Download - Data-Powered Algorithms Bernard Chazelle Princeton University Bernard Chazelle Princeton University
![Page 1: Data-Powered Algorithms Bernard Chazelle Princeton University Bernard Chazelle Princeton University](https://reader030.vdocuments.us/reader030/viewer/2022032800/56649d445503460f94a20f07/html5/thumbnails/1.jpg)
Data-Powered AlgorithmsData-Powered AlgorithmsData-Powered AlgorithmsData-Powered Algorithms
Bernard ChazelleBernard Chazelle
Princeton UniversityPrinceton University
Bernard ChazelleBernard Chazelle
Princeton UniversityPrinceton University
![Page 2: Data-Powered Algorithms Bernard Chazelle Princeton University Bernard Chazelle Princeton University](https://reader030.vdocuments.us/reader030/viewer/2022032800/56649d445503460f94a20f07/html5/thumbnails/2.jpg)
![Page 3: Data-Powered Algorithms Bernard Chazelle Princeton University Bernard Chazelle Princeton University](https://reader030.vdocuments.us/reader030/viewer/2022032800/56649d445503460f94a20f07/html5/thumbnails/3.jpg)
Linear ProgrammingLinear Programming Linear ProgrammingLinear Programming
![Page 4: Data-Powered Algorithms Bernard Chazelle Princeton University Bernard Chazelle Princeton University](https://reader030.vdocuments.us/reader030/viewer/2022032800/56649d445503460f94a20f07/html5/thumbnails/4.jpg)
![Page 5: Data-Powered Algorithms Bernard Chazelle Princeton University Bernard Chazelle Princeton University](https://reader030.vdocuments.us/reader030/viewer/2022032800/56649d445503460f94a20f07/html5/thumbnails/5.jpg)
![Page 6: Data-Powered Algorithms Bernard Chazelle Princeton University Bernard Chazelle Princeton University](https://reader030.vdocuments.us/reader030/viewer/2022032800/56649d445503460f94a20f07/html5/thumbnails/6.jpg)
![Page 7: Data-Powered Algorithms Bernard Chazelle Princeton University Bernard Chazelle Princeton University](https://reader030.vdocuments.us/reader030/viewer/2022032800/56649d445503460f94a20f07/html5/thumbnails/7.jpg)
![Page 8: Data-Powered Algorithms Bernard Chazelle Princeton University Bernard Chazelle Princeton University](https://reader030.vdocuments.us/reader030/viewer/2022032800/56649d445503460f94a20f07/html5/thumbnails/8.jpg)
![Page 9: Data-Powered Algorithms Bernard Chazelle Princeton University Bernard Chazelle Princeton University](https://reader030.vdocuments.us/reader030/viewer/2022032800/56649d445503460f94a20f07/html5/thumbnails/9.jpg)
![Page 10: Data-Powered Algorithms Bernard Chazelle Princeton University Bernard Chazelle Princeton University](https://reader030.vdocuments.us/reader030/viewer/2022032800/56649d445503460f94a20f07/html5/thumbnails/10.jpg)
![Page 11: Data-Powered Algorithms Bernard Chazelle Princeton University Bernard Chazelle Princeton University](https://reader030.vdocuments.us/reader030/viewer/2022032800/56649d445503460f94a20f07/html5/thumbnails/11.jpg)
![Page 12: Data-Powered Algorithms Bernard Chazelle Princeton University Bernard Chazelle Princeton University](https://reader030.vdocuments.us/reader030/viewer/2022032800/56649d445503460f94a20f07/html5/thumbnails/12.jpg)
N constraints and d variablesN constraints and d variables
![Page 13: Data-Powered Algorithms Bernard Chazelle Princeton University Bernard Chazelle Princeton University](https://reader030.vdocuments.us/reader030/viewer/2022032800/56649d445503460f94a20f07/html5/thumbnails/13.jpg)
N constraints and d variablesN constraints and d variables
![Page 14: Data-Powered Algorithms Bernard Chazelle Princeton University Bernard Chazelle Princeton University](https://reader030.vdocuments.us/reader030/viewer/2022032800/56649d445503460f94a20f07/html5/thumbnails/14.jpg)
Dimension ReductionDimension Reduction
1000010000 2525Images (face recognition)Images (face recognition) Signals (voice recognition)Signals (voice recognition)Text (NLP)Text (NLP). . . . . .
