296:
67:
548:
413:
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219:
371:
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568:
339:
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195:
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155:
135:
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95:
513:
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computable examples of such graphs with good parameters. Algorithms that compute extractor (and disperser) graphs have found many applications in
681:
222:
301:
in the natural way. With this view it turns out that the extractor property is equivalent to: for any source of randomness
676:
637:
it is easy to show that extractor graphs with really good parameters exist. The challenge is to find explicit or
226:
671:
245:
22:
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627:
521:
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418:
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642:
350:
655:
465:
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74:
593:
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324:
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180:
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140:
120:
100:
80:
492:
665:
198:
345:
620:
233:
177:, the distribution on right vertices obtained by choosing a random node in
137:
neighbors (on the right), which has the added property that for any subset
518:
Extractors are interesting when they can be constructed with small
239:
An equivalent way to view an extractor is as a bivariate function
341:
616:
Extractor functions were originally researched as a way to
613:(the total randomness in the input sources) as possible.
117:
nodes on the right such that each node on the left has
596:
576:
556:
524:
495:
468:
441:
421:
379:
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183:
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103:
83:
25:
605:
582:
562:
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481:
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427:
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365:
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290:
213:
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169:
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61:
8:
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352:
326:
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182:
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122:
102:
82:
24:
157:of the left vertices of size at least
291:{\displaystyle E:\times \rightarrow }
201:to get a node x on the right side is
7:
489:denotes the uniform distribution on
62:{\displaystyle (N,M,D,K,\epsilon )}
14:
656:Recent developments in extractors
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496:
402:
383:
285:
279:
276:
273:
267:
261:
255:
56:
26:
1:
543:{\displaystyle K,D,\epsilon }
682:Theoretical computer science
623:from weakly random sources.
197:and then following a random
698:
408:{\displaystyle E(X,U_{D})}
16:Bipartite graph with nodes
428:{\displaystyle \epsilon }
214:{\displaystyle \epsilon }
227:total variation distance
607:
584:
564:
544:
509:
483:
456:
429:
409:
367:
366:{\displaystyle \log K}
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315:
292:
215:
191:
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131:
111:
97:nodes on the left and
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63:
608:
585:
565:
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510:
484:
482:{\displaystyle U_{T}}
457:
455:{\displaystyle U_{M}}
430:
410:
368:
336:
316:
293:
236:is a related graph.
216:
192:
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152:
132:
112:
92:
64:
635:probabilistic method
628:randomness extractor
594:
574:
554:
522:
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466:
439:
419:
377:
351:
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305:
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223:uniform distribution
205:
181:
161:
141:
121:
101:
81:
23:
373:, the distribution
606:{\displaystyle KD}
603:
580:
560:
540:
505:
479:
452:
425:
405:
363:
331:
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288:
211:
187:
167:
147:
127:
107:
87:
59:
583:{\displaystyle M}
563:{\displaystyle N}
334:{\displaystyle n}
314:{\displaystyle X}
190:{\displaystyle A}
170:{\displaystyle K}
150:{\displaystyle A}
130:{\displaystyle D}
110:{\displaystyle M}
90:{\displaystyle N}
689:
677:Pseudorandomness
654:Ronen Shaltiel,
643:computer science
612:
610:
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604:
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586:
581:
569:
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541:
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508:{\displaystyle }
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220:
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196:
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116:
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108:
96:
94:
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68:
66:
65:
60:
697:
696:
692:
691:
690:
688:
687:
686:
662:
661:
651:
639:polynomial time
592:
591:
590:is as close to
572:
571:
552:
551:
520:
519:
491:
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469:
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463:
442:
437:
436:
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349:
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179:
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159:
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75:bipartite graph
21:
20:
17:
12:
11:
5:
695:
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672:Graph families
664:
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602:
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579:
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527:
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330:
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221:-close to the
210:
186:
166:
146:
126:
106:
86:
58:
55:
52:
49:
46:
43:
40:
37:
34:
31:
28:
15:
13:
10:
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6:
4:
3:
2:
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683:
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669:
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640:
636:
631:
629:
626:
622:
619:
614:
600:
597:
577:
557:
537:
534:
531:
528:
525:
516:
499:
474:
470:
447:
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422:
397:
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360:
357:
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343:
328:
308:
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270:
264:
258:
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241:
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208:
200:
184:
164:
144:
124:
104:
84:
76:
72:
53:
50:
47:
44:
41:
38:
35:
32:
29:
632:
624:
617:
615:
550:relative to
517:
300:
238:
231:
225:in terms of
70:
18:
346:min-entropy
321:that gives
666:Categories
658:- a survey
649:References
633:Using the
621:randomness
435:-close to
538:ϵ
423:ϵ
358:
277:→
265:×
234:disperser
209:ϵ
71:extractor
54:ϵ
462:, where
618:extract
344:with
77:with
73:is a
570:and
342:bits
199:edge
625:See
415:is
355:log
19:An
668::
645:.
630:.
515:.
232:A
229:.
601:D
598:K
578:M
558:N
535:,
532:D
529:,
526:K
503:]
500:T
497:[
475:T
471:U
448:M
444:U
403:)
398:D
394:U
390:,
387:X
384:(
381:E
361:K
329:n
309:X
286:]
283:M
280:[
274:]
271:D
268:[
262:]
259:N
256:[
253::
250:E
185:A
165:K
145:A
125:D
105:M
85:N
69:-
57:)
51:,
48:K
45:,
42:D
39:,
36:M
33:,
30:N
27:(
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