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… Extractors, and the closely related “Dispersers”, exhibit some of the most “random-like” properties of explicitly constructed combinatorial structures. In turn, extractors and dispersers have many applications in “removing randomness” in various settings, and in making randomized constructions explicit …
The Story of Randomized Computations and Weak Random Sources
Sampling, Simulations, Algorithms (e.g. Approximated TSP).
Hey Santa, you can use my cat !!
almost uniform output
source of biased correlated bits
Assumeb1 b2 …bi … are i.i.d. 0/1 variablesand bi =1with someprobabilityp < 1then translate
10 0Deterministic Extraction
Other “easy” sources: markov chains[vN51,Eli72,Blu84],two independent sources[SV84,Vaz85,CG85] , bit-fixing sources[CGH+85,BBR85,BL85,LLS87,CDH+00],some efficiently samplable sources[TV00].
p = p(b1 b2 …bi-1)[½-, ½+]
cannot deterministically extract even a single bit !!
A single SV-Source is sufficient to simulate BPPCan this Work for all Sources?
x, Ext(x,y)Weak Sources in (Space Bounced) Computations
Other results in the high min-entropy case [GW], low min-entropy case [GW,SZ], dispersers [SSZ,TaS]
Ha yes ... and there is a very nice one based on the NW generatorsConstructions of Extractors (cont.)
XA Construction in Search of Many Applications [WZ]