Nearest neighbor searchingNearest neighbor searchingClusteringClustering. . .. . .
![Page 15: Data-Powered Algorithms Bernard Chazelle Princeton University Bernard Chazelle Princeton University](https://reader030.vdocuments.us/reader030/viewer/2022032800/56649d445503460f94a20f07/html5/thumbnails/15.jpg)
Dimension reductionDimension reduction
All pairwise distances nearly preserved
![Page 16: Data-Powered Algorithms Bernard Chazelle Princeton University Bernard Chazelle Princeton University](https://reader030.vdocuments.us/reader030/viewer/2022032800/56649d445503460f94a20f07/html5/thumbnails/16.jpg)
Johnson-Lindenstrauss Transform (JLT)
c log nc log n22
dd
Random OrthogonalMatrix
vv dd
![Page 17: Data-Powered Algorithms Bernard Chazelle Princeton University Bernard Chazelle Princeton University](https://reader030.vdocuments.us/reader030/viewer/2022032800/56649d445503460f94a20f07/html5/thumbnails/17.jpg)
Friendly JLTFriendly JLT
c log nc log n22
dd
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)
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)
![Page 18: Data-Powered Algorithms Bernard Chazelle Princeton University Bernard Chazelle Princeton University](https://reader030.vdocuments.us/reader030/viewer/2022032800/56649d445503460f94a20f07/html5/thumbnails/18.jpg)
Friendlier JLTFriendlier JLT
c log nc log n22
dd
11++-- 11++-- 11++-- 11++--11++--11++--11++--11++--
11++--11++-- 11++--
11++-- 11++--11++-- 11++--
11++--
d log nd log n 22 = =
![Page 19: Data-Powered Algorithms Bernard Chazelle Princeton University Bernard Chazelle Princeton University](https://reader030.vdocuments.us/reader030/viewer/2022032800/56649d445503460f94a20f07/html5/thumbnails/19.jpg)
Sparse JLTSparse JLT? ?
c log nc log n22
11++--11++--11++--
11++-- 11++--11++--
11++--
00
00
00
00
00
00
0000
00
dd
11 dd
00
00
00
00
. .
..
. .
. .
..
. .
o(1)-Fraction non-o(1)-Fraction non-zeroszeros
![Page 20: Data-Powered Algorithms Bernard Chazelle Princeton University Bernard Chazelle Princeton University](https://reader030.vdocuments.us/reader030/viewer/2022032800/56649d445503460f94a20f07/html5/thumbnails/20.jpg)
Main Tool: Uncertainty Main Tool: Uncertainty PrinciplePrinciple
TimeTime
FrequencyFrequency
HeisenbergHeisenberg
![Page 21: Data-Powered Algorithms Bernard Chazelle Princeton University Bernard Chazelle Princeton University](https://reader030.vdocuments.us/reader030/viewer/2022032800/56649d445503460f94a20f07/html5/thumbnails/21.jpg)
Fast Johnson-Lindenstrauss Transform (FJLT)Fast Johnson-Lindenstrauss Transform (FJLT)
1+- 1+- 1+-
1+-
dd
DiscreteFourier
Transform
dddd
c log nc log n22
. . .
0N(0,1)
= =OO+ d log d + d + d log d + d loglog33 n n 22
dd
OptimalOptimal?? ??
