32,403 research outputs found
Extended Learning Graphs for Triangle Finding
We present new quantum algorithms for Triangle Finding improving its best previously known quantum query complexities for both dense and sparse instances. For dense graphs on n vertices, we get a query complexity of O(n^(5/4)) without any of the extra logarithmic factors present in the previous algorithm of Le Gall [FOCS\u2714]. For sparse graphs with m >= n^(5/4) edges, we get a query complexity of O(n^(11/12) m^(1/6) sqrt(log n)), which is better than the one obtained by Le Gall and Nakajima [ISAAC\u2715] when m >= n^(3/2). We also obtain an algorithm with query complexity O(n^(5/6) (m log n)^(1/6) + d_2 sqrt(n)) where d_2 is the variance of the degree distribution.
Our algorithms are designed and analyzed in a new model of learning graphs that we call extended learning graphs. In addition, we present a framework in order to easily combine and analyze them. As a consequence we get much simpler algorithms and analyses than previous algorithms of Le Gall based on the MNRS quantum walk framework [SICOMP\u2711]
Improved Quantum Algorithm for Triangle Finding via Combinatorial Arguments
In this paper we present a quantum algorithm solving the triangle finding
problem in unweighted graphs with query complexity , where
denotes the number of vertices in the graph. This improves the previous
upper bound recently obtained by Lee, Magniez and
Santha. Our result shows, for the first time, that in the quantum query
complexity setting unweighted triangle finding is easier than its edge-weighted
version, since for finding an edge-weighted triangle Belovs and Rosmanis proved
that any quantum algorithm requires queries.
Our result also illustrates some limitations of the non-adaptive learning graph
approach used to obtain the previous upper bound since, even over
unweighted graphs, any quantum algorithm for triangle finding obtained using
this approach requires queries as well. To
bypass the obstacles characterized by these lower bounds, our quantum algorithm
uses combinatorial ideas exploiting the graph-theoretic properties of triangle
finding, which cannot be used when considering edge-weighted graphs or the
non-adaptive learning graph approach.Comment: 17 pages, to appear in FOCS'14; v2: minor correction
On the maximal number of real embeddings of minimally rigid graphs in , and
Rigidity theory studies the properties of graphs that can have rigid
embeddings in a euclidean space or on a sphere and which in
addition satisfy certain edge length constraints. One of the major open
problems in this field is to determine lower and upper bounds on the number of
realizations with respect to a given number of vertices. This problem is
closely related to the classification of rigid graphs according to their
maximal number of real embeddings.
In this paper, we are interested in finding edge lengths that can maximize
the number of real embeddings of minimally rigid graphs in the plane, space,
and on the sphere. We use algebraic formulations to provide upper bounds. To
find values of the parameters that lead to graphs with a large number of real
realizations, possibly attaining the (algebraic) upper bounds, we use some
standard heuristics and we also develop a new method inspired by coupler
curves. We apply this new method to obtain embeddings in . One of
its main novelties is that it allows us to sample efficiently from a larger
number of parameters by selecting only a subset of them at each iteration.
Our results include a full classification of the 7-vertex graphs according to
their maximal numbers of real embeddings in the cases of the embeddings in
and , while in the case of we achieve this
classification for all 6-vertex graphs. Additionally, by increasing the number
of embeddings of selected graphs, we improve the previously known asymptotic
lower bound on the maximum number of realizations. The methods and the results
concerning the spatial embeddings are part of the proceedings of ISSAC 2018
(Bartzos et al, 2018)
Quantum Algorithms for Finding Constant-sized Sub-hypergraphs
We develop a general framework to construct quantum algorithms that detect if
a -uniform hypergraph given as input contains a sub-hypergraph isomorphic to
a prespecified constant-sized hypergraph. This framework is based on the
concept of nested quantum walks recently proposed by Jeffery, Kothari and
Magniez [SODA'13], and extends the methodology designed by Lee, Magniez and
Santha [SODA'13] for similar problems over graphs. As applications, we obtain a
quantum algorithm for finding a -clique in a -uniform hypergraph on
vertices with query complexity , and a quantum algorithm for
determining if a ternary operator over a set of size is associative with
query complexity .Comment: 18 pages; v2: changed title, added more backgrounds to the
introduction, added another applicatio
Mathematics, understanding the score : improving practice in mathematics teaching at secondary level
- …