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By Webber R.E.

During this thesis, a number of facets of quadtree representations are analyzed. The quadtree is a hierarchical variable-resolution facts constitution appropriate for representing the geometric items of special effects, the polygonal maps of machine cartography, and the digitized photographs of desktop vision.The research of quadtrees is gifted in 3 parts:A) a proper semantics for quadtree algorithms,B) greater algorithms for manipulating the traditional area quadtree, andC) diversifications of the quadtree method to the duty of representing polygonal maps.

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4. edu. 5. A. Czumaj, P. Krysta, and B. V¨ ocking. Selfish traffic allocation for server farms. In Proc. 34th SOTC, ACM Press, 2002, pp. 287–296. 6. A. Czumaj and B. V¨ ocking. Tight bounds for worst-case equilibria. In Proc. of 13th ACM-SIAM Sympos. Discrete Algorithms, ACM Press, 2002, pp. 413–420. 7. J. M. Karp. Theoretical improvement in algorithmic efficiency for network flow problems. Journal of the ACM 19 (2) (1972), 248–264. 8. E. Even-Dar, A. Kesselman, and Y. Mansour. Convergence time to Nash equilibria.

In section 3 we show how the framework of Garg and Konemann can be adapted to our setting where multiple primal problems are being approximated using a single dual subroutine. e. C1 ≤ C2 ≤ . . ≤ Cn . We first use lemma 1 to describe the j-th candidate dual solution, x(j) : (j) (j) (j) x1 = x2 = . . = xj = j 1 j i=1 (j) Ci , and xi = 0 for i > j. Let Cj denote the quantity i=1 Ci . , xj = 1/Cj . This upper envelope solution has several desirable properties. First, it dominates all the candidate dual solutions Bandwidth Allocation in Networks 35 and hence, Pk (x) ≥ βk (l) for all 1 ≤ k ≤ n.

Fast fair and frugal bandwidth allocation in atm networks. Proceedings of the 10th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 92–101, 1999. 5. R. Bhargava, A. Goel, and A. Meyerson. Using approximate majorization to characterize protocol fairness. Proceedings of ACM Sigmetrics, pages 330–331, June 2001. (Poster paper). Bandwidth Allocation in Networks 41 6. N. Garg and J. Konemann. Faster and simpler algorithms for multicommodity flow and other fractional packing problems. 39th Annual Symposium on Foundations of Computer Science, pages 300–309, 1998.

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