2005
DOI: 10.1007/10991541_10
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Collision Free Motion Planning on Graphs

Abstract: Abstract. A topological theory initiated in [4], [5] uses methods of algebraic topology to estimate numerically the character of instabilities arising in motion planning algorithms. The present paper studies random motion planning algorithms and reveals how the topology of the robot's configuration space influences their structure. We prove that the topological complexity of motion planning TC(X) coincides with the minimal n such that there exists an nvalued random motion planning algorithm for the system; her… Show more

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Cited by 29 publications
(44 citation statements)
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“…By Proposition 4.5(1) & (3), these equivalence classes correspond to the only elements of the standard basis for H 1 (U D 4 T min ; Z/2Z) which might have non-trivial cup products. In fact, checking labels of edges, it is not difficult to see that cases (1)-(5) and(8) are all (distinct) critical cells, and thus correspond to linearly independent elements of the standard basis for H 2 (U D 4 T min ; Z/2Z). We now consider the remaining edges.…”
mentioning
confidence: 98%
“…By Proposition 4.5(1) & (3), these equivalence classes correspond to the only elements of the standard basis for H 1 (U D 4 T min ; Z/2Z) which might have non-trivial cup products. In fact, checking labels of edges, it is not difficult to see that cases (1)-(5) and(8) are all (distinct) critical cells, and thus correspond to linearly independent elements of the standard basis for H 2 (U D 4 T min ; Z/2Z). We now consider the remaining edges.…”
mentioning
confidence: 98%
“…The second part was claimed in [4,Theorems 9,10]. Both parts easily follow from our local step-by-step computations.…”
Section: Two-point Braid Groups Of Graphs In the Unordered Casementioning
confidence: 56%
“…vertices with valence at least 3). Farber showed that TC(Conf n (T )) = 2|V ≥3 | whenever n ≥ 2|V ≥3 |; see [Far05] and also Farber's survey article [Far17]. In particular, TC(Conf n (T )) doesn't depend on n within that range.…”
Section: Introductionmentioning
confidence: 99%