson trees

2000
*
Electronic Journal of Probability
*

Decompose the population at t according to the particles' degree

doi:10.1214/ejp.v5-73
fatcat:xvluendy45d6bb34j3zdn32vya
*of*relationship*with**a*distinguished particle picked purely at*random*from those alive at t. ... Keeping track*of*the times when the different*families*grow out*of*the distinguished line*of*descent and the related*family*sizes at t, we represent this relationship structure as*a*point process in*a*... Let T be the*random**family**tree*obtained by having one*edge*for each particle produced*with*the*length**of*an*edge*being the particle's lifetime. ...
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