2211:
Roughly speaking, in order to solve the undirected s-t connectivity problem in logarithmic space, the input graph is transformed, using a combination of powering and the zigzag product, into a constant-degree regular graph with a logarithmic diameter. The power product increases the expansion (hence
2191:
In 2002 Omer
Reingold, Salil Vadhan, and Avi Wigderson gave a simple, explicit combinatorial construction of constant-degree expander graphs. The construction is iterative, and needs as a basic building block a single, expander of constant size. In each iteration the zigzag product is used in order
2195:
From the properties of the zigzag product mentioned above, we see that the product of a large graph with a small graph, inherits a size similar to the large graph, and degree similar to the small graph, while preserving its expansion properties from both, thus enabling to increase the size of the
1735:
1642:
945:
864:
783:
701:
2044:
254:, and connects the vertices by moving a small step (zig) inside a cloud, followed by a big step (zag) between two clouds, and finally performs another small step inside the destination cloud.
2208:
problem, the problem of testing whether there is a path between two given vertices in an undirected graph, using only logarithmic space. The algorithm relies heavily on the zigzag product.
1503:
1417:
1014:
620:
261:. When the zig-zag product was first introduced, it was used for the explicit construction of constant degree expanders and extractors. Later on, the zig-zag product was used in
488:
387:
1543:
1268:
1956:
1769:
1328:
514:
212:
91:
579:
2076:
1444:
1358:
1075:
541:
65:
2192:
to generate another graph whose size is increased but its degree and expansion remains unchanged. This process continues, yielding arbitrarily large expanders.
2176:
2156:
2136:
2116:
2096:
1976:
1898:
1878:
1832:
1812:
1789:
1647:
1308:
1288:
1227:
1207:
1175:
1155:
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1115:
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1048:
427:
407:
323:
303:
252:
232:
183:
159:
135:
115:
1924:
1858:
453:
349:
2247:
2212:
reduces the diameter) at the price of increasing the degree, and the zigzag product is used to reduce the degree while preserving the expansion.
1548:
1209:
is a “good enough” expander (has a large spectral gap) then the expansion of the zigzag product is close to the original expansion of
2313:
262:
626:
1981:
872:
791:
710:
1449:
1363:
1177:
the product in fact splits the edges of each original vertex between the vertices of the cloud that replace it.
584:
2339:
458:
357:
138:
2245:(2000), "Entropy waves, the zig-zag graph product, and new constant-degree expanders and extractors",
1508:
1310:
vertices, whose second largest eigenvalue (of the associated random walk) has absolute value at most
1189:, with an important property of the zigzag product the preservation of the spectral gap. That is, if
953:
1235:
2279:
2252:
1929:
1748:
1313:
493:
191:
70:
546:
2318:
2288:
2262:
2221:
2049:
94:
1422:
1336:
1053:
519:
2205:
1730:{\displaystyle f(\lambda _{1},\lambda _{2})<\lambda _{1}+\lambda _{2}+\lambda _{2}^{2}}
266:
165:, then the expansion of the resulting graph is only slightly worse than the expansion of
44:
17:
2161:
2141:
2121:
2101:
2081:
1961:
1883:
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1774:
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412:
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308:
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237:
217:
168:
162:
144:
120:
100:
1903:
1837:
432:
328:
2333:
2304:
2300:
2274:
2242:
2234:
1030:
It is immediate from the definition of the zigzag product that it transforms a graph
137:) and produces a graph that approximately inherits the size of the large one but the
39:
2308:
2238:
1186:
352:
31:
2266:
2322:
2292:
2311:(2006), "Pseudorandom walks on regular digraphs and the RL vs. L problem",
141:
of the small one. An important property of the zig-zag product is that if
2204:
In 2005 Omer
Reingold introduced an algorithm that solves the undirected
1637:{\displaystyle (N_{1}\cdot D_{1},D_{2}^{2},f(\lambda _{1},\lambda _{2}))}
2257:
2200:
Solving the undirected s-t connectivity problem in logarithmic space
2248:
Proc. 41st IEEE Symposium on
Foundations of Computer Science (FOCS)
258:
2314:
Proc. 38th ACM Symposium on Theory of
Computing (STOC)
696:{\displaystyle \mathrm {Rot} _{G\circ H}((v,a),(i,j))}
2164:
2144:
2124:
2104:
2084:
2052:
1984:
1964:
1932:
1906:
