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Unformatted text preview: ENT SET Input: graph G Output: independent set I I 0 Mark all nodes of G of degree >= 9 while some nodes remain unmarked do Choose an unmarked node v Mark v and all neighbors of v I I U {v} triangulate. Icosahedron Schlegel diagram: 5 triangles meet at each vertex. There exist at least n/2 vertices of degree at most 8 (derivation). An independent set of a polytope graph of n vertices produced by INDEPENDENT SET has size at least n/18 (derivation). Extremal Polytope Queries Details (continued) i i +1 Extremal Polytope Queries Details (continued) Extremal Polytope Queries Main Idea (repeated in more detail) To use nested polytope hierarchy to answer an extreme point query: Find extreme with respect to inner polytope Pk (brute-force search) Move from polytope Pi+1 to Pi (Assume w.l.o.g. vertical query direction) Yields bounds on # edges to check. Let ai and ai+1 be uniquely highest vertices of Pi and Pi+1. Then either ai = ai+1 or ai+1 is the highest among the vertices adjacent to ai ai is the highest among the parents of the extreme edges Li+1 and Ri+1 L'i+1 = R'i+1 (See proof sketch in future slide.) (See definition of L, R in future slide.) ' P'i+1 tetrahedron projected onto xz plane (Plane effect : see next slide) Extremal Polytope Queries Details (continued) For maximal z query: Moving from Pi+1 to Pi is like raising a plane orthogonal to z axis from ai+1 to ai...a0 Extremal Polytope Queries Main Idea (continued) Algorithm: EXTREME POINT of a POLYTOPE Input: polytope P and direction vector u Output: vertex a of P extreme in u direction Construct nested polytope hierarchy P = P0, P1,..., Pk ak vertex of Pk extreme in u direction Compute Lk and Rk for i = k - 1, k - 2, ...,1, 0 do ai extreme vertex among ai+1 and parents of Li+1 and Ri+1 for all edges incident to ai do save extreme edges Li and Ri if ai = ai+1 then u Pk P0 polytope extreme-point queries can be answered in O(log n) time each else (ai = ai+1) compute and space preprocessing, After O(n) time Li from Li+1 etc... Extremal Polytope Queries Details (continued) Lemma 7.10.2: (repeated) Let ai and ai+1 be uniquely highest vertices of Pi...
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This note was uploaded on 02/13/2012 for the course CS 91.504 taught by Professor Daniels during the Spring '10 term at UMass Lowell.

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