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lec5 - R d 4 Fixed d,generically classify no solution...

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Geometric Constraint Lecture - Higher Dimension Instructor: Meera Sitharam, Recorded by Heping Gao Thuresday, Feb 09, 2006 1 Introduction Dimension is variable. In which dimension the constraint graph is realizable? What’s the minimum dimension the constraint graph realizable? What kind of constraint graph is realizable in any dimension? 2 Pointes and Distances Considering points and distances, HGCS(Higer Dimension Constraint System):(G, P ) 1. (Generically) What’s the minimum d so that it is realizable in R d , C d ? 2. Is (G, P ) realizable in any ±nite dimension d? a. If (G, P ) is not a metric space (Geometry inequality doesn’t hold ), there doesn’t exist d s.t. (G, P ) is realizable in R d . b. Exact characterization? — Caley Menger (Algebraic) — Bnurgain/Milmau/Kigie/Sehechtman /Jolnson-Hindenstruss 3. Is G d-realizable? Def: A ( G, P ) [G] is d-realizable (generically) i² any Euclidean realization in any dimension implies it can be realizable in
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Unformatted text preview: R d . 4. Fixed d,generically classify : no solution, unique solution or nitely many solution. Note, in 3d, the above question is still open. Particularly, for what special class of graphs in d Dimension, we can classify it? For question 1, a randomized algorithm with high probability (1 ( | a | )), nds an appropriate P + realization in approximately (1+ ) d min dimension d by running time T ( | a | , 1 /, 1 , 1 / ). 1 3 Non-Distance Constraint Very specif graphs (regular graphs) 4 Questions and Discussion 1. How to generate a random subspace oF dimension d in R n ? or example, we can sample two angles For 3D. How to sample 2-d in R 4 ? 2. In 2-d, the lasgest m s.t. (1). m points and m lines s.t. every pair oF points is seperated by exactly m/2 lines; (2). or maximum m, fnd all such realization? Modulo what? 2...
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