1,969 research outputs found
Looking for vertex number one
Given an instance of the preferential attachment graph , we
would like to find vertex 1, using only 'local' information about the graph;
that is, by exploring the neighborhoods of small sets of vertices. Borgs et. al
gave an an algorithm which runs in time , which is local in the
sense that at each step, it needs only to search the neighborhood of a set of
vertices of size . We give an algorithm to find vertex 1, which
w.h.p. runs in time and which is local in the strongest sense
of operating only on neighborhoods of single vertices. Here
is any function that goes to infinity with .Comment: As accepted for AA
The topology of competitively constructed graphs
We consider a simple game, the -regular graph game, in which players take
turns adding edges to an initially empty graph subject to the constraint that
the degrees of vertices cannot exceed . We show a sharp topological
threshold for this game: for the case a player can ensure the resulting
graph is planar, while for the case , a player can force the appearance of
arbitrarily large clique minors.Comment: 9 pages, 2 figure
On the insertion time of random walk cuckoo hashing
Cuckoo Hashing is a hashing scheme invented by Pagh and Rodler. It uses
distinct hash functions to insert items into the hash table. It has
been an open question for some time as to the expected time for Random Walk
Insertion to add items. We show that if the number of hash functions
is sufficiently large, then the expected insertion time is per item.Comment: 9 page
On a Connectivity Threshold for Colorings of Random Graphs and Hypergraphs
Let Omega_q=Omega_q(H) denote the set of proper [q]-colorings of the hypergraph H. Let Gamma_q be the graph with vertex set Omega_q where two vertices are adjacent iff the corresponding colorings differ in exactly one vertex. We show that if H=H_{n,m;k}, k >= 2, the random k-uniform hypergraph with V=[n] and m=dn/k hyperedges then w.h.p. Gamma_q is connected if d is sufficiently large and q >~ (d/log d)^{1/(k-1)}. This is optimal to the first order in d. Furthermore, with a few more colors, we find that the diameter of Gamma_q is O(n) w.h.p, where the hidden constant depends on d. So, with this choice of d,q, the natural Glauber Dynamics Markov Chain on Omega_q is ergodic w.h.p
Long paths in random Apollonian networks
We consider the length of the longest path in a randomly generated
Apollonian Network (ApN) . We show that w.h.p. for any constant
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