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d(A) is used as a shorthand notation for d(A, T ). d(Ai ) signi es the fact that point A is located on the boundary of the ith obstacle met by MA on its way to point T . P is the total length of the path generated by MA on its way from S to T . pi is the perimeter of ith obstacle encountered by MA. i pi is the sum of perimeters of obstacles met by MA on its way to T , or of obstacles contained in a speci c area of the scene; this quantity will be used to assess performance of a path planning algorithm or to compare path planning algorithms.
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3.2 UNIVERSAL LOWER BOUND FOR THE PATH PLANNING PROBLEM This lower bound, formulated in Theorem 3.2.1 below, informs us what performance can be expected in the worst case from any path planning algorithm operating within our model. The bound is formulated in terms of the length of paths generated by MA on its way from point S to point T . We will see later that the bound is a powerful means for measuring performance of different path planning procedures. Theorem 3.2.1 ([58]). For any path planning algorithm satisfying the assumptions of our model, any (however large) P > 0, any (however small) D > 0, and any (however small) > 0, there exists a scene in which the algorithm will generate a path of length no less than P , P D+
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where D is the distance between points S and T , and pi are perimeters of obstacles intersecting the disk of radius D centered at point T . Proof: We want to prove that for any known or unknown algorithm X a scene can be designed such that the length of the path generated by X in it will satisfy (3.1).2 Algorithm X can be of any type: It can be deterministic or random; its intermediate steps may or may not depend on intermediate results; and so on. The only thing we know about X is that it operates within the framework of our model above. The proof consists of designing a scene with a special set of obstacles and then proving that this scene will force X to generate a path not shorter than P in (3.1).
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In Section 3.5 we will learn of a lower bound that is better and tighter: P D + 1.5 i pi . The proof of that bound is somewhat involved, so in order to demonstrate the underlying ideas we prefer to consider in detail the easier proof of the bound (3.1).
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UNIVERSAL LOWER BOUND FOR THE PATH PLANNING PROBLEM
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We will use the following scheme to design the required scene (called the resultant scene). The scene is built in two stages. At the rst stage, a virtual obstacle is introduced. Parts of this obstacle or the whole of it, but not more, will eventually become, when the second stage is completed, the actual obstacle(s) of the resultant scene. Consider a virtual obstacle shown in Figure 3.1a. It presents a corridor of nite width 2W > and of nite length L. The top end of the corridor is closed. The corridor is positioned such that the point S is located at the middle point of its closed end; the corridor opens in the direction opposite to the line (S, T ). The thickness of the corridor walls is negligible compared to its other dimensions. Still in the rst stage, MA is allowed to walk from S to T along the path prescribed by the algorithm X. Depending on the X s procedure, MA may or may not touch the virtual obstacle. When the path is complete, the second stage starts. A segment of the virtual obstacle is said to be actualized if all points of the inside wall of the segment have been touched by MA. If MA has contiguously touched the inside wall of the virtual obstacle at some length l, then the actualized segment is exactly of length l. If MA touched the virtual obstacle at a point and then bounced back, the corresponding actualized area is considered to be a wall segment of length around the point of contact. If two segments of the MA s path along the virtual obstacle are separated by an area of the virtual obstacle that MA has not touched, then MA is said to have actualized two separate segments of the virtual obstacle. We produce the resultant scene by designating as actual obstacles only those areas of the virtual obstacle that have been actualized. Thus, if an actualized
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