Rectangular Grid Graph Research Articles

A substring–substring LCS data structure

For any rectangular grid graph with grid units each potentially having a diagonal edge, we propose a data structure that is constructible in time linear in the number of grid units in the graph and supports linear-time queries of a shortest path between any pair of vertices in the graph. Setting the position of grid units having a diagonal edge appropriately, we can use this data structure to solve a local variant of the longest common subsequence problem or certain related problems in linear time.

Longest (s, t)-paths in L-shaped grid graphs

The longest path problem, that is, finding a simple path with the maximum number of vertices, is a well-known NP-hard problem with many applications. However, for some classes of graphs, including solid grid graphs and grid graphs with some holes, it is open. An L-shaped grid graph is a special kind of a rectangular grid graph with a rectangular hole. In this paper, we show that a longest path between two given vertices s and t of an L-shaped grid graph can be computed in linear time.

Maximal independent sets on a grid graph

An independent vertex set of a graph is a set of vertices of the graph in which no two vertices are adjacent, and a maximal independent set is one that is not a proper subset of any other independent set. In this paper we count the number of maximal independent sets of vertices on a complete rectangular grid graph. More precisely, we provide a recursive matrix-relation producing the partition function with respect to the number of vertices. The asymptotic behavior of the maximal hard square entropy constant is also provided. We adapt the state matrix recursion algorithm, recently invented by the author to answer various two-dimensional regular lattice model problems in enumerative combinatorics and statistical mechanics.

Untangling Planar Curves

Any generic closed curve in the plane can be transformed into a simple closed curve by a finite sequence of local transformations called homotopy moves. We prove that simplifying a planar closed curve with n self-crossings requires \(\Theta (n^{3/2})\) homotopy moves in the worst case. Our algorithm improves the best previous upper bound \(O(n^2)\), which is already implicit in the classical work of Steinitz; the matching lower bound follows from the construction of closed curves with large defect, a topological invariant of generic closed curves introduced by Aicardi and Arnold. Our lower bound also implies that \(\Omega (n^{3/2})\) facial electrical transformations are required to reduce any plane graph with treewidth \(\Omega (\sqrt{n})\) to a single vertex, matching known upper bounds for rectangular and cylindrical grid graphs. More generally, we prove that transforming one immersion of k circles with at most n self-crossings into another requires \(\Theta (n^{3/2} + nk + k^2)\) homotopy moves in the worst case. Finally, we prove that transforming one non-contractible closed curve to another on any orientable surface requires \(\Omega (n^2)\) homotopy moves in the worst case; this lower bound is tight if the curve is homotopic to a simple closed curve.

A linear-time algorithm for finding Hamiltonian (s,t)-paths in even-sized rectangular grid graphs with a rectangular hole

The Hamiltonian path problem for general grid graphs is NP-complete. In this paper, we give the necessary conditions for the existence of a Hamiltonian path between two given vertices in a rectangular grid graph with a rectangular hole; where the size of graph is even. In addition, we show that the Hamiltonian path in these graphs can be computed in linear-time.

A linear-time algorithm for finding Hamiltonian (s, t)-paths in odd-sized rectangular grid graphs with a rectangular hole

A grid graph $$G_{\mathrm{g}}$$ is a finite vertex-induced subgraph of the two-dimensional integer grid $$G^\infty $$ . A rectangular grid graph R(m, n) is a grid graph with horizontal size m and vertical size n. A rectangular grid graph with a rectangular hole is a rectangular grid graph R(m, n) such that a rectangular grid subgraph R(k, l) is removed from it. The Hamiltonian path problem for general grid graphs is NP-complete. In this paper, we give necessary conditions for the existence of a Hamiltonian path between two given vertices in an odd-sized rectangular grid graph with a rectangular hole. In addition, we show that how such paths can be computed in linear time.

Embedding cycles and paths on solid grid graphs

Despite many algorithms for embedding graphs on unbounded grids, only a few results on embedding graphs on restricted grids have been published. In this paper, we study the problem of embedding paths and cycles on solid grid graphs. We show that a cycle of length k is unit-length embeddable on a solid grid graph G if k is an even integer between four and the length of the longest cycle of G. In addition, our result shows that a path of length k is unit-length embeddable on G, between its two given vertices s and t, if $$k\le L$$k≤L and $$k\equiv L (\mathrm{mod}\ 2)$$kźL(mod2), in which L is the length of the longest path between s and t. Our presented two algorithms show that such embeddings can be found in linear time for cycles and quadratic time for paths, with respect to the size of graph G. In the case of rectangular grid graphs, the running time of the algorithms can be improved to O(k) and O$$(k^2)$$(k2), respectively. In addition, we extend our results to $$m\times n\times o$$m×n×o 3D grids. A application of our result is in the interconnection network mapping in parallel processing.

