Search results for "combinatoric"
showing 10 items of 1776 documents
Local inhomogeneous weighted summary statistics for marked point processes
2023
We introduce a family of local inhomogeneous mark-weighted summary statistics, of order two and higher, for general marked point processes. Depending on how the involved weight function is specified, these summary statistics capture different kinds of local dependence structures. We first derive some basic properties and show how these new statistical tools can be used to construct most existing summary statistics for (marked) point processes. We then propose a local test of random labelling. This procedure allows us to identify points, and consequently regions, where the random labelling assumption does not hold, e.g.~when the (functional) marks are spatially dependent. Through a simulatio…
Binary jumbled string matching for highly run-length compressible texts
2012
The Binary Jumbled String Matching problem is defined as: Given a string $s$ over $\{a,b\}$ of length $n$ and a query $(x,y)$, with $x,y$ non-negative integers, decide whether $s$ has a substring $t$ with exactly $x$ $a$'s and $y$ $b$'s. Previous solutions created an index of size O(n) in a pre-processing step, which was then used to answer queries in constant time. The fastest algorithms for construction of this index have running time $O(n^2/\log n)$ [Burcsi et al., FUN 2010; Moosa and Rahman, IPL 2010], or $O(n^2/\log^2 n)$ in the word-RAM model [Moosa and Rahman, JDA 2012]. We propose an index constructed directly from the run-length encoding of $s$. The construction time of our index i…
Forrelation
2014
We achieve essentially the largest possible separation between quantum and classical query complexities. We do so using a property-testing problem called Forrelation, where one needs to decide whether one Boolean function is highly correlated with the Fourier transform of a second function. This problem can be solved using 1 quantum query, yet we show that any randomized algorithm needs Ω(√(N)log(N)) queries (improving an Ω(N[superscript 1/4]) lower bound of Aaronson). Conversely, we show that this 1 versus Ω(√(N)) separation is optimal: indeed, any t-query quantum algorithm whatsoever can be simulated by an O(N[superscript 1-1/2t])-query randomized algorithm. Thus, resolving an open questi…
Time and space efficient quantum algorithms for detecting cycles and testing bipartiteness
2016
We study space and time efficient quantum algorithms for two graph problems -- deciding whether an $n$-vertex graph is a forest, and whether it is bipartite. Via a reduction to the s-t connectivity problem, we describe quantum algorithms for deciding both properties in $\tilde{O}(n^{3/2})$ time and using $O(\log n)$ classical and quantum bits of storage in the adjacency matrix model. We then present quantum algorithms for deciding the two properties in the adjacency array model, which run in time $\tilde{O}(n\sqrt{d_m})$ and also require $O(\log n)$ space, where $d_m$ is the maximum degree of any vertex in the input graph.
Completely independent spanning trees in some regular graphs
2014
International audience; Let k >= 2 be an integer and T-1,..., T-k be spanning trees of a graph G. If for any pair of vertices {u, v} of V(G), the paths between u and v in every T-i, 1 <= i <= k, do not contain common edges and common vertices, except the vertices u and v, then T1,... Tk are completely independent spanning trees in G. For 2k-regular graphs which are 2k-connected, such as the Cartesian product of a complete graph of order 2k-1 and a cycle, and some Cartesian products of three cycles (for k = 3), the maximum number of completely independent spanning trees contained in these graphs is determined and it turns out that this maximum is not always k. (C) 2016 Elsevier B.V. All righ…
Almost disjoint spanning trees: relaxing the conditions for completely independent spanning trees
2017
International audience; The search of spanning trees with interesting disjunction properties has led to the introduction of edge-disjoint spanning trees, independent spanning trees and more recently completely independent spanning trees. We group together these notions by dening (i, j)-disjoint spanning trees, where i (j, respectively) is the number of vertices (edges, respectively) that are shared by more than one tree. We illustrate how (i, j)-disjoint spanning trees provide some nuances between the existence of disjoint connected dominating sets and completely independent spanning trees. We prove that determining if there exist two (i, j)-disjoint spanning trees in a graph G is NP-comple…
Grand Dyck paths with air pockets
2022
Grand Dyck paths with air pockets (GDAP) are a generalization of Dyck paths with air pockets by allowing them to go below the $x$-axis. We present enumerative results on GDAP (or their prefixes) subject to various restrictions such as maximal/minimal height, ordinate of the last point and particular first return decomposition. In some special cases we give bijections with other known combinatorial classes.
On building 4-critical plane and projective plane multiwheels from odd wheels
2012
We build unbounded classes of plane and projective plane multiwheels that are 4-critical that are received summing odd wheels as edge sums modulo two. These classes can be considered as ascending from single common graph that can be received as edge sum modulo two of the octahedron graph O and the minimal wheel W3. All graphs of these classes belong to 2n-2-edges-class of graphs, among which are those that quadrangulate projective plane, i.e., graphs from Gr\"otzsch class, received applying Mycielski's Construction to odd cycle.
The General Routing Problem polyhedron: Facets from the RPP and GTSP polyhedra
1998
[EN] In this paper we study the polyhedron associated with the General Routing Problem (GRP). This problem, first introduced by Orloff in 1974, is a generalization of both the Rural Postman Problem (RPP) and the Graphical Traveling Salesman Problem (GTSP) and, thus, is NP -hard. We describe a formulation of the problem such that from every non-trivial facet-inducing inequality for the RPP and GTSP polyhedra, we obtain facet-inducing inequalities for the GRP polyhedron, We describe a new family of facet-inducing inequalities for the GRP, the honeycomb constraints, which seem to be very useful for solving GRP and RPP instances. Finally, new classes of facets obtained by composition of facet-i…
An efficient algorithm for stopping on a sink in a directed graph
2013
Abstract Vertices of an unknown directed graph of order n are revealed one by one in some random permutation. At each point, we know the subgraph induced by the revealed vertices. Our goal is to stop on a sink, a vertex with no out-neighbors. We show that if a sink exists this can be achieved with probability Θ ( 1 / n ) , which is best possible.