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Below we rst review those rst graph-searching approaches, switching then to the proper prior work on robot motion planning. 2.9.1 The Beginnings Leonhard Euler, perhaps the most famous mathematician of all time, was born on April 15, 1707, in Basel, Switzerland, and spent most of his career, between 1727 and 1741 and then from 1766 until his death on September 18, 1783, in St. Petersburg, Russia, holding a prestigious position of academician that is, a full member of Russian Academy of Sciences. As his fame grew, in 1733 he succeeded Daniel Bernoulli to the chair of mathematics in the Academy. It was at this rst period of his St. Petersburg career, in 1736, that Euler proposed and solved a problem that was to become famous under the name K enigsberg Bridge o Problem. This work marked the beginning of two new mathematical disciplines, graph theory and topology. It also gives an important insight into the robot motion planning problem. The city of K enigsberg (called today Kaliningrad and being a part of Russia) o was divided into two parts by the Pregel River, with the Island of Kneiphof in the middle. Seven bridges connected the island with the rest of the city (see Figure 2.18a). The question posed to Euler by the city s residents was this: Can a pedestrian, starting at some point, pass all seven bridges and return to the starting point so that he will traverse each bridge exactly once To nd the solution for the puzzled residents of K enigsberg, Euler decided o to rst reduce the problem to an equivalent abstract problem. In a leap of imagination, he said that the shapes and dimensions of the masses of lands that the bridges connect (A, B, C, D, Figure 2.18a) are immaterial for the problem. What matters are the connectivity properties of the scene, what today we would call the topological properties of space. This argument became the beginning of the discipline of topology. Euler denoted the land masses as vertices of a diagram, and he denoted the bridges as edges connecting the vertices (Figure 2.18b); hence the graph theory was born.
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C c d g c A e f D A a B (a) (b) b f d D e C g
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The K enigsberg Bridge Problem. o
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The path that the K enigsberg citizens wanted is called today a Euler path, o and the graph it corresponds to is called a Euler graph. De ne the number of edges incident to a graph vertex as the vertex degree. In today s formulation the related theorem sounds as follows: Theorem 2.9.1 (Euler [38]). A nite graph G is an Euler graph if and only if (a) G is connected and (b) every vertex of G is of an even degree. In the case of distinct starting and nal points (which is the situation typical for the robot motion planning problems), exactly two vertices must be of an odd degree. For the K enigsberg Bridge Problem the answer to the question posed to o Euler is therefore no because all the vertices on the corresponding graph are of an odd degree. As we saw above in the role of connectivity graphs in the Piano Mover s model, graph theory became an important tool in designing motion planning strategies with complete information. We will see in s 3, 5, and 6 that topology became a no less important tool in sensor-based motion planning. Neither Euler nor many of his followers asked explicitly what information about the scene was available to the traveler in the K enigsberg Bridge Problem.6 o It was implicitly assumed that the traveler had complete information.7 What if he didn t What if at any moment of the trip the traveler s knowledge was limited by what he could see around him plus whatever he remembered from the path he had traversed already What if this more realistic situation took place No live creature counts on knowing in advance all the objects on its journey, or calculates the precise path in advance. Algorithmically, the question about available input information puts the problem squarely into the domain of sensor-based motion planning, presenting it as a maze-searching problem. Clearly, even if the K enigsberg bridges made a Euler graph, and if the traveler had no picture as in o Figure 2.18, we can doubt that he would pass every bridge exactly once, except perhaps by sheer chance. If not, what would be the traveler s performance say, with the best algorithm possible Can a strategy be designed that will guarantee at least passing the whole graph when one starts with a zero knowledge about it If so, how about doing it in some reasonably ef cient manner We will return to these questions later in this chapter. This branch of motion planning which can be formulated as moving in a graph without prior information about it started long before Euler. Since the times of Theseus of Athens, people had great interest in labyrinths (mazes). After Theseus slew the Minotaur, he used the thread of glittering jewels given to him by Ariadne to nd his way in the passages of the Labyrinth of Knossos. Many
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Eventually this question did appear in graph theory, though much later, as a question of existence of local algorithms [39]. 7 Interestingly, when in the 1960s and 1970s researchers turned to the problem of robot motion planning, the question of input information was not raised either. There seems to be something in human psychology that, unless told otherwise, between two choices minimum and maximum of input information we implicitly assume the latter.
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