1. Data-Powered Algorithms Bernard Chazelle Princeton University Bernard Chazelle Princeton University

3. Linear Programming Linear Programming

12. N constraints and d variables

14. Dimension Reduction  10000  25 Images (face recognition) Signals (voice recognition) Text (NLP)...... Nearest neighbor searching Clustering...

15. Dimension reduction All pairwise distances nearly preserved

16. Johnson-Lindenstrauss Transform (JLT) c log n  2 d Random Orthogonal Matrix v d

17. Friendly JLT c log n  2 d N(0,1)N(0,1)N(0,1) N(0,1) N(0,1)N(0,1)N(0,1) N(0,1) N(0,1)N(0,1)N(0,1) N(0,1) N(0,1)N(0,1)N(0,1) N(0,1)

18. Friendlier JLT c log n  2 d1+ -1+ -1+ -1+ - 1+ - 1+ - 1+ -1+ - 1+ - 1+ - 1+ - 1+ -1+ - 1+ -1+ - 1+ - d log n  2  2 =  

19. Sparse JLT ? c log n  2 1+ - 1+ -1+ - 1+ -1+ - 1+ - 1+ - 0 0 0 0 0 0 0 0 0 d 1 d 0 0 0 0... o(1)-Fraction non-zeros

20. Main Tool: Uncertainty Principle Time Frequency Heisenberg

21. Fast Johnson-Lindenstrauss Transform (FJLT) 1 + - 1 + - 1 + - 1 + - d Discrete Fourier Transform dd c log n  2... 0 N(0,1) = O  + d log d + d  log 3 n  2  2d Optimal ??

23. theory experimentation

24. computation theory experimentation

25. computation theory experimentation

26. input output Most interesting problems are too hard !! Most interesting problems are too hard !!

27. input output randomization approximation So, we change the model… So, we change the model…

28. input output randomization approximation PTAS for ETSP

29. input output randomization approximation Impossible to approximate chromatic chromatic number within a factor of… Impossible to approximate chromatic chromatic number within a factor of…

30. input output randomization approximation Property Testing [RS’96, GGR’96] Property Testing [RS’96, GGR’96] Berkeley “school” (program checking & probabilistic proofs) Berkeley “school” (program checking & probabilistic proofs)

32. Distance is 3

33. Distance is 4

34. nono yesyes bipartitebipartite

35. nono yesyes bipartitebipartite anythinganything [GR’97][GR’97]

38. Birthday paradox 6262 1818 77 polylog cycles 1717 Mixing case

39. [M’89][M’89] Nonmixing implies small cuts Non-mixing case

40. Dense graphs [GGR98, AK99] Hofstadter. Godel, Escher, Bach. Is graph k-colorable?

41. Main tool Szemerédi’s Regularity Lemma Far from k-colorable Lots of witnesses

42. Property Testing  Graph algorithms  connectivity  acyclicity  k-way cuts  clique  Distributions  independence  entropy  monotonicity  distances  Geometry  convexity  disjointness  delaunay  plane EMST http://www.cs.princeton.edu/~chazelle/http://www.cs.princeton.edu/~chazelle/