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Register a new userCovering Paths for Planar Point Sets
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Added by
Adrian Dumitrescu
Publication date
2013
fields
Informatics Engineering
and research's language is
English
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Given $n$ points in the plane, a emph{covering path} is a polygonal path that visits all the points. If no three points are collinear, every covering path requires at least $n/2$ segments, and $n-1$ straight line segments obviously suffice even if the covering path is required to be noncrossing. We show that every set of $n$ points in the plane admits a (possibly self-crossi ng) covering path consisting of $n/2 +O(n/log{n})$ straight line segments. If the path is required to be noncrossing, we prove that $(1-eps)n$ straight line segments suffice for a small constant $eps>0$, and we exhibit $n$-element point sets that require at least $5n/9 -O(1)$ segments in every such path. Further, the analogous question for noncrossing emph{covering trees} is considered and similar bounds are obtained. Finally, it is shown that computing a noncrossing covering path for $n$ points in the plane requires $Omega(n log{n})$ time in the worst case.
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A family $mathcal F$ has covering number $tau$ if the size of the smallest set intersecting all sets from $mathcal F$ is equal to $s$. Let $m(n,k,tau)$ stand for the size of the largest intersecting family $mathcal F$ of $k$-element subsets of ${1,ldots,n}$ with covering number $tau$. It is a classical result of ErdH os and Lovasz that $m(n,k,k)le k^k$ for any $n$. In this short note, we explore the behaviour of $m(n,k,tau)$ for $n<k^2$ and large $tau$. The results are quite surprising: For example, we show that $m(k^{3/2},k,tau) = (1-o(1)){n-1choose k-1}$ for $taule k-k^{3/4+o(1)}$. At the same time, $m(k^{3/2},k,tau)<e^{-ck^{1/2}}{nchoose k}$ if $tau>k-frac 12k^{1/2}$.
For generalized Dyck paths (i.e., directed lattice paths with any finite set of jumps), we analyse their local time at zero (i.e., the number of times the path is touching or crossing the abscissa). As we are in a discrete setting, the event we analyse here is invisible to the tools of Brownian motion theory. It is interesting that the key tool for analysing directed lattice paths, which is the kernel method, is not directly applicable here. Therefore, we introduce a variant of this kernel method to get the trivariate generating function (length, final altitude, local time): this leads to an expression involving symmetric and algebraic functions. We apply this analysis to different types of constrained lattice paths (meanders , excursions, bridges,. . .). Then, we illustrate this approach on basketball walks which are walks defined by the jumps --2, --1, 0, +1, +2. We use singularity analysis to prove that the limit laws for the local time are (depending on the drift and the type of walk) the geometric distribution, the negative binomial distribution, the Rayleigh distribution, or the half-normal distribution (a universal distribution up to now rarely encountered in analytic combinatorics).
Suppose that the vertices of a graph $G$ are colored with two colors in an unknown way. The color that occurs on more than half of the vertices is called the majority color (if it exists), and any vertex of this color is called a majority vertex. We study the problem of finding a majority vertex (or show that none exists) if we can query edges to learn whether their endpoints have the same or different colors. Denote the least number of queries needed in the worst case by $m(G)$. It was shown by Saks and Werman that $m(K_n)=n-b(n)$, where $b(n)$ is the number of 1s in the binary representation of $n$. In this paper, we initiate the study of the problem for general graphs. The obvious bounds for a connected graph $G$ on $n$ vertices are $n-b(n)le m(G)le n-1$. We show that for any tree $T$ on an even number of vertices we have $m(T)=n-1$ and that for any tree $T$ on an odd number of vertices, we have $n-65le m(T)le n-2$. Our proof uses results about the weighted version of the problem for $K_n$, which may be of independent interest. We also exhibit a sequence $G_n$ of graphs with $m(G_n)=n-b(n)$ such that $G_n$ has $O(nb(n))$ edges and $n$ vertices.
It is an open problem whether the 3-coloring problem can be solved in polynomial time in the class of graphs that do not contain an induced path on $t$ vertices, for fixed $t$. We propose an algorithm that, given a 3-colorable graph without an induced path on $t$ vertices, computes a coloring with $max{5,2lceil{frac{t-1}{2}}rceil-2}$ many colors. If the input graph is triangle-free, we only need $max{4,lceil{frac{t-1}{2}}rceil+1}$ many colors. The running time of our algorithm is $O((3^{t-2}+t^2)m+n)$ if the input graph has $n$ vertices and $m$ edges.
This paper studies problems related to visibility among points in the plane. A point $x$ emph{blocks} two points $v$ and $w$ if $x$ is in the interior of the line segment $bar{vw}$. A set of points $P$ is emph{$k$-blocked} if each point in $P$ is assigned one of $k$ colours, such that distinct points $v,win P$ are assigned the same colour if and only if some other point in $P$ blocks $v$ and $w$. The focus of this paper is the conjecture that each $k$-blocked set has bounded size (as a function of $k$). Results in the literature imply that every 2-blocked set has at most 3 points, and every 3-blocked set has at most 6 points. We prove that every 4-blocked set has at most 12 points, and that this bound is tight. In fact, we characterise all sets ${n_1,n_2,n_3,n_4}$ such that some 4-blocked set has exactly $n_i$ points in the $i$-th colour class. Amongst other results, for infinitely many values of $k$, we construct $k$-blocked sets with $k^{1.79...}$ points.
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