![Page 22: Data-Powered Algorithms Bernard Chazelle Princeton University Bernard Chazelle Princeton University](https://reader030.vdocuments.us/reader030/viewer/2022032800/56649d445503460f94a20f07/html5/thumbnails/22.jpg)
![Page 23: Data-Powered Algorithms Bernard Chazelle Princeton University Bernard Chazelle Princeton University](https://reader030.vdocuments.us/reader030/viewer/2022032800/56649d445503460f94a20f07/html5/thumbnails/23.jpg)
theory experimentation
![Page 24: Data-Powered Algorithms Bernard Chazelle Princeton University Bernard Chazelle Princeton University](https://reader030.vdocuments.us/reader030/viewer/2022032800/56649d445503460f94a20f07/html5/thumbnails/24.jpg)
computation
theory experimentation
![Page 25: Data-Powered Algorithms Bernard Chazelle Princeton University Bernard Chazelle Princeton University](https://reader030.vdocuments.us/reader030/viewer/2022032800/56649d445503460f94a20f07/html5/thumbnails/25.jpg)
computation
theory experimentation
![Page 26: Data-Powered Algorithms Bernard Chazelle Princeton University Bernard Chazelle Princeton University](https://reader030.vdocuments.us/reader030/viewer/2022032800/56649d445503460f94a20f07/html5/thumbnails/26.jpg)
inputinput outputoutput
Most interestingMost interestingproblems areproblems are
too hard !!too hard !!
Most interestingMost interestingproblems areproblems are
too hard !!too hard !!
![Page 27: Data-Powered Algorithms Bernard Chazelle Princeton University Bernard Chazelle Princeton University](https://reader030.vdocuments.us/reader030/viewer/2022032800/56649d445503460f94a20f07/html5/thumbnails/27.jpg)
inputinput outputoutput
randomizationrandomization
approximationapproximation
So, we change So, we change the model…the model…
So, we change So, we change the model…the model…
![Page 28: Data-Powered Algorithms Bernard Chazelle Princeton University Bernard Chazelle Princeton University](https://reader030.vdocuments.us/reader030/viewer/2022032800/56649d445503460f94a20f07/html5/thumbnails/28.jpg)
inputinput outputoutput
randomizationrandomization
approximationapproximationPTAS for ETSPPTAS for ETSPPTAS for ETSPPTAS for ETSP
![Page 29: Data-Powered Algorithms Bernard Chazelle Princeton University Bernard Chazelle Princeton University](https://reader030.vdocuments.us/reader030/viewer/2022032800/56649d445503460f94a20f07/html5/thumbnails/29.jpg)
inputinput outputoutput
randomizationrandomization
approximationapproximation
Impossible toImpossible toapproximateapproximate chromatic chromatic
number withinnumber withina factor of… a factor of…
Impossible toImpossible toapproximateapproximate chromatic chromatic
number withinnumber withina factor of… a factor of…
![Page 30: Data-Powered Algorithms Bernard Chazelle Princeton University Bernard Chazelle Princeton University](https://reader030.vdocuments.us/reader030/viewer/2022032800/56649d445503460f94a20f07/html5/thumbnails/30.jpg)
inputinput outputoutput
randomizationrandomization
approximationapproximationProperty Property TestingTesting
[RS’96, [RS’96, GGR’96]GGR’96]
Property Property TestingTesting
[RS’96, [RS’96, GGR’96]GGR’96]
Berkeley “school”Berkeley “school”(program checking &(program checking &probabilistic proofs)probabilistic proofs)
Berkeley “school”Berkeley “school”(program checking &(program checking &probabilistic proofs)probabilistic proofs)
![Page 31: Data-Powered Algorithms Bernard Chazelle Princeton University Bernard Chazelle Princeton University](https://reader030.vdocuments.us/reader030/viewer/2022032800/56649d445503460f94a20f07/html5/thumbnails/31.jpg)
![Page 32: Data-Powered Algorithms Bernard Chazelle Princeton University Bernard Chazelle Princeton University](https://reader030.vdocuments.us/reader030/viewer/2022032800/56649d445503460f94a20f07/html5/thumbnails/32.jpg)
