434 research outputs found
Robust Coin Flipping
Alice seeks an information-theoretically secure source of private random
data. Unfortunately, she lacks a personal source and must use remote sources
controlled by other parties. Alice wants to simulate a coin flip of specified
bias , as a function of data she receives from sources; she seeks
privacy from any coalition of of them. We show: If , the
bias can be any rational number and nothing else; if , the bias
can be any algebraic number and nothing else. The proof uses projective
varieties, convex geometry, and the probabilistic method. Our results improve
on those laid out by Yao, who asserts one direction of the case in his
seminal paper [Yao82]. We also provide an application to secure multiparty
computation.Comment: 22 pages, 1 figur
Faster Separators for Shallow Minor-Free Graphs via Dynamic Approximate Distance Oracles
Plotkin, Rao, and Smith (SODA'97) showed that any graph with edges and
vertices that excludes as a depth -minor has a
separator of size and that such a separator can be
found in time. A time bound of for
any constant was later given (W., FOCS'11) which is an
improvement for non-sparse graphs. We give three new algorithms. The first has
the same separator size and running time O(\mbox{poly}(h)\ell
m^{1+\epsilon}). This is a significant improvement for small and .
If for an arbitrarily small chosen constant
, we get a time bound of O(\mbox{poly}(h)\ell n^{1+\epsilon}).
The second algorithm achieves the same separator size (with a slightly larger
polynomial dependency on ) and running time O(\mbox{poly}(h)(\sqrt\ell
n^{1+\epsilon} + n^{2+\epsilon}/\ell^{3/2})) when . Our third algorithm has running time
O(\mbox{poly}(h)\sqrt\ell n^{1+\epsilon}) when . It finds a separator of size O(n/\ell) + \tilde
O(\mbox{poly}(h)\ell\sqrt n) which is no worse than previous bounds when
is fixed and . A main tool in obtaining our results
is a novel application of a decremental approximate distance oracle of Roditty
and Zwick.Comment: 16 pages. Full version of the paper that appeared at ICALP'14. Minor
fixes regarding the time bounds such that these bounds hold also for
non-sparse graph
An Efficient Partitioning Oracle for Bounded-Treewidth Graphs
Partitioning oracles were introduced by Hassidim et al. (FOCS 2009) as a
generic tool for constant-time algorithms. For any epsilon > 0, a partitioning
oracle provides query access to a fixed partition of the input bounded-degree
minor-free graph, in which every component has size poly(1/epsilon), and the
number of edges removed is at most epsilon*n, where n is the number of vertices
in the graph.
However, the oracle of Hassidimet al. makes an exponential number of queries
to the input graph to answer every query about the partition. In this paper, we
construct an efficient partitioning oracle for graphs with constant treewidth.
The oracle makes only O(poly(1/epsilon)) queries to the input graph to answer
each query about the partition.
Examples of bounded-treewidth graph classes include k-outerplanar graphs for
fixed k, series-parallel graphs, cactus graphs, and pseudoforests. Our oracle
yields poly(1/epsilon)-time property testing algorithms for membership in these
classes of graphs. Another application of the oracle is a poly(1/epsilon)-time
algorithm that approximates the maximum matching size, the minimum vertex cover
size, and the minimum dominating set size up to an additive epsilon*n in graphs
with bounded treewidth. Finally, the oracle can be used to test in
poly(1/epsilon) time whether the input bounded-treewidth graph is k-colorable
or perfect.Comment: Full version of a paper to appear in RANDOM 201
Linear-Space Approximate Distance Oracles for Planar, Bounded-Genus, and Minor-Free Graphs
A (1 + eps)-approximate distance oracle for a graph is a data structure that
supports approximate point-to-point shortest-path-distance queries. The most
relevant measures for a distance-oracle construction are: space, query time,
and preprocessing time. There are strong distance-oracle constructions known
for planar graphs (Thorup, JACM'04) and, subsequently, minor-excluded graphs
(Abraham and Gavoille, PODC'06). However, these require Omega(eps^{-1} n lg n)
space for n-node graphs. We argue that a very low space requirement is
essential. Since modern computer architectures involve hierarchical memory
(caches, primary memory, secondary memory), a high memory requirement in effect
may greatly increase the actual running time. Moreover, we would like data
structures that can be deployed on small mobile devices, such as handhelds,
which have relatively small primary memory. In this paper, for planar graphs,
bounded-genus graphs, and minor-excluded graphs we give distance-oracle
constructions that require only O(n) space. The big O hides only a fixed
constant, independent of \epsilon and independent of genus or size of an
excluded minor. The preprocessing times for our distance oracle are also faster
than those for the previously known constructions. For planar graphs, the
preprocessing time is O(n lg^2 n). However, our constructions have slower query
times. For planar graphs, the query time is O(eps^{-2} lg^2 n). For our
linear-space results, we can in fact ensure, for any delta > 0, that the space
required is only 1 + delta times the space required just to represent the graph
itself
What does the local structure of a planar graph tell us about its global structure?
