Given a grid of cells, each of which is assigned a numerical value quantifying its suitability for a certain use, one problem in geographic information science concerns the selection of a region, i.e. a connected set of cells, with a specified size that maximizes the sum of all their values. This task can be cast as a combinatorial optimization problem called the maximum value region problem, and exact and heuristic methods exist for its solution. While those solutions are guaranteed to be feasible (if not optimal), they may not be desirable for practical use if they contain too narrow segments (down to the width of a single cell). In this paper, we present a new variation of the maximum value region problem—the maximum value wide region problem—that requires a region to be at least as wide as a specified width. We offer a heuristic method for its solution which models a region as a set of neighborhoods and test its performance through computational experiments. Results demonstrate that the method generates good feasible solutions in terms of connectedness, size, width, and value, but requires more computing time than methods for maximum value regions without minimum width requirements.

Selection of optimal paths or sequences of cells from a grid of cells is one of the most basic functions of raster-based geographic information systems. For this function to work, it is often assumed that the optimality of a path can be evaluated by the sum of the weighted lengths of all its segments – weighted, i.e. by the underlying cell values. The validity of this assumption must be questioned, however, if those values are measured on a scale that does not permit arithmetic operations. Through computational experiments with randomly generated artificial landscapes, this paper compares two models, minisum and minimax path models, which aggregate the values of the cells associated with a path using the sum function and the maximum function, respectively. Results suggest that the minisum path model is effective if the path search can be translated into the conventional least-cost path problem, which aims to find a path with the minimum cost-weighted length between two terminuses on a ratio-scaled raster cost surface. On the other hand, the minimax path model is found mathematically sounder if the cost values are measured on an ordinal scale and practically useful if the problem is concerned not with the minimization of cost but with the maximization of some desirable condition such as suitability.

Given a grid of cells, each having a value indicating its cost per unit area, a variant of the least-cost path problem is to find a corridor of a specified width connecting two termini such that its cost-weighted area is minimized. A computationally efficient method exists for finding such corridors, but as is the case with conventional raster-based least-cost paths, their incremental orientations are limited to a fixed number of (typically eight orthogonal and diagonal) directions, and therefore, regardless of the grid resolution, they tend to deviate from those conceivable on the Euclidean plane. In this paper, we propose a method for solving the raster-based least-cost corridor problem with reduced distortion by adapting a distortion reduction technique originally designed for least-cost paths and applying it to an efficient but distortion-prone least-cost corridor algorithm. The proposed method is, in theory, guaranteed to generate no less accurate solutions than the existing one in polynomial time and, in practice, expected to generate more accurate solutions, as demonstrated experimentally using synthetic and real-world data.

Planners who are involved in locational decision-making often use raster-based geographic information systems to quantify the value of land in terms of suitability or cost for a certain use. From a computational point of view, this process can be seen as a transformation of one or more sets of values associated with a grid of cells into another set of such values through a function reflecting one or more criteria. While it is generally anticipated that different transformations lead to different ‘best’ locations, little has been known on how such differences arise (or do not arise). The paper attempts to answer this question in the context of path planning through a series of computational experiments using a number of random landscape grids with a variety of spatial and nonspatial structures. In the experiments, we generated least-cost paths on a number of cost grids transformed from the landscape grids using a variety of transformation parameters and analyzed the locations and (weighted) lengths of those paths. Results show that the same pair of terminal cells may well be connected by different least-cost paths on different cost grids though derived from the same landscape grid and that the variation among those paths is affected by how given values are distributed in the landscape grid as well as by how derived values are distributed in the cost grids. Most significantly, the variation tends to be smaller when the landscape grid contains more distinct patches of cells potentially attracting or distracting cost-saving passage or when the cost grid contains a smaller number of low-cost cells.

Given a grid of cells each having an associated cost value, a raster version of the least-cost path problem seeks a sequence of cells connecting two specified cells such that its total accumulated cost is minimized. Identifying least-cost paths is one of the most basic functions of raster-based geographic information systems. Existing algorithms are useful if the path width is assumed to be zero or negligible compared to the cell size. This assumption, however, may not be valid in many real-world applications ranging from wildlife corridor planning to highway alignment. This paper presents a method to solve a raster-based least-cost path problem whose solution is a path having a specified width in terms of Euclidean distance (rather than by number of cells). Assuming that all cell values are positive, it does so by transforming the given grid into a graph such that each node represents a neighborhood of a certain form determined by the specified path width, and each arc represents a possible transition from one neighborhood to another. An existing shortest path algorithm is then applied to the graph. This method is highly efficient, as the number of nodes in the transformed graph is not more than the number of cells in the given grid and decreases with the specified path width. However, a shortcoming of this method is the possibility of generating a self-intersecting path which occurs only when the given grid has an extremely skewed distribution of cost values.

This paper considers the problem of planning a path in a circumstance where its origin is given, but its destination is not specified and is to be selected from among a set of candidate destinations during a trip. A situation like this may be experienced by a group of people who have different preferred destinations, as well as by an individual who is simply indecisive about where to go. To resolve such an uncertainty, one may stay at the origin until he decides on a destination, or choose to proceed on some path that does not overly deviate from a shortest path, whichever destination is eventually chosen, and make a decision on the way. The latter action is sensible when the risk of traveling longer is outweighed by the benefit of buying more time for a better destination decision. The problem of finding such a time-buying path is formulated and a simple algorithm is developed for its solution. Some extensions and applications are also discussed.

From a single-attribute raster layer in which each cell is assigned a numerical value, a connected set of a specified number of cells that has the maximum (or minimum) total value is selected. This is a highly common decision problem in the context of raster-based geographic information systems (GIS) and seems general enough to deserve inclusion in the standard functionality of such systems. Yet it is a computationally difficult optimization problem, for which no efficient exact solution method has been found. This article presents a new dynamic programming-based heuristic method for the problem. Its performance is tested with randomly generated raster layers with various degrees of spatial autocorrelation. Results suggest that the proposed heuristic is a promising alternative to the existing integer programming-based exact method, as it can handle significantly larger raster data with fair accuracy.

Abstract A new variant of the shortest path problem involves a bicycle traveling from an origin to a destination through a network situated on a hilly geography. Determining a path that takes the least amount of pedaling work involves a conservative force, gravity, and a nonconservative force, friction, acting on the bicycle. The cyclist's pedaling work to overcome the friction of each arc varies with the bicycle's kinetic and gravitational potential energies, which transform to one another. Although geometric characteristics of the network are invariable, arc weights representing required pedaling work are variable. This problem is formulated as a quadratic integer program and an approximation procedure is presented. © 2008 Wiley Periodicals, Inc. NETWORKS, 2008