Abstract
The introductory chapter explains the difference between shortest paths in finite graphs and shortest paths in Euclidean geometry, which is also called ‘the common geometry of our world’. The chapter demonstrates the diversity of such problems, defined between points in a plane, on a surface, or in the 3-dimensional space.
Ptolemy once asked Euclid whether there was any shorter way to a knowledge of geometry than by a study of the Elements, whereupon Euclid answered that there was no royal road to geometry.
Proclus Diadochus (410…412–485)
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Notes
- 1.
Read “if and only if”; abbreviation proposed by Paul Richard Halmos (1916–2006).
- 2.
Carl Friedrich Gauss (1777–1855) proved this theorem in his PhD thesis, published in 1799.
- 3.
Named after Èvariste Galois (1811–1832).
- 4.
Shown by Bartel Leendert Van der Waerden (1903–1996).
- 5.
Euclid of Alexandria (see Fig. 1.1, right) was living around −300. He wrote either as an individual, or as the leader of a team of mathematicians a multi-volume book Elements that established Euclidean geometry and number theory.
- 6.
Introduced by René Descartes (in Latin: Cartesius; 1596–1650).
- 7.
Named after Hermann Minkowski (1864–1909).
- 8.
Adjacency relations and metrics on grid points were introduced by the US computer scientist and mathematician Azriel Rosenfeld (1931–2004), a pioneer of computer vision.
- 9.
From http://hdl.handle.net/1911/9343, available as specified on http://creativecommons.org/licenses/by/2.5/.
- 10.
- 11.
The Dutch scientist Edger Wybe Dijkstra (1930–2002) has made many substantial contributions to computer science.
- 12.
This was defined by Camille Jordan (1838–1922) in 1893, and γ is also called a Jordan curve today.
- 13.
Used by Johann Benedict Listing (1808–1882) in 1861 when illustrating the skeletonisation of shapes in ℝ3.
- 14.
References
Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M.: Complexity and Approximation. Springer, New York (1999)
Bhowmick, P., Pal, O., Klette, R.: A linear-time algorithm for the generation of random digital curves. In: Proc. PSIVT, pp. 168–173. IEEE Comput. Soc., Los Alamitos (2010)
Canny, J., Reif, J.H.: New lower bound techniques for robot motion planning problems. In: Proc. IEEE Conf. Foundations Computer Science, pp. 49–60 (1987)
Choi, J., Sellen, J., Yap, C.-K.: Approximate Euclidean shortest path in 3-space. In: Proc. ACM Conf. Computational Geometry, pp. 41–48 (1994)
Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 2nd edn. MIT Press, Cambridge (2001)
Dijkstra, E.W.: A note on two problems in connection with graphs. Numer. Math. 1, 269–271 (1959)
Hart, P.E., Nilsson, N.J., Raphael, B.: A formal basis for the heuristic determination of minimum cost paths. IEEE Trans. Syst. Sci. Cybern. 4, 100–107 (1968)
Hochbaum, D.S. (ed.): Approximation Algorithms for NP-Hard Problems. PWS, Boston (1997)
Hromkovič, J.: Algorithms for Hard Problems. Springer, Berlin (2001)
Kleinberg, J., Tardos, E.: Algorithm Design. Pearson Education, Toronto (2005)
Latombe, J.-C.: Robot Motion Planning. Kluwer Academic, Boston (1991)
LaValle, S.M.: Planning Algorithms. Cambridge University Press, Cambridge, UK (2006)
Li, T.-Y., Chen, P.-F., Huang, P.-Z.: Motion for humanoid walking in a layered environment. In: Proc. Conf. Robotics Automation, vol. 3, pp. 3421–3427 (2003)
Mayr, E.W., Prömel, H.J., Steger, A. (eds.): Lectures on Proof Verification and Approximation Algorithms. Springer, Berlin (1998)
Mitchell, J.S.B.: Geometric shortest paths and network optimization. In: Handbook of Computational Geometry, pp. 633–701. Elsevier, Amsterdam (2000)
Mitchell, J.S.B., Sharir, M.: New results on shortest paths in three dimensions. In: Proc. SCG, pp. 124–133 (2004)
Moore, E.F.: The shortest path through a maze. In: Proc. Int. Symp. Switching Theory, vol. 2, pp. 285–292. Harvard University Press, Cambridge (1959)
Rabani, Y.: Approximation algorithms. http://www.cs.technion.ac.il/~rabani/236521.04.wi.html (2004). Accessed 28 October 2004
Skiena, S.S.: The Algorithm Design Manual. Springer, New York (1998)
Vazirani, V.V.: Approximation Algorithms. Springer, Berlin (2001)
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Li, F., Klette, R. (2011). Euclidean Shortest Paths. In: Euclidean Shortest Paths. Springer, London. https://doi.org/10.1007/978-1-4471-2256-2_1
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