Monotone Path Research Articles

2-Convex Polyominoes: Non-Empty Corners

A polyomino P is called 2-convex if for every two cells there exists a monotone path included in P with at most two changes of direction. This paper studies the geometrical properties of a sub-class of 2-convex polyominoes called where the upper left corner and the lower right corner of the polyomino each contains only one cell.

Free energy in the Potts spin glass

We study the Potts spin glass model, which generalizes the Sherrington–Kirkpatrick model to the case when spins take more than two values but their interactions are counted only if the spins are equal. We obtain the analogue of the Parisi variational formula for the free energy, with the order parameter now given by a monotone path in the set of positive-semidefinite matrices. The main idea of the paper is a novel synchronization mechanism for blocks of overlaps. This mechanism can be used to solve a more general version of the Sherrington–Kirkpatrick model with vector spins interacting through their scalar product, which includes the Potts spin glass as a special case. As another example of application, one can show that Talagrand’s bound for multiple copies of the mixed $p$-spin model with constrained overlaps is asymptotically sharp. We will consider these problems in the subsequent paper and illustrate the main new idea on the technically more transparent case of the Potts spin glass.

A SAT attack on the Erdős–Szekeres conjecture

A classical conjecture of Erdős and Szekeres states that, for every integer k≥2, every set of 2k−2+1 points in the plane in general position contains k points in convex position. In 2006, Peters and Szekeres introduced the following stronger conjecture: every red-blue coloring of the edges of the ordered complete 3-uniform hypergraph on 2k−2+1 vertices contains an ordered subhypergraph with k vertices and k−2 edges, which is a union of a red monotone path and a blue monotone path that are vertex disjoint except for their two common end-vertices.Applying a state of art SAT solver, we refute the conjecture of Peters and Szekeres. We also apply techniques of Erdős, Tuza, and Valtr to refine the Erdős–Szekeres conjecture in order to tackle it with SAT solvers.

Computing Contour Trees for 2D Piecewise Polynomial Functions

AbstractContour trees are extensively used in scalar field analysis. The contour tree is a data structure that tracks the evolution of level set topology in a scalar field. Scalar fields are typically available as samples at vertices of a mesh and are linearly interpolated within each cell of the mesh. A more suitable way of representing scalar fields, especially when a smoother function needs to be modeled, is via higher order interpolants. We propose an algorithm to compute the contour tree for such functions. The algorithm computes a local structure by connecting critical points using a numerically stable monotone path tracing procedure. Such structures are computed for each cell and are stitched together to obtain the contour tree of the function. The algorithm is scalable to higher degree interpolants whereas previous methods were restricted to quadratic or linear interpolants. The algorithm is intrinsically parallelizable and has potential applications to isosurface extraction.

Signature inversion for monotone paths

The aim of this article is to provide a simple sampling procedure to reconstruct any monotone path from its signature. For every $N$, we sample a lattice path of $N$ steps with weights given by the coefficient of the corresponding word in the signature. We show that these weights on lattice paths satisfy the large deviations principle. In particular, this implies that the probability of picking up a “wrong” path is exponentially small in $N$. The argument relies on a probabilistic interpretation of the signature for monotone paths.

Geometric Path Problems with Violations

In this paper, we study variants of the classical geometric shortest path problem inside a simple polygon, where we allow a part of the path to go outside the polygon. Let P be a simple polygon consisting of n vertices and let s, t be a pair of points in P. Let $$\text{ int }(P)$$ represent the interior of P and let $$\overline{P}$$ represent the exterior of P, i.e. $$\text{ int }(P)=P{\setminus }\partial (P)$$ and $$\overline{P}=\mathbb {R}^2{\setminus }\text{ int }(P)$$ . For an integer $$k\ge 0$$ , we define a k-violation path from s to t to be a path connecting s and t such that its intersection with $$\overline{P}$$ consists of at most k segments. There is no restriction in terms of the number of segments of the path within P. The objective is to compute a path of minimum Euclidean length among all possible $$({\le } k)$$ -violation paths from s to t. In this paper, we study this problem for $$k=1$$ and propose an algorithm that computes the shortest one-violation path in $$O(n^3)$$ time. We show that for rectilinear polygons, the minimum length rectilinear one-violation path can be computed in $$O(n\log n)$$ time. We extend the concept of one-violation path to a one-stretch violation path. In this case, the path between s and t is composed of (a) a path in P from s to a vertex u of P, (b) a path in $$\overline{P}$$ between u and a vertex v of P, and (c) a path in P between v and t. We show that a minimum length one-stretch violation path can be computed in $$O(n\log n\log \log n)$$ time. Next, we introduce one- and two-violation monotone rectilinear path problems among a set of n disjoint axis-parallel rectangular objects. Let s, t be two points in $$\mathbb {R}^2$$ that are not in the interior of any of the objects. In the case of one-violation monotone path problem, the desired rectilinear path from s to t consists of horizontal edges that are directed towards the right and vertical edges that are directed upwards, except for at most one edge. Similarly, in the case of a two-violation monotone path problem, all horizontal edges are directed towards the right except at most one and all vertical edges are directed upwards except at most one. Our algorithms for both of these problems run in $$O(n\log n)$$ time.

