Production matrices for geometric graphs

LetS be a set ofn points in ℝ d . A setW is aweak e-net for (convex ranges of)S if, for anyT⊆S containing en points, the convex hull ofT intersectsW. We show the existence of weak e-nets of size\(O((1/\varepsilon ^d )\log ^{\beta _d } (1/\varepsilon ))\), whereβ2=0,β3=1, andβ d ≈0.149·2d-1(d-1)!, improving a previous bound of Alonet al. Such a net can be computed effectively. We also consider two special cases: whenS is a planar point set in convex position, we prove the existence of a net of sizeO((1/e) log1.6(1/e)). In the case whereS consists of the vertices of a regular polygon, we use an argument from hyperbolic geometry to exhibit an optimal net of sizeO(1/e), which improves a previous bound of Capoyleas.

We present a new game, Dots & Polygons, played on a planar point set. We prove that its NP-hard and discuss strategies for the case when the point set is in convex position.

This paper starts the investigation of a constrained version of the point-set embed-dability problem. Let G = (V,E) be a planar graph with n vertices, G′ = (V′,E′) a subgraph of G, and S a set of n distinct points in the plane. We study the problem of computing a point-set embedding of G on S subject to the constraint that G′ is drawn with straight-line edges. Different drawing algorithms are presented that guarantee small curve complexity of the resulting drawing, i.e. a small number of bends per edge. It is proved that: • If G′ is an outerplanar graph and S is any set of points in convex position, a point-set embedding of G on S can be computed such that the edges of E\E′ have at most 4 bends each. • If S is any set of points in general position and G′ is a face of G or if it is a simple path, the curve complexity of the edges of E\E′ is at most 8. • If S is in general position and G′ is a set of k disjoint paths, the curve complexity of the edges of E \ E′ is O(2k).

Given an n-vertex outer-planar graph G and a set P of n points in the plane, we present an O(n log3n) time and O(n) space algorithm to compute a straight-line embedding of G in P, improving upon the algorithm in [GMPP91, CU96] that requires O(n2) time. Our algorithm is near-optimal as there is an Ω(n log n) lower bound for the problem [BMS95]. We present a simpler O(nd) time and O(n) space algorithm to compute a straight-line embedding of G in P where log n ≤ d ≤ 2n is the length of the longest vertex disjoint path in the dual of G. Therefore, the time complexity of the simpler algorithm varies between O(n log n) and O(n2) depending on the value of d. More efficient algorithms are presented for certain restricted cases. If the dual of G is a path, then an optimal Θ(n log n) time algorithm is presented. If the given point set is in convex position then we show that O(n) time suffices.

We study the following family of problems: Given a set of $n$ points in convex position, what is the maximum number triangles one can create having these points as vertices while avoiding certain sets of forbidden configurations. As forbidden configurations we consider all 8 ways in which a pair of triangles in such a point set can interact. This leads to 256 extremal Turán-type questions. We give nearly tight (within a $\log n$ factor) bounds for 248 of these questions and show that the remaining 8 questions are all asymptotically equivalent to Stein's longstanding tripod packing problem.

Let $S$ be a set of $n$ points in the plane that is in convex position. In this paper, using the well-known path-greedy spanner algorithm, we present an algorithm that constructs a plane $\frac{3+4\pi}{3}$-spanner $G$ of degree 3 on the point set $S$. Recently, Biniaz et al. ({\it Towards plane spanners of degree 3, Journal of Computational Geometry, 8 (1), 2017}) have proposed an algorithm that constructs a degree 3 plane $\frac{3+4\pi}{3}$-spanner $G'$ for $S$. We show that there is no upper bound with a constant factor on the total weight of $G'$, but the total weight of $G$ is asymptotically equal to the total weight of the minimum spanning tree of $S$.

