scholarly journals Random Horn formulas and propagation connectivity for directed hypergraphs

2012 ◽  
Vol Vol. 14 no. 2 (Combinatorics) ◽  
Author(s):  
Robert H. Sloan ◽  
Despina Stasi ◽  
György Turán
Keyword(s):  
High Probability ◽  
Forward Chaining ◽  
Horn Formulas ◽  
International Audience ◽  
Directed Hypergraphs

Combinatorics International audience We consider the property that in a random definite Horn formula of size-3 clauses over n variables, where every such clause is included with probability p, there is a pair of variables for which forward chaining produces all other variables. We show that with high probability the property does not hold for p <= 1/(11n ln n), and does hold for p >= (5 1n ln n)/(n ln n).

scholarly journals VC-dimensions of random function classes

2008 ◽  
Vol Vol. 10 no. 1 (Combinatorics) ◽  
Author(s):  
Bernard Ycart ◽  
Joel Ratsaby
Keyword(s):  
Random Function ◽  
Classical Result ◽  
High Probability ◽  
Necessary Condition ◽  
Vc Dimension ◽  
Uniform Sampling ◽  
Function Classes ◽  
International Audience ◽  
Asymptotic Probability ◽  
Binary Functions

Combinatorics International audience For any class of binary functions on [n]={1, ..., n} a classical result by Sauer states a sufficient condition for its VC-dimension to be at least d: its cardinality should be at least O(nd-1). A necessary condition is that its cardinality be at least 2d (which is O(1) with respect to n). How does the size of a 'typical' class of VC-dimension d compare to these two extreme thresholds ? To answer this, we consider classes generated randomly by two methods, repeated biased coin flips on the n-dimensional hypercube or uniform sampling over the space of all possible classes of cardinality k on [n]. As it turns out, the typical behavior of such classes is much more similar to the necessary condition; the cardinality k need only be larger than a threshold of 2d for its VC-dimension to be at least d with high probability. If its expected size is greater than a threshold of O(&log;n) (which is still significantly smaller than the sufficient size of O(nd-1)) then it shatters every set of size d with high probability. The behavior in the neighborhood of these thresholds is described by the asymptotic probability distribution of the VC-dimension and of the largest d such that all sets of size d are shattered.

scholarly journals Hydras: Directed hypergraphs and Horn formulas

2017 ◽  
Vol 658 ◽  
pp. 417-428 ◽  
Author(s):  
Robert H. Sloan ◽  
Despina Stasi ◽  
György Turán
Keyword(s):  
Horn Formulas ◽  
Directed Hypergraphs

scholarly journals Blocks in Constrained Random Graphs with Fixed Average Degree

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Konstantinos Panagiotou
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
High Probability ◽  
Average Degree ◽  
Additional Restriction ◽  
Structural Constraints ◽  
Analytic Framework ◽  
Discrete Algorithms ◽  
International Audience ◽  
Sharp Concentration

International audience This work is devoted to the study of typical properties of random graphs from classes with structural constraints, like for example planar graphs, with the additional restriction that the average degree is fixed. More precisely, within a general analytic framework, we provide sharp concentration results for the number of blocks (maximal biconnected subgraphs) in a random graph from the class in question. Among other results, we discover that essentially such a random graph belongs with high probability to only one of two possible types: it either has blocks of at most logarithmic size, or there is a \emphgiant block that contains linearly many vertices, and all other blocks are significantly smaller. This extends and generalizes the results in the previous work [K. Panagiotou and A. Steger. Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '09), pp. 432-440, 2009], where similar statements were shown without the restriction on the average degree.

scholarly journals Analysis of biclusters with applications to gene expression data

2005 ◽  
Vol DMTCS Proceedings vol. AD,... (Proceedings) ◽  
Author(s):  
Gahyun Park ◽  
Wojciech Szpankowski
Keyword(s):  
Gene Expression ◽  
Gene Expression Data ◽  
High Probability ◽  
Bipartite Graphs ◽  
Finite Alphabet ◽  
Expression Data ◽  
Original Matrix ◽  
International Audience ◽  
String Pattern ◽  
Clique Problem

International audience For a given matrix of size $n \times m$ over a finite alphabet $\mathcal{A}$, a bicluster is a submatrix composed of selected columns and rows satisfying a certain property. In microarrays analysis one searches for largest biclusters in which selected rows constitute the same string (pattern); in another formulation of the problem one tries to find a maximally dense submatrix. In a conceptually similar problem, namely the bipartite clique problem on graphs, one looks for the largest binary submatrix with all '1'. In this paper, we assume that the original matrix is generated by a memoryless source over a finite alphabet $\mathcal{A}$. We first consider the case where the selected biclusters are square submatrices and prove that with high probability (whp) the largest (square) bicluster having the same row-pattern is of size $\log_Q^2 n m$ where $Q^{-1}$ is the (largest) probability of a symbol. We observe, however, that when we consider $\textit{any}$ submatrices (not just $\textit{square}$ submatrices), then the largest area of a bicluster jumps to $A_n$ (whp) where $A$ is an explicitly computable constant. These findings complete some recent results concerning maximal biclusters and maximum balanced bicliques for random bipartite graphs.