On the exhaustive generation of k-convex polyominoes

The degree of convexity of a convex polyomino P is the smallest integer k such that any two cells of P can be joined by a monotone path inside P with at most k changes of direction. In this paper we present a simple algorithm for computing the degree of convexity of a convex polyomino and we show how it can be used to design an algorithm that generates, given an integer k, all k-convex polyominoes of area n in constant amortized time, using space O(n). Furthermore, by applying few changes, we are able to generate all convex polyominoes whose degree of convexity is exactly k.

A SAT attack on the Erdős–Szekeres conjecture

A classical conjecture of Erdős and Szekeres states that every set of 2k−2+1 points in the plane in general position contains k points in convex position. In 2006, Peters and Szekeres introduced the following stronger conjecture: every red-blue coloring of the edges of the ordered complete 3-uniform hypergraph on 2k−2+1 vertices contains an ordered k-vertex hypergraph consisting of a red and a blue monotone path that are vertex disjoint except for the common end-vertices.Applying the state of art SAT solver, we refute the conjecture of Peters and Szekeres. We also apply techniques of Erdős, Tuza, and Valtr to refine the Erdős–Szekeres conjecture in order to tackle it with SAT solvers.

Straight-line monotone grid drawings of series–parallel graphs

A monotone drawing of a planar graph G is a planar straight-line drawing of G where a monotone path exists between every pair of vertices of G in some direction. Recently monotone drawings of graphs have been discovered as a new standard for visualizing graphs. In this paper we study monotone drawings of series–parallel graphs in a variable embedding setting. We show that a series–parallel graph of n vertices has a straight-line planar monotone drawing on a grid of size O(n) × O(n2) and such a drawing can be found in linear time.

On Computing the Degree of Convexity of Polyominoes

In this paper we present an algorithm which has as input a convex polyomino $P$ and computes its degree of convexity, defined as the smallest integer $k$ such that any two cells of $P$ can be joined by a monotone path inside $P$ with at most $k$ changes of direction. The algorithm uses space $O(m + n)$ to represent a polyomino $P$ with $n$ rows and $m$ columns, and has a running time $O(min(m; r k))$, where $r$ is the number of corners of $P$. Moreover, the algorithm leads naturally to a decomposition of $P$ into simpler polyominoes.

The saturation number for the length of degree monotone paths

A degree monotone path in a graph $G$ is a path $P$ such that the sequence of degrees of the vertices in the order in which they appear on $P$ is monotonic. The length of the longest degree monotone path in $G$ is denoted by $mp(G)$. This parameter, inspired by the well-known Erdos-Szekeres theorem, has been studied by the authors in two earlier papers. Here we consider a saturation problem for the parameter $mp(G)$. We call $G$ saturated if, for every edge $e$ added to $G$, $mp(G+e) >mp(G)$, and we define $h(n,k)$ to be the least possible number of edges in a saturated graph $G$ on $n$ vertices with $mp(G) < k$, while $mp(G+e) \geq k$ for every new edge $e$. We obtain linear lower and upper bounds for $h(n,k)$, we determine exactly the values of $h(n,k)$ for $k=3$ and $4$, and we present constructions of saturated graphs.