We present an explicit formula for the transition matrix $\\mathcal{C}$ from\nthe type $C_n$ degeneration of the Koornwinder polynomials\n$P_{(1^r)}(x\\,|\\,a,-a,c,-c\\,|\\,q,t)$ with one column diagrams, to the type\n$C_n$ monomial symmetric polynomials $m_{(1^{r})}(x)$. The entries of the\nmatrix $\\mathcal{C}$ enjoy a set of three term recursion relations, which can\nbe regarded as a $(a,c,t)$-deformation of the one for the Catalan triangle or\nballot numbers. Some transition matrices are studied associated with the type\n$(C_n,C_n)$ Macdonald polynomials $P^{(C_n,C_n)}_{(1^r)}(x\\,|\\,b;q,t)=\nP_{(1^r)}\\big(x\\,|\\,b^{1/2},-b^{1/2},q^{1/2}b^{1/2},-q^{1/2}b^{1/2}\\,|\\,q,t\\big)$.\nIt is also shown that the $q$-ballot numbers appear as the Kostka polynomials,\nnamely in the transition matrix from the Schur polynomials\n$P^{(C_n,C_n)}_{(1^r)}(x\\,|\\,q;q,q)$ to the Hall-Littlewood polynomials\n$P^{(C_n,C_n)}_{(1^r)}(x\\,|\\,t;0,t)$.\n

For each integer k?1, we define an algorithm which associates to a partition whose maximal value is at most k a certain subset of all partitions. In the case when we begin with a partition ? which is square-bounded, i.e. ?=(? 1?????? k ) with ? 1=k and ? k =1, applying the algorithm ? times gives rise to a set whose cardinality is either the Catalan number c ??k+1 (the self dual case) or twice that Catalan number. The algorithm defines a tree and we study the propagation of the tree, which is not in the isomorphism class of the usual Catalan tree. The algorithm can also be modified to produce a two-parameter family of sets and the resulting cardinalities of the sets are the ballot numbers. Finally, we give a conjecture on the rank of a particular module for the ring of symmetric functions in 2?+m variables.

On the stretch factor of Delaunay triangulations of points in convex position

Enumerating several aspects of non-decreasing Dyck paths

Documenting and treating a dysregulated gut-immune-brain axis in schizophrenia

In this paper we prove a $q$-analogue of Koshy's formula in terms of the Narayana polynomial due to Lassalle and a $q$-analogue of Koshy's formula in terms of $q$-hypergeometric series due to Andrews by applying the inclusion-exclusion principle on Dyck paths and on partitions. We generalize these two $q$-analogues of Koshy's formula for $q$-Catalan numbers to that for $q$-Ballot numbers. This work also answers an open question by Lassalle and two questions raised by Andrews in 2010. We conjecture that if $n$ is odd, then for $m\ge n\ge 1$, the polynomial $(1+q^n){m\brack n-1}_q$ is unimodal. If $n$ is even, for any even $j\ne 0$ and $m\ge n\ge 1$, the polynomial $(1+q^n)[j]_q{m\brack n-1}_q$ is unimodal. This implies the answer to the second problem posed by Andrews.

Given n red and n blue points in convex position in the plane, we show that there exists a noncrossing alternating path of length \(n+c\sqrt{n\over \log n}\). We disprove a conjecture of Erdős by constructing an example without any such path of length greater than \({4\over 3}n+c'\sqrt{n}\).KeywordsPlanar GraphOuter SegmentHamiltonian CycleAlternate PathBlue PointThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This paper considers an unconstrained collaborative optimization of a sum of convex functions, where agents make decisions using local information in the presence of random interconnection topologies. We recast the problem as minimization of the sum of convex functions over a constraint set defined as the set of fixed-value points of a random operator derived from weighted matrices of random graphs. We show that the derived random operator has nonexpansivity property; therefore, this formulation does not need the distribution of random communication topologies. Hence, it includes random networks with/without asynchronous protocols. As an extension of the problem, we define a novel optimization problem, namely minimization of a convex function over the fixed-value point set of a nonexpansive random operator. We propose a discrete-time algorithm using diminishing step size for converging almost surely and in mean square to the global solution of the optimization problem under suitable assumptions. Consequently, as a special case, it reduces to a totally asynchronous algorithm for the distributed optimization problem. We show that fixed-value point is a bridge from deterministic analysis to random analysis of the algorithm. Finally, a numerical example illustrates the convergence of the proposed algorithm.

Lower bounds on the maximum number of non-crossing acyclic graphs