scholarly journals Near―perfect non-crossing harmonic matchings in randomly labeled points on a circle

2005 ◽  
Vol DMTCS Proceedings vol. AD,... (Proceedings) ◽  
Author(s):  
József Balogh ◽  
Boris Pittel ◽  
Gelasio Salazar
Keyword(s):  
High Probability ◽  
Perfect Matching ◽  
Matching Algorithm ◽  
Convex Position ◽  
International Audience

International audience Consider a set $S$ of points in the plane in convex position, where each point has an integer label from $\{0,1,\ldots,n-1\}$. This naturally induces a labeling of the edges: each edge $(i,j)$ is assigned label $i+j$, modulo $n$. We propose the algorithms for finding large non―crossing $\textit{harmonic}$ matchings or paths, i. e. the matchings or paths in which no two edges have the same label. When the point labels are chosen uniformly at random, and independently of each other, our matching algorithm with high probability (w.h.p.) delivers a nearly―perfect matching, a matching of size $n/2 - O(n^{1/3}\ln n)$.

scholarly journals Synchronizing random automata

2010 ◽  
Vol Vol. 12 no. 4 ◽  
Author(s):  
Evgeny Skvortsov ◽  
Yulia Zaks
Keyword(s):  
High Probability ◽  
Special Issue ◽  
International Audience ◽  
Synchronizing Automaton ◽  
Random Automaton ◽  
Random Automata

special issue dedicated to the second edition of the conference AutoMathA: from Mathematics to Applications International audience Conjecture that any synchronizing automaton with n states has a reset word of length (n - 1)(2) was made by. Cerny in 1964. Notwithstanding the numerous attempts made by various researchers this conjecture hasn't been definitively proven yet. In this paper we study a random automaton that is sampled uniformly at random from the set of all automata with n states and m(n) letters. We show that for m(n) > 18 ln n any random automaton is synchronizing with high probability. For m(n) > n(beta), beta > 1/2 we also show that any random automaton with high probability satisfies the. Cerny conjecture.

scholarly journals Cover time of a random graph with given degree sequence

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Mohammed Abdullah ◽  
Colin Cooper ◽  
Alan Frieze
Keyword(s):  
Random Graph ◽  
High Probability ◽  
Minimum Degree ◽  
Degree Sequence ◽  
Average Degree ◽  
Cover Time ◽  
Vertex Set ◽  
International Audience

International audience In this paper we establish the cover time of a random graph $G(\textbf{d})$ chosen uniformly at random from the set of graphs with vertex set $[n]$ and degree sequence $\textbf{d}$. We show that under certain restrictions on $\textbf{d}$, the cover time of $G(\textbf{d})$ is with high probability asymptotic to $\frac{d-1}{ d-2} \frac{\theta}{ d}n \log n$. Here $\theta$ is the average degree and $d$ is the $\textit{effective minimum degree}$. The effective minimum degree is the first entry in the sorted degree sequence which occurs order $n$ times.

scholarly journals Finding hidden cliques in linear time

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Uriel Feige ◽  
Dorit Ron
Keyword(s):  
High Probability ◽  
Linear Time ◽  
Maximum Clique ◽  
Spectral Approach ◽  
Input Graph ◽  
Random Choice ◽  
Spectral Techniques ◽  
Vertex Graph ◽  
International Audience ◽  
Linear Time Algorithms

International audience In the hidden clique problem, one needs to find the maximum clique in an $n$-vertex graph that has a clique of size $k$ but is otherwise random. An algorithm of Alon, Krivelevich and Sudakov that is based on spectral techniques is known to solve this problem (with high probability over the random choice of input graph) when $k \geq c \sqrt{n}$ for a sufficiently large constant $c$. In this manuscript we present a new algorithm for finding hidden cliques. It too provably works when $k > c \sqrt{n}$ for a sufficiently large constant $c$. However, our algorithm has the advantage of being much simpler (no use of spectral techniques), running faster (linear time), and experiments show that the leading constant $c$ is smaller than in the spectral approach. We also present linear time algorithms that experimentally find even smaller hidden cliques, though it remains open whether any of these algorithms finds hidden cliques of size $o(\sqrt{n})$.

scholarly journals Permutations with short monotone subsequences

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Dan Romik
Keyword(s):  
High Probability ◽  
Young Tableaux ◽  
Limit Shape ◽  
Simple Description ◽  
Monotone Subsequence ◽  
International Audience

International audience We consider permutations of $1,2,...,n^2$ whose longest monotone subsequence is of length $n$ and are therefore extremal for the Erdős-Szekeres Theorem. Such permutations correspond via the Robinson-Schensted correspondence to pairs of square $n \times n$ Young tableaux. We show that all the bumping sequences are constant and therefore these permutations have a simple description in terms of the pair of square tableaux. We deduce a limit shape result for the plot of values of the typical such permutation, which in particular implies that the first value taken by such a permutation is with high probability $(1+o(1))n^2/2$.

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