8,752 research outputs found
Universal Communication, Universal Graphs, and Graph Labeling
We introduce a communication model called universal SMP, in which Alice and Bob receive a function f belonging to a family ?, and inputs x and y. Alice and Bob use shared randomness to send a message to a third party who cannot see f, x, y, or the shared randomness, and must decide f(x,y). Our main application of universal SMP is to relate communication complexity to graph labeling, where the goal is to give a short label to each vertex in a graph, so that adjacency or other functions of two vertices x and y can be determined from the labels ?(x), ?(y). We give a universal SMP protocol using O(k^2) bits of communication for deciding whether two vertices have distance at most k in distributive lattices (generalizing the k-Hamming Distance problem in communication complexity), and explain how this implies a O(k^2 log n) labeling scheme for deciding dist(x,y) ? k on distributive lattices with size n; in contrast, we show that a universal SMP protocol for determining dist(x,y) ? 2 in modular lattices (a superset of distributive lattices) has super-constant ?(n^{1/4}) communication cost. On the other hand, we demonstrate that many graph families known to have efficient adjacency labeling schemes, such as trees, low-arboricity graphs, and planar graphs, admit constant-cost communication protocols for adjacency. Trees also have an O(k) protocol for deciding dist(x,y) ? k and planar graphs have an O(1) protocol for dist(x,y) ? 2, which implies a new O(log n) labeling scheme for the same problem on planar graphs
Near-Optimal Induced Universal Graphs for Bounded Degree Graphs
A graph is an induced universal graph for a family of graphs if every
graph in is a vertex-induced subgraph of . For the family of all
undirected graphs on vertices Alstrup, Kaplan, Thorup, and Zwick [STOC
2015] give an induced universal graph with vertices,
matching a lower bound by Moon [Proc. Glasgow Math. Assoc. 1965].
Let . Improving asymptotically on previous results by
Butler [Graphs and Combinatorics 2009] and Esperet, Arnaud and Ochem [IPL
2008], we give an induced universal graph with vertices for the family of graphs with vertices of maximum degree
. For constant , Butler gives a lower bound of
. For an odd constant , Esperet et al.
and Alon and Capalbo [SODA 2008] give a graph with
vertices. Using their techniques for any
(including constant) even values of gives asymptotically worse bounds than
we present.
For large , i.e. when , the previous best
upper bound was due to Adjiashvili and
Rotbart [ICALP 2014]. We give upper and lower bounds showing that the size is
. Hence the optimal size is
and our construction is within a factor of
from this. The previous results were
larger by at least a factor of .
As a part of the above, proving a conjecture by Esperet et al., we construct
an induced universal graph with vertices for the family of graphs with
max degree . In addition, we give results for acyclic graphs with max degree
and cycle graphs. Our results imply the first labeling schemes that for any
are at most bits from optimal
Sparser Johnson-Lindenstrauss Transforms
We give two different and simple constructions for dimensionality reduction
in via linear mappings that are sparse: only an
-fraction of entries in each column of our embedding matrices
are non-zero to achieve distortion with high probability, while
still achieving the asymptotically optimal number of rows. These are the first
constructions to provide subconstant sparsity for all values of parameters,
improving upon previous works of Achlioptas (JCSS 2003) and Dasgupta, Kumar,
and Sarl\'{o}s (STOC 2010). Such distributions can be used to speed up
applications where dimensionality reduction is used.Comment: v6: journal version, minor changes, added Remark 23; v5: modified
abstract, fixed typos, added open problem section; v4: simplified section 4
by giving 1 analysis that covers both constructions; v3: proof of Theorem 25
in v2 was written incorrectly, now fixed; v2: Added another construction
achieving same upper bound, and added proof of near-tight lower bound for DKS
schem
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