Bafna, V., Kulyanasundaram, B., and Pruhs, K., "Not All Insertion Methods Yield Constant Approximate Tours in the Euclidean Plane", Theoretical Computer Science, 125(2), pp.345-353, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... other edge of a tour on S # that respects the order given by TSP(S) Edges of this tour correspond to disjoint sections of TSP(S) so its length is at most TSP(S) After #log 2 n# stages we get down to a tour on just two vertices, which again has length at most TSP(S) Bafna et al. BKP94] and Azar [Aza94] see also [AA93] independently adapted a construction of Bentley and Saxe [BS80] to show that the bound of Theorem 8.1 is tight up to a factor of O(loglogn) We use a slightly simpler adaptation [CKT94] below. THEOREM 8.2 Some insertion orders produce tours of length ....

V. Bafna, B. Kalyanasundaram, and K. Pruhs. Not all insertion methods yield constant approximate tours in the Euclidean plane. Theoretical Computer Science, 125:345--353, 1994.

....This method performs better in practice than the other methods. The best known lower bound for this method is constant [Hu] there is a metric space for which it is 6:5, and it is 2:43 for the plane. Theorem 1. 1 was proved independently and at the same time by Bafna, Kalyanasundaram and Pruhs [BKP]. The basic approach in this paper resembles the one of Bentley and Saxe in [BS] for the nearest neighbor algorithm and of Alon and Azar [AA] for on line Steiner trees, but some different ideas are required. 2 The lower bound proofs We first prove Theorem 1.1. The metric space considered is the ....

V. Bafna, B. Kalyanasundaram and K. Pruhs, Not all insertion methods yields constant approximate tours in the Euclidean plane, Manuscript.

....metric spaces. Further, they showed that nearest insertion and cheapest insertion lead to a 2 approximation. It remained open for some time whether or not an insertion order exists that does not achieve a constant factor approximation. Independently, Azar [47] and Bafna, Kalyanasundaram, and Pruhs [49] showed that indeed an insertion order exists that has worst case factor Omega Gammacto n= log log n) even for instances in the Euclidean plane. Furthermore, Azar shows that the worst case factor for random insertion (add the sites in random order) is Omega Gamma 22 log n= log log log n) also ....

V. Bafna, B. Kalyanasundaram, and K. Pruhs. Not all insertion methods yield constant approximate tours in the Euclidean plane. Theoret. Comput. Sci., 125:345--353, 1994.

....that it is possible to maintain a O(log n) competitive spanning tree under the point by point scenario. Alon and Azar [1] give an Omega Gamma 18 n= log log n) lower bound on the competitiveness achievable for constructing a Steiner tree point by point in the plane. Further results can be found in [2, 4, 14]. The main difference between this scenario and the fixed graph scenario is in what information in known to the online algorithm about explored vertices. In the point by point scenario the online algorithm may not be aware of all edges incident to revealed points. More precisely, the online ....

V. Bafna, B. Kalyanasundaram, and K. Pruhs, Not all insertion methods yield constant approximate tours in the plane, Technical Report, Computer Science Department, University of Pittsburgh, 1992.

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Bafna, V., Kulyanasundaram, B., and Pruhs, K., "Not All Insertion Methods Yield Constant Approximate Tours in the Euclidean Plane", Theoretical Computer Science, 125(2), pp.345-353, 1994.

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