Dynamic Pricing Strategies in the Presence of Demand Shifts

Many factors introduce the prospect of changes in the demand environment that a firm faces, with the specifics of such changes not necessarily known in advance. If and when realized, such changes affect the delicate balance between demand and supply and thus current prices should account for these future possibilities. We study the dynamic pricing problem of a retailer facing the prospect of a change in the demand function during a finite selling season with no inventory replenishment opportunity. In particular, the time of the change and the postchange demand function are unknown upfront, and we focus on the fundamental trade-off between collecting revenues from current demand and doing so for postchange demand, with the capacity constraint introducing the main tension. We develop a formulation that allows for isolating the role of dynamic pricing in balancing inventory consumption throughout the horizon. We establish that, in many settings, optimal pricing policies follow a monotone path up to the change in demand. We show how one may compare upfront the attractiveness of pre- and postchange demand conditions and how such a comparison depends on the problem primitives. We further analyze the impact of the model inputs on the optimal policy and its structure, ranging from the impact of model parameter changes to the impact of different representations of uncertainty about future demand.

Collective rationality and monotone path division rules

We impose the axiom Independence of Irrelevant Alternatives on division rules for the conflicting claims problem. With the addition of Consistency and Resource Monotonicity, this characterizes a family of rules which can be described in three different but intuitive ways. First, a rule is identified with a fixed monotone path in the space of awards, and for a given claims vector, the path of awards for that claims vector is simply the monotone path truncated by the claims vector. Second, a rule is identified with a set of parametric functions indexed by the claimants, and for a given claims problem, each claimant receives the value of his parametric function at a common parameter value, but truncated by his claim. Third, a rule is identified with an additively separable, strictly concave social welfare function, and for a given claims problem, the amount awarded is the maximizer of the social welfare function subject to the constraint of choosing a feasible award.

Ramsey Theory, integer partitions and a new proof of the Erdős–Szekeres Theorem

Let H be a k-uniform hypergraph whose vertices are the integers 1,…,N. We say that H contains a monotone path of length n if there are x1<x2<⋯<xn+k−1 so that H contains all n edges of the form {xi,xi+1,…,xi+k−1}. Let Nk(q,n) be the smallest integer N so that every q-coloring of the edges of the complete k-uniform hypergraph on N vertices contains a monochromatic monotone path of length n. While the study of Nk(q,n) for specific values of k and q goes back (implicitly) to the seminal 1935 paper of Erdős and Szekeres, the problem of bounding Nk(q,n) for arbitrary k and q was studied by Fox, Pach, Sudakov and Suk.Our main contribution here is a novel approach for bounding the Ramsey-type numbers Nk(q,n), based on establishing a surprisingly tight connection between them and the enumerative problem of counting high-dimensional integer partitions. Some of the concrete results we obtain using this approach are the following:•We show that for every fixed q we have N3(q,n)=2Θ(nq−1), thus resolving an open problem raised by Fox et al.•We show that for every k≥3, Nk(2,n)=2⋅⋅2(2−o(1))n where the height of the tower is k−2, thus resolving an open problem raised by Eliáš and Matoušek.•We give a new pigeonhole proof of the Erdős–Szekeres Theorem on cups-vs-caps, similar to Seidenberg's proof of the Erdős–Szekeres Lemma on increasing/decreasing subsequences.

Local Solarity of Suns in Normed Linear Spaces

The paper is concerned with solarity of intersections of suns with bars (in particular, with closed balls and extreme hyperstrips) in normed linear spaces. A sun in a finite-dimensional (BM)-space (in particular, in ℓ 1(n)) is shown to be monotone path connected. A nonempty intersection of an m-connected set (in particular, a sun in a two-dimensional space or in a finite-dimensional (BM)-space) with a bar is shown to be a monotone path-connected sun. Similar results are obtained for boundedly compact subsets of infinite-dimensional spaces. A nonempty intersection of a monotone path-connected subset of a normed space with a bar is shown to be a monotone path-connected α-sun.

Monotone Descent Path Queries on Dynamic Terrains

Monotone paths are useful in many engineering design applications. In this paper, we address the problem of answering monotone descent path queries on terrains that are continually changing. A terrain can be represented by a unique contour tree. Such a contour tree belongs to a class of graphs called arbitrarily directed trees (ADTs). Let T be an ADT with n nodes. In this paper, we present a new linear time preprocessing algorithm for decomposing a static ADT T into a forest F, with which we can answer lowest common descendent (LCA) queries in O(1) time. This is useful in answering monotone path queries on the corresponding terrain. We show how to maintain this data structure, and thereby answer LCA queries efficiently, for dynamic ADTs. We also show how to maintain the data structure of dynamic terrains, while simultaneously maintaining the corresponding contour tree. This allows us to efficiently answer monotone path queries between any two points on dynamic terrains.

