Cost distance analysis
The various problems, algorithms, and tools of cost distance analysis operate over an unconstrained two-dimensional space, meaning that a path could be of any shape.
Although they are similar in principle, the problems in network space require very different (usually simpler) algorithms to solve, largely adopted from graph theory.
Historic, even ancient, roads show patterns similar to what modern computational algorithms would generate, traveling straight across flat spaces, but curving around mountains, canyons, and thick vegetation.
In 1957, during the Quantitative revolution in Geography, with its propensity to adopt principles or mathematical formalisms from the "hard" sciences (known as social physics), William Warntz used refraction as an analogy for how minimizing travel cost will make transportation routes change direction at the boundary between two landscapes with very different friction of distance (e.g., emerging from a forest into a prairie).
[2] His principle of "parsimonious movement," changing direction to minimize cost, was widely accepted, but the refraction analogy and mathematics (Snell's law) was not, largely because it does not scale well to normally complex geographic situations.
[3] Warntz and others then adopted another analogy that proved much more successful in the common situation where travel cost varies continuously over space, by comparing it to terrain.
[5][6] Additional lines of research in the 1960s further developed the nature of the cost rate field as a manifestation of the concept of friction of distance, studying how it was affected by various geographic features.
Raster GIS provided the first feasible platform for implementing the theoretical solution by converting the continuous integration into a discrete summation procedure.
Other costs are much more difficult to measure due to their qualitative or subjective nature, such as political protest or ecological impact; these typically require operationalization through the creation of a scale.
The typical solution algorithm is a discrete raster implementation of the cost integration strategy of Warntz and Lindgren,[5] which is a deterministic (NP-complete) optimization.
The algorithm to derive this corridor field is created by generating two cost accumulation grids: one using the source as described above.
[14] The second solution is to first run the basic accumulation algorithm, then use the backlink grid to determine the source into which each cell "flows."
