10,184 research outputs found
A network model of interpersonal alignment in dialog
In dyadic communication, both interlocutors adapt to each other linguistically, that is, they align interpersonally. In this article, we develop a framework for modeling interpersonal alignment in terms of the structural similarity of the interlocutorsâ dialog lexica. This is done by means of so-called two-layer time-aligned network series, that is, a time-adjusted graph model. The graph model is partitioned into two layers, so that the interlocutorsâ lexica are captured as subgraphs of an encompassing dialog graph. Each constituent network of the series is updated utterance-wise. Thus, both the inherent bipartition of dyadic conversations and their gradual development are modeled. The notion of alignment is then operationalized within a quantitative model of structure formation based on the mutual information of the subgraphs that represent the interlocutorâs dialog lexica. By adapting and further developing several models of complex network theory, we show that dialog lexica evolve as a novel class of graphs that have not been considered before in the area of complex (linguistic) networks. Additionally, we show that our framework allows for classifying dialogs according to their alignment status. To the best of our knowledge, this is the first approach to measuring alignment in communication that explores the similarities of graph-like cognitive representations. Keywords: alignment in communication; structural coupling; linguistic networks; graph distance measures; mutual information of graphs; quantitative network analysi
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
Distance labeling schemes for trees
We consider distance labeling schemes for trees: given a tree with nodes,
label the nodes with binary strings such that, given the labels of any two
nodes, one can determine, by looking only at the labels, the distance in the
tree between the two nodes.
A lower bound by Gavoille et. al. (J. Alg. 2004) and an upper bound by Peleg
(J. Graph Theory 2000) establish that labels must use
bits\footnote{Throughout this paper we use for .}. Gavoille et.
al. (ESA 2001) show that for very small approximate stretch, labels use
bits. Several other papers investigate various
variants such as, for example, small distances in trees (Alstrup et. al.,
SODA'03).
We improve the known upper and lower bounds of exact distance labeling by
showing that bits are needed and that bits are sufficient. We also give ()-stretch labeling
schemes using bits for constant .
()-stretch labeling schemes with polylogarithmic label size have
previously been established for doubling dimension graphs by Talwar (STOC
2004).
In addition, we present matching upper and lower bounds for distance labeling
for caterpillars, showing that labels must have size . For simple paths with nodes and edge weights in , we show that
labels must have size
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
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