4. Properties of least-cost distances

With most LCP software, the user can select a cost function to assign costs to the links in the graph in terms of energy or time required. The cost distance of the LCP is then the sum of the costs of the links forming the LCP. In mathematics, a metric or distance function has some properties that apply also to cost distances (Worboys and Duckham 2004, 123–6; Herzog 2013a). General properties of a metric are: (i) non-negative, (ii) identity of indiscernibles, (iii) triangle inequality, and (iv) symmetry.

For a cost distance CostDist, properties (i) and (ii) hold. Covering the distance between two well-separated locations A and B always involves some positive costs in time or energy, i.e. CostDist(A,B) > 0, for all B≠A . Although this statement seems to be quite intuitive, cost functions that assign zero or negative costs to some links were proposed and applied in the past (e.g. Bell and Lock 2000; Polla 2009). In an area with zero costs, long detours accumulate the same costs as straight-line connections and, even worse, in areas of negative costs, long detours decrease the overall costs. Moreover, Dijkstra's algorithm requires positive weights for each link.

If the LCP between A and B is forced through another location C, this path will never be shorter than the true LCP connecting A and B: CostDist(A,B) ≤ CostDist(A,C) + CostDist(C,B) for all locations A, B, and C. This property is known as triangle inequality.

A quasimetric obeys the three properties discussed above. The symmetry property must be fulfilled so that CostDist is a distance according to the definition in mathematics: CostDist(A,B) = CostDist(B,A). However, if the route from A to B is on a river with strong currents, or if slopes are to be climbed, then the effort and time required will be different for the return path. In Section 3.1, the term isotropic costs was introduced for costs independent of the direction of movement, and the term anisotropic costs is used in a situation where CostDist(A,B) ≠ CostDist(B,A), for some locations A and B. An anisotropic cost function can be converted to an isotropic function by averaging the costs of the links in both directions (De Silva and Pizziolo 2001; Fábrega Álvarez and Parcero Oubiña 2007; Llobera and Sluckin 2007). This will ensure that the LCP from A to B coincides with the return LCP and that CostDist is symmetric. Such an approach is recommended when replacing map distance by cost distance in spatial data analysis, i.e. for site catchments, least-cost buffers, clustering, and autocorrelation calculations. However, in some situations it seems advisable to use a different route when returning. Such one-way systems were recorded both for animal trails (Ganskopp et al. 2000) and for Roman roads (Grewe 2004, 30–4). In this case, a distance in mathematical terms can be defined by BiCostDist(A,B) = CostDist(A,B) + CostDist(B,A). The difference to the approach described first is that the return path may differ from the initial path.

Moreover, small changes in the environment in general should not result in sudden jumps of the cost function, i.e. cost functions are expected to be continuous. Exceptions are streams, which can be crossed easily when their width is small but present substantial difficulties when their depth and/or width exceeds a certain limit. Another example might be vehicles climbing slopes, which is not possible for very steep slopes because the power of the vehicle is not sufficient to do so safely (Rowe and Ross 1990).

Classifying a continuous cost function so that all costs within a certain range are set to a constant value (e.g. Zakšek et al. 2008) may result in LCPs that prefer the upper limits of the cost ranges selected, which certainly is an undesired effect. In fact, classifying a continuous cost function is an example of a monotonically increasing transformation, and even a strictly monotone transformation of the initial cost function will often result in another least-cost route (Herzog 2013a). The only transformation of a cost function that does not alter the LCP outcome is multiplication by a positive constant; such a transformation is equivalent to changing the measurement system for instance from joule to kilocalories.

A consequence of the fact that strictly increasing monotone transformations of cost functions often result in different LCPs is the fact that in general, unmodified ordinal or interval scale variables should not be used for LCP calculations (Conolly and Lake 2006, 255), but should be converted to calibrated cost values.