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Figure 3.1 Illustration for Theorem 3.2.1. Actualized segments of the virtual obstacle are shown in solid black. S, start point; T , target point.
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segment is of length l, then the perimeter of the corresponding actual obstacle is equal to 2l; this takes into account the inside and outside walls of the segment and also the fact that the thickness of the wall is negligible (see Figure 3.1). This method of producing the resultant scene is justi ed by the fact that, under the accepted model, the behavior of MA is affected only by those obstacles that it touches along its way. Indeed, under algorithm X the very same path would have been produced in two different scenes: in the scene with the virtual obstacle and in the resultant scene. One can therefore argue that the areas of the virtual obstacle that MA has not touched along its way might have never existed, and that algorithm X produced its path not in the scene with the virtual obstacle but in the resultant scene. This means the performance of MA in the resultant scene can be judged against (3.1). This completes the design of the scene. Note that depending on the MA s behavior under algorithm X, zero, one, or more actualized obstacles can appear in the scene (Figure 3.1b). We now have to prove that the MA s path in the resultant scene satis es inequality (3.1). Since MA starts at a distance D = d(S, T ) from point T , it obviously cannot avoid the term D in (3.1). Hence we concentrate on the second term in (3.1). One can see by now that the main idea behind the described process of designing the resultant scene is to force MA to generate, for each actual obstacle, a segment of the path at least as long as the total length of that obstacle s boundary. Note that this characteristic of the path is independent of the algorithm X. The MA s path in the scene can be divided into two parts, P 1 and P 2; P 1 corresponds to the MA s traveling inside the corridor, and P 2 corresponds to its traveling outside the corridor. We use the same notation to indicate the length of the corresponding part. Both parts can become intermixed since, after having left the corridor, MA can temporarily return into it. Since part P 2 starts at the exit point of the corridor, then P2 L + C (3.2)
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where C = D 2 + W 2 is the hypotenuse AT of the triangle ATS (Figure 3.1a). As for part P 1 of the path inside the corridor, it will be, depending on the algorithm X, some curve. Observe that in order to defeat the bound that is, produce a path shorter than the bound (3.1) algorithm X has to decrease the path per obstacle ratio as much as possible. What is important for the proof is that, from the path per obstacle standpoint, every segment of P 1 that does not result in creating an equivalent segment of the actualized obstacle makes the path worse. All possible alternatives for P 1 can be clustered into three groups. We now consider these groups separately. 1. Part P 1 of the path never touches walls of the virtual obstacle (Figure 3.1a). As a result, no actual obstacles will be created in this case, i pi = 0. Then the resulting path is P > D, and so for an algorithm X that produces this kind of path the theorem holds. Moreover, at the nal evaluation, where
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