Elementos aplicables de la Arquitectura Empresarial TI a la Secretaría TIC

This section discusses the details of this approach, simultaneously proving various nice properties of the Fermat heuristic. To do so, we will need to recall the following properties of A* search [67][35]:

• A* with an admissible heuristic is guaranteed to find the optimal solution, if it

terminates

• A* with a consistentheuristic never “opens” the same node twice

• If A* uses an heuristic consisting of admissible heuristic weighted by some factor

greater than one, then it is guaranteed to terminate with a solution within that factor of the optimum solution

We will first see that the Fermat heuristic is consistent and admissible, as long as we construct it carefully.

Definition 8.1.1 (Fermat heuristic). Consider a constrained path planning problem with

cost function C(x), and suppose we are given a cost function Bd(x) :RN →R+ satisfying

Bd(x)≤ min

xd+1,xd+2,...,xN

C(x1, . . . , xd, xd+1, . . . , xN) (8.1.1)

and ∂Bd/∂xi = 0, ∀i > d,∀x. Then a d-th order Fermat heuristic hd(x;Bd) for this problem is given by the optimal value of the unconstrained path planning problem with

cost function Bd(x).

HereBd(x) is a function chosen to lower-bound the original cost function by minimizing

C(x) over all but the firstddimensions. Hence,Bd(x) is a low-dimensional approximation

of C(x) whose optimal paths are hopefully similar to those of C(x). Figure 8.1 gives a

concrete example of this approximation.

Theorem 8.1.2. Any Fermat heuristic hd(x;Bd) of a path planning problem is an admis-

sible heuristic for that problem.

Proof. Denote byVC(x) the value function of the original constrained problem, and denote

Figure 8.1: Example showing a one-dimensional approximationB1(x) of a two-dimensional

cost function C(x). B1(x) is formed by minimizing C(x) over x2. Dashed lines show

optimal paths with respect to corresponding cost functions.

∀x, B_d(x) ≤ C(x), and since the value of the unconstrained problem must be less than that of the constrained problem, we have

hd(x;Bd)≤VU(x)≤VC(x)

Another nice property of the Fermat heuristic is that it is strictly better than the standard Euclidean distance heuristic, in the following sense:

Theorem 8.1.3. The greatest 0-th order Fermat heuristic of a particular problem is ex-

actly equal to the Euclidean heuristic for that problem, and the greatest d-th order Fermat heuristic is always greater than or equal to the Euclidean heuristic.

Proof. By “greatest d-th order heuristic” in the statement above, we mean the heuristic

that uses the greatest lower boundBd possible. First note that in the cased= 0, Bdis a

constant. The greatest 0-th order Fermat heuristic therefore has

B0(x) = min

x C(x) =B0

The value of the heuristic is consequently determined as the optimal value function of a 0-dimensional cost function; i.e., constant cost. The path minimizing this cost is trivially

greatest 0-th order Fermat heuristic is therefore given by h0(x;B0) =kx−ykB0

which is exactly equal to the Euclidean heuristic for this problem. Note that the scaling

by B0 ensures that the Euclidean heuristic is as tight as possible.

The second part of the theorem follows easily from the fact that the value of the path

planning problem with cost function Bd+1 is always greater than or equal to the value of

the path planning problem with cost functionBd.

Finally, the Fermat heuristic is consistent, as shown by the following.

Theorem 8.1.4. Let the discrete cost function C0(x, y) be equal to the cost of some path

passing through x and y under cost function C. Any Fermat heuristic hd(x;Bd) of a problem with cost function C is a consistent heuristic for the problem when A* is applied with discrete cost function C0.

Proof. Note that in the statement of the theorem a distinction is made between the differ- ential cost function used in the continuous problem and the discrete cost function used in the A* solution of the problem.

Denote by hd(x;y, Bd) the value of the Fermat heuristic at x with a different goal y.

First, note that

hd(x;Bd)≤hd(x;y, Bd) +hd(y;Bd)

since h is a value function that consequently obeys the triangle inequality. Denoting by

V(x;y) the optimal cost to travel fromx toy in the original problem, we then have

hd(x;Bd) ≤ V(x;y) +hd(y;Bd)

≤ C0(x, y) +hd(y;Bd) (8.1.2)

The consistency and admissibility of the Fermat heuristic when employed in the manner described above ensure that A* is guaranteed to find the optimal solution (up to sampling resolution) of our problem without “backtracking.” In practice, however, we might find that we are willing to give up these guarantees of optimality in exchange for decreased computation time. The last property of A* mentioned above is therefore particularly

interesting. It provides a quantitative suboptimality guarantee in case we decide toinflate

(scale up) the heuristic. Inflating the heuristic biases A* to explore the states with lower heuristic values, generally causing it to terminate faster, assuming that the heuristic does

decrease monotonically and continuously towards the goal. Note that were it not for the admissibility of the Fermat heuristic, we would not have this reassuring suboptimality guarantee.