Distance is 3Distance is 3Distance is 3Distance is 3
![Page 33: Data-Powered Algorithms Bernard Chazelle Princeton University Bernard Chazelle Princeton University](https://reader030.vdocuments.us/reader030/viewer/2022032800/56649d445503460f94a20f07/html5/thumbnails/33.jpg)
Distance is 4Distance is 4Distance is 4Distance is 4
![Page 34: Data-Powered Algorithms Bernard Chazelle Princeton University Bernard Chazelle Princeton University](https://reader030.vdocuments.us/reader030/viewer/2022032800/56649d445503460f94a20f07/html5/thumbnails/34.jpg)
nononono
yesyesyesyes
bipartitebipartitebipartitebipartite
![Page 35: Data-Powered Algorithms Bernard Chazelle Princeton University Bernard Chazelle Princeton University](https://reader030.vdocuments.us/reader030/viewer/2022032800/56649d445503460f94a20f07/html5/thumbnails/35.jpg)
nononono
yesyesyesyesbipartitebipartitebipartitebipartite
anythinganythinganythinganything
[GR’97][GR’97][GR’97][GR’97]
![Page 36: Data-Powered Algorithms Bernard Chazelle Princeton University Bernard Chazelle Princeton University](https://reader030.vdocuments.us/reader030/viewer/2022032800/56649d445503460f94a20f07/html5/thumbnails/36.jpg)
![Page 37: Data-Powered Algorithms Bernard Chazelle Princeton University Bernard Chazelle Princeton University](https://reader030.vdocuments.us/reader030/viewer/2022032800/56649d445503460f94a20f07/html5/thumbnails/37.jpg)
![Page 38: Data-Powered Algorithms Bernard Chazelle Princeton University Bernard Chazelle Princeton University](https://reader030.vdocuments.us/reader030/viewer/2022032800/56649d445503460f94a20f07/html5/thumbnails/38.jpg)
Birthday paradox Birthday paradox Birthday paradox Birthday paradox
62626262
181818187777
polylog cyclespolylog cyclespolylog cyclespolylog cycles
17171717
MixingMixing casecaseMixingMixing casecase
![Page 39: Data-Powered Algorithms Bernard Chazelle Princeton University Bernard Chazelle Princeton University](https://reader030.vdocuments.us/reader030/viewer/2022032800/56649d445503460f94a20f07/html5/thumbnails/39.jpg)
[M’89[M’89
]][M’89[M’89
]]Nonmixing implies small cutsNonmixing implies small cutsNonmixing implies small cutsNonmixing implies small cuts
Non-mixingNon-mixing casecaseNon-mixingNon-mixing casecase
![Page 40: Data-Powered Algorithms Bernard Chazelle Princeton University Bernard Chazelle Princeton University](https://reader030.vdocuments.us/reader030/viewer/2022032800/56649d445503460f94a20f07/html5/thumbnails/40.jpg)
Dense graphsDense graphsDense graphsDense graphs
[GGR98, AK99][GGR98, AK99][GGR98, AK99][GGR98, AK99]
Hofstadter. Godel, Escher, Bach.
Is graph k-colorable?Is graph k-colorable?Is graph k-colorable?Is graph k-colorable?
1010001
0101011
1101100
1010011
1101101
0010110
1011001
![Page 41: Data-Powered Algorithms Bernard Chazelle Princeton University Bernard Chazelle Princeton University](https://reader030.vdocuments.us/reader030/viewer/2022032800/56649d445503460f94a20f07/html5/thumbnails/41.jpg)
Main Main tooltoolMain Main tooltool
Szemerédi’s Regularity Lemma Szemerédi’s Regularity Lemma Szemerédi’s Regularity Lemma Szemerédi’s Regularity Lemma
Far from k-colorableFar from k-colorableFar from k-colorableFar from k-colorable
Lots of Lots of witnesseswitnesses
Lots of Lots of witnesseswitnesses
![Page 42: Data-Powered Algorithms Bernard Chazelle Princeton University Bernard Chazelle Princeton University](https://reader030.vdocuments.us/reader030/viewer/2022032800/56649d445503460f94a20f07/html5/thumbnails/42.jpg)
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/http://www.cs.princeton.edu/~chazelle/~chazelle/
http://www.cs.princeton.edu/http://www.cs.princeton.edu/~chazelle/~chazelle/