A linear-time algorithm for the longest path problem in rectangular grid graphs

The longest path problem is a well-known NP-hard problem and so far it has been solved polynomially only for a few classes of graphs. In this paper, we give a linear-time algorithm for finding a longest path between any two given vertices in a rectangular grid graph.

Bicolored graph partitioning, or: gerrymandering at its worst

This study is motivated by an electoral application where we look into the following question: how much biased can the assignment of parliament seats be in a majority system under the effect of vicious gerrymandering when the two competing parties have the same electoral strength? To give a first theoretical answer to this question, we introduce a stylized combinatorial model, where the territory is represented by a rectangular grid graph, the vote outcome by a “balanced” red/blue node bicoloring and a district map by a connected partition of the grid whose components all have the same size. We constructively prove the existence in cycles and grid graphs of a balanced bicoloring and of two antagonist “partisan” district maps such that the discrepancy between their number of “red” (or “blue”) districts for that bicoloring is extremely large, in fact as large as allowed by color balance.

A NoteConcerning Hamilton Cycles in Some Classes of Grid Graphs

A graph G is called hamiltonian if it contains a Hamilton cycle, i.e. a cycle containing all vertices. Deciding whether a given graph has a Hamilton cycle is an NP-complete problem. But, it is a polynomial problem within some special graph classes. In this paper we consider three classes of grid graphs, namely rectangular grid graph, eta grid graph and omega grid graph. Our main result characterizes which of these graphs are hamiltonian.

On a 2-dimensional equipartition problem

The present paper deals with the following problem, which arises in a variety of applications: given a node-weighted rectangular grid graph, perform p horizontal full cuts and q vertical ones so as to make the weights of the resulting ( p+1)( q+1) rectangular subgrids “as close as possible”. Computational complexity aspects are discussed. Several heuristic algorithms for obtaining partitions which minimize the maximum weight of a subgrid are developed, and computational results are reported. Theoretical bounds on the approximation error are also given.

Max-min partitioning of grid graphs into connected components

The partitioning of a rectangular grid graph with weighted vertices into p connected components such that the component of smallest weight is as heavy as possible (the max-min problem) is considered. It is shown that the problem is NP-hard for rectangles with at least three rows. A shifting algorithm is given which approximates the optimal solution. Bounds for the relative error are determined under a posteriori hypotheses. A further shifting algorithm is also given which allows for error estimates under a priori hypotheses and for asymptotic error estimates. A similar approach can be taken with the problem of finding the partition whose largest component is as small as possible (the min-max problem). The case of rectangles with two rows has a polynomial algorithm and is dealt with in another paper. © 1998 John Wiley & Sons, Inc. Networks 32: 115–125, 1998

Maximum lower bound on embedding areas of general graphs

AbstractIn designing VLSI, the problem of how large the chip area will be is of considerable interest. Thus, the primary objective in investigating this problem is to consider formally processing elements and wires of a circuit to be vertices and edges of an undirected graph, and then clarify the area sufficient or necessary to embed this graph onto a rectangular grid graph according to a certain embedding method (this is called an embedding model). In this paper, vertices correspond to a rectangular graph such that [length of long side]/[length of short side] ≤ constant. It is shown that under an embedding model, which forbids images of edges (wires) from passing through the inside of the vertex region of the rectangle, the maximum lower bound of the area corresponding to a general n‐vertex d‐degree graph is ωd2n2. As the result, it is seen that the conventionally known upper bound O(d2n2) cannot be improved any further as long as all n‐vertex d‐degree graphs are considered.

Hamilton Paths in Grid Graphs

A grid graph is a node-induced finite subgraph of the infinite grid. It is rectangular if its set of nodes is the product of two intervals. Given a rectangular grid graph and two of its nodes, we give necessary and sufficient conditions for the graph to have a Hamilton path between these two nodes. In contrast, the Hamilton path (and circuit) problem for general grid graphs is shown to be NP-complete. This provides a new, relatively simple, proof of the result that the Euclidean traveling salesman problem is NP-complete.