The local k-neighborhood of a vertex v in an unweighted graph G = (V,E) with vertex set V and edge set E is the subgraph induced by all vertices of distance at most k from v. The rooted k-neighborhood of v is also called a k-disk around vertex v. If a graph has maximum degree bounded by a constant d, and k is also constant, the number of isomorphism classes of k-disks is constant as well. We can describe the local structure of a bounded-degree graph G by counting the number of isomorphic copies in G of each possible k-disk. We can summarize this information in form of a vector that has an entry for each isomorphism class of k-disks. The value of the entry is the number of isomorphic copies of the corresponding k-disk in G. We call this vector frequency vector of k-disks. If we only know this vector, what does it tell us about the structure of G?
In this paper we will survey a series of papers in the area of Property Testing that leads to the following result (stated informally): There is a k = k(ε,d) such that for any planar graph G its local structure (described by the frequency vector of k-disks) determines G up to insertion and deletion of at most εd n edges (and relabelling of vertices)
Small grid embeddings of 3-polytopes
We introduce an algorithm that embeds a given 3-connected planar graph as a
convex 3-polytope with integer coordinates. The size of the coordinates is
bounded by . If the graph contains a triangle we can
bound the integer coordinates by . If the graph contains a
quadrilateral we can bound the integer coordinates by . The
crucial part of the algorithm is to find a convex plane embedding whose edges
can be weighted such that the sum of the weighted edges, seen as vectors,
cancel at every point. It is well known that this can be guaranteed for the
interior vertices by applying a technique of Tutte. We show how to extend
Tutte's ideas to construct a plane embedding where the weighted vector sums
cancel also on the vertices of the boundary face
Valiant's model: from exponential sums to exponential products
12 pagesWe study the power of big products for computing multivariate polynomials in a Valiant-like framework. More precisely, we define a new class \vpip as the set of families of polynomials that are exponential products of easily computable polynomials. We investigate the consequences of the hypothesis that these big products are themselves easily computable. For instance, this hypothesis would imply that the nonuniform versions of P and NP coincide. Our main result relates this hypothesis to Blum, Shub and Smale's algebraic version of P versus NP. Let be a field of characteristic 0. Roughly speaking, we show that in order to separate \p_K from \np_K using a problem from a fairly large class of ``simple'' problems, one should first be able to show that exponential products are not easily computable. The class of ``simple'' problems under consideration is the class of NP problems in the structure , in which multiplication is not allowed
A separator theorem for string graphs and its applications
A string graph is the intersection graph of a collection of continuous arcs in the plane. We show that any string graph with in edges can be separated into two parts of roughly equal size by the removal of O(m(3/4)root log m) vertices. This result is then used to deduce that every string graph with n vertices and no complete bipartite subgraph K-t,K-t has at most c(t)n edges, where c(t) is a constant depending only on t. Another application shows that locally tree-like string graphs are globally tree-like: for any epsilon > 0, there is an integer g(epsilon) such that every string graph with n vertices and girth at least g(epsilon) has at most (1 + epsilon)n edges. Furthermore, the number of such labelled graphs is at most (1 + epsilon)(n) T(n), where T(n) = n(n-2) is the number of labelled trees on n vertices
Engineering Planar Separator Algorithms
We consider classical linear-time planar separator
algorithms, determining for a given planar graph a
small subset of the nodes whose removal separates the
graph into two components of similar size. These algorithms
are based upon Planar Separator Theorems, which
guarantee separators of size asymptotically in the
square root of the number of nodes n and remaining
components of size less than 2n/3. In this work, we
present a comprehensive experimental study of the
algorithms applied to a large variety of graphs, where
the main goal is to find separators that do not only
satisfy upper bounds but also possess other desirable
qualities with respect to separator size and component
balance. We propose the usage of fundamental cycles,
whose size is at most twice the diameter of the graph, as planar
separators: For graphs of small diameter the
guaranteed bound is better than the bounds of the classical
algorithms, and it turns out that this simple strategy almost
always outperforms the other algorithms, even for graphs with
large diameter
Do Israelis understand the Hebrew bible?
The Hebrew Bible should be taught like a foreign language in Israel too, argues Ghil'ad Zuckermann, inter alia endorsing Avraham Ahuvia’s recently-launched translation of the Old Testament into what Zuckermann calls high-register 'Israeli'. According to Zuckermann, Tanakh RAM fulfills the mission of 'red 'el ha'am' not only in its Hebrew meaning (Go down to the people) but also – more importantly – in its Yiddish meaning ('red' meaning 'speak!', as opposed to its colorful communist sense). Ahuvia's translation is most useful and dignified. Given its high register, however, Zuckermann predicts that the future promises consequent translations into more colloquial forms of Israeli, a beautifully multi-layered and intricately multi-sourced language, of which to be proud.Ghil'ad Zuckerman
- …