Non-crossing Paths Research Articles

The smallest pair of noncrossing paths in a rectilinear polygon

Smallest rectilinear paths are rectilinear paths with simultaneous minimum numbers of bends and minimum lengths. Given two pairs of terminals within a rectilinear polygon, the authors derive an algorithm to find a pair of noncrossing rectilinear paths within the polygon such that the total number of bends and the total length are both minimized. Although a smallest rectilinear path between two terminals in a rectilinear polygon always exists, they show that such a smallest pair may not exist for some problem instances. In that case, the algorithm presented will find, among all noncrossing paths with a minimum total number of bends, a pair whose total length is the shortest, or find, among all noncrossing paths with a minimum total length, a pair whose total number of bends is minimized. They provide a simple linear time and space algorithm based on the fact that there are only a limited number of configurations of such a solution pair.

A non-commutative version of Jacobi's equality on the cofactors of a matrix

We give a combinatorial proof of Jacobi's equality relating a cofactor of a matrix with the complementary cofactor of its inverse. This result unifies two previous approaches of the combinatorial interpretation of determinants: generating functions of weighted permutations and generating functions of families (configurations) of non-crossing paths. We show that Jacobi's equality is valid with the same choice of non-commutative entries as in Foata's proof of matrix inversion by cofactors.

Shortest noncrossing paths in plane graphs

Let G be an undirected plane graph with nonnegative edge length, and letk terminal pairs lie on two specified face boundaries. This paper presents an algorithm for findingk “noncrossing paths” inG, each connecting a terminal pair, and whose total length is minimum. Noncrossing paths may share common vertices or edges but do not cross each other in the plane. The algorithm runs in timeO(n logn) wheren is the number of vertices inG andk is an arbitrary integer.

Shortest Noncrossing Paths in Plane Graphs

Shortest Noncrossing Paths in Plane Graphs

Algorithms for finding noncrossing paths with minimum total length in plane graphs

AbstractAssume that G is an undirected planar graph and the edge length of G is a nonnegative real number. When k terminal pairs are specified on two specified face boundaries, this paper gives an algorithm that derives the “noncrossing paths” with the minimum sum of lengths that connects the respective terminal pairs. By the noncrossing paths is meant the paths which do not cross on the plane, although the point or the edge may be shared. the computation time of the proposed algorithm is O(n log n), where n is the number of points on the planar graph G; k need not be a constant.

Enumeration of generalized Young tableaux with bounded height

We consider Young tableaux strictly increasing in rows, weakly increasing in columns. We show that the number of such tableaux with entries between 1 and n, with p columns having an odd number of elements and having at most 2 k rows, it the product n p 2k+p −1 p n+2k+p p П 1⩽i⩽j⩽n⩽ 2k+i+j i+j . The proof is mainly bijective, using configurations of noncrossing paths.

Noncrossing Subgraphs in Topological Layouts

The computational complexity of the following type of problems is studied. Given a topological layout (i.e., a drawing in the plane) of a graph, does it contain a noncrossing subgraph of a given type? It is conjectured that such problems are always NP-hard (provided planar subgraphs are looked for) regardless of the complexity of their nonplanar versions. This conjecture is verified for several cases in a very strong sense. In particular, it is shown that deciding the existence of a noncrossing path connecting two given vertices in a given topological layout of a 3-regular subgraph, as well as deciding the existence of a noncrossing cycle in such a layout, are NP-complete problems. It is also proved that deciding the existence of a noncrossing k-factor in a topological layout of a $( k + 1 )$-regular graph is NP-complete for $k = 2,3,4,5$. For $k = 1$, this question is NP-complete in layouts of 3-regular graphs, while it is polynomial solvable for layouts of graphs with maximum degree two.

Coincident probabilities and applications in combinatorics

For a strong Markov process on the line with continuous paths the Karlin–McGregor determinant formula of coincidence probabilities for multiple particle systems is extended to allow the individual component processes to start at variable times and run for variable durations. The extended formula is applied to a variety of combinatorial problems including counts of non-crossing paths in the plane with variable start and end points, dominance orderings, numbers of dominated majorization orderings, and time-inhomogeneous random walks.

Coincident probabilities and applications in combinatorics

For a strong Markov process on the line with continuous paths the Karlin–McGregor determinant formula of coincidence probabilities for multiple particle systems is extended to allow the individual component processes to start at variable times and run for variable durations. The extended formula is applied to a variety of combinatorial problems including counts of non-crossing paths in the plane with variable start and end points, dominance orderings, numbers of dominated majorization orderings, and time-inhomogeneous random walks.

A lattice path lemma and an application in enzyme kinetics

Abstract We start by proving that if a lattice path from (0,0) to (a,b) is chosen at random it has probability 1 (k+1) of passing underneath k points in the rectangle picked at random but with the ordinates and abscissas both in increasing order. We next translate this result into results about pairs of noncrossing paths. Consider a row of M enzymes each of which is either reduced (1) or oxidized (0) so that the row can be represented by a zero-one sequence. The basic transition rule is that an oxygen atom can be passed from left to right so that a 01 subsequence becomes 10 independently of the rest of the sequence. The steady state distributions are found by a surprising connection with the results on lattice paths.