Dynamic Pricing Strategies in the Presence of Demand Shifts

Many factors introduce the prospect of changes in the demand environment that a firm faces, with the specifics of such changes not necessarily known in advance. If and when realized, such changes affect the delicate balance between demand and supply and thus current prices should account for these future possibilities. We study the dynamic pricing problem of a retailer facing the prospect of a change in the demand function during a finite selling season with no inventory replenishment opportunity. In particular, the time of the change and the postchange demand function are unknown upfront, and we focus on the fundamental trade-off between collecting revenues from current demand and doing so for postchange demand, with the capacity constraint introducing the main tension. We develop a formulation that allows for isolating the role of dynamic pricing in balancing inventory consumption throughout the horizon. We establish that, in many settings, optimal pricing policies follow a monotone path up to the change in demand. We show how one may compare upfront the attractiveness of pre- and postchange demand conditions and how such a comparison depends on the problem primitives. We further analyze the impact of the model inputs on the optimal policy and its structure, ranging from the impact of model parameter changes to the impact of different representations of uncertainty about future demand.

The number of $k$-parallelogram polyominoes

A convex polyomino is $k$-$\textit{convex}$ if every pair of its cells can be connected by means of a $\textit{monotone path}$, internal to the polyomino, and having at most $k$ changes of direction. The number $k$-convex polyominoes of given semi-perimeter has been determined only for small values of $k$, precisely $k=1,2$. In this paper we consider the problem of enumerating a subclass of $k$-convex polyominoes, precisely the $k$-$\textit{convex parallelogram polyominoes}$ (briefly, $k$-$\textit{parallelogram polyominoes}$). For each $k \geq 1$, we give a recursive decomposition for the class of $k$-parallelogram polyominoes, and then use it to obtain the generating function of the class, which turns out to be a rational function. We are then able to express such a generating function in terms of the $\textit{Fibonacci polynomials}$. Un polyomino convexe est dit $k$-$\textit{convexe}$ lorsqu’on peut relier tout couple de cellules par un chemin monotone ayant au plus $k$ changements de direction. Le nombre de polyominos $k$-convexes n’est connu que pour les petites valeurs de $k = 1,2$. Dans cet article, nous énumérons la sous-classe des polyominos $k$-convexes qui sont également parallélogramme, que nous appelons $k$-$\textit{parallélogrammes}$. Nous donnons une décomposition récursive de la classe des polyominos $k$-parallélogrammes pour chaque $k$, et en déduisons la fonction génératrice, rationnelle, selon le demi-périmètre. Nous donnons enfin une expression de cette fonction génératrice en termes des $\textit{polynômes de Fibonacci}$.

Erdős-Szekeres-type theorems for monotone paths and convex bodies

For any sequence of positive integers j_1 < j_2 < ... < j_n, the k-tuples (j_i,j_{i + 1},...,j_{i + k-1}), i=1, 2,..., n - k+1, are said to form a monotone path of length n. Given any integers n\ge k\ge 2 and q\ge 2, what is the smallest integer N with the property that no matter how we color all k-element subsets of [N]=\{1,2,..., N\} with q colors, we can always find a monochromatic monotone path of length n? Denoting this minimum by N_k(q,n), it follows from the seminal 1935 paper of Erd\H os and Szekeres that N_2(q,n)=(n-1)^q+1 and N_3(2,n) = {2n -4\choose n-2} + 1. Determining the other values of these functions appears to be a difficult task. Here we show that 2^{(n/q)^{q-1}} \leq N_3(q,n) \leq 2^{n^{q-1}\log n}, for q \geq 2 and n \geq q+2. Using a stepping-up approach that goes back to Erdos and Hajnal, we prove analogous bounds on N_k(q,n) for larger values of k, which are towers of height k-1 in n^{q-1}. As a geometric application, we prove the following extension of the Happy Ending Theorem. Every family of at least M(n)=2^{n^2 \log n} plane convex bodies in general position, any pair of which share at most two boundary points, has n members in convex position, that is, it has n members such that each of them contributes a point to the boundary of the convex hull of their union.

A monotone path connected Chebyshev set is a sun

A monotone path connected Chebyshev set is a sun