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629:
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435:
415:
395:
360:
331:
311:
291:
240:
220:
194:
171:
147:
123:
103:
73:
47:
2039:{\displaystyle G|_{S}\circ H=G\circ H|_{S\times D}}
1771:operates separately on each connected component of
2170:
2150:
2130:
2110:
2090:
2070:
2038:
1970:
1950:
1918:
1892:
1872:
1852:
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1806:
1783:
1763:
1729:
1636:
1537:
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1411:
1352:
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1302:
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1262:
1221:
1201:
1169:
1149:
1129:
1109:
1089:
1069:
1042:
1008:
939:
858:
777:
695:
614:
573:
535:
508:
482:
447:
421:
401:
381:
343:
317:
297:
246:
226:
206:
177:
153:
129:
109:
85:
59:
1137:. Roughly speaking, by amplifying each vertex of
2277:(2008), "Undirected connectivity in log-space",
1185:The expansion of a graph can be measured by its
1117:, the zigzag product will reduce the degree of
940:{\displaystyle (b,j')=\mathrm {Rot} _{H}(b',j)}
859:{\displaystyle (w,b')=\mathrm {Rot} _{G}(v,a')}
778:{\displaystyle (a',i')=\mathrm {Rot} _{H}(a,i)}
8:
1498:{\displaystyle (D_{1},D_{2},\lambda _{2})}
1412:{\displaystyle (N_{1},D_{1},\lambda _{1})}
2256:
2187:Construction of constant degree expanders
2163:
2143:
2123:
2103:
2083:
2062:
2057:
2051:
2024:
2019:
1994:
1989:
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1963:
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1649:
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1609:
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1559:
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1529:
1516:
1510:
1486:
1473:
1460:
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1430:
1424:
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1387:
1374:
1365:
1344:
1338:
1315:
1295:
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1237:
1214:
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1162:
1142:
1122:
1102:
1082:
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1055:
1035:
955:
911:
900:
874:
830:
819:
793:
754:
743:
712:
642:
631:
628:
615:{\displaystyle \mathrm {Rot} _{G\circ H}}
600:
589:
586:
548:
527:
521:
495:
474:
463:
460:
434:
414:
394:
373:
362:
359:
330:
310:
290:
239:
219:
193:
170:
146:
122:
102:
72:
46:
274:
259:Reingold, Vadhan & Wigderson (2000)
2196:expander without deleterious effects.
188:Roughly speaking, the zig-zag product
1794:Formally speaking, given two graphs:
257:The zigzag product was introduced by
7:
2138:which contains all of the edges in
907:
904:
901:
826:
823:
820:
750:
747:
744:
638:
635:
632:
596:
593:
590:
483:{\displaystyle \mathrm {Rot} _{H}}
470:
467:
464:
382:{\displaystyle \mathrm {Rot} _{G}}
369:
366:
363:
25:
1538:{\displaystyle G_{1}\circ G_{2}}
27:Binary operation in graph theory
1097:is a significantly larger than
1009:{\displaystyle ((w,b),(j',i'))}
263:computational complexity theory
2058:
2020:
1990:
1945:
1939:
1913:
1907:
1847:
1841:
1680:
1654:
1631:
1628:
1602:
1552:
1492:
1453:
1406:
1367:
1263:{\displaystyle (N,D,\lambda )}
1257:
1239:
1003:
1000:
978:
972:
960:
957:
934:
917:
893:
876:
853:
836:
812:
795:
772:
760:
736:
714:
690:
687:
675:
669:
657:
654:
568:
562:
556:
550:
442:
436:
338:
332:
1:
1958:is a connected component of
1157:into a cloud of the size of
2287:(4): Article 17, 24 pages,
1951:{\displaystyle S\subseteq }
97:which takes a large graph (
2356:
1741:Connectivity preservation
1181:Spectral gap preservation
18:Zig-zag product of graphs
2267:10.1109/SFCS.2000.892006
1764:{\displaystyle G\circ H}
1323:{\displaystyle \lambda }
1050:to a new graph which is
509:{\displaystyle G\circ H}
214:replaces each vertex of
207:{\displaystyle G\circ H}
86:{\displaystyle G\circ H}
2323:10.1145/1132516.1132583
2293:10.1145/1391289.1391291
1026:Reduction of the degree
574:{\displaystyle \times }
234:with a copy (cloud) of
2172:
2152:
2132:
2112:
2092:
2072:
2071:{\displaystyle G|_{S}}
2040:
1972:
1952:
1920:
1894:
1874:
1854:
1828:
1808:
1785:
1765:
1731:
1638:
1539:
1499:
1440:
1413:
1354:
1324:
1304:
1284:
1264:
1223:
1203:
1171:
1151:
1131:
1111:
1091:
1071:
1044:
1010:
941:
860:
779:
697:
616:
575:
537:
510:
490:. The zig-zag product
484:
449:
423:
403:
383:
345:
319:
299:
248:
228:
208:
179:
155:
131:
111:
87:
61:
2173:
2153:
2133:
2113:
2093:
2073:
2041:
1973:
1953:
1921:
1895:
1875:
1855:
1829:
1809:
1786:
1766:
1732:
1639:
1540:
1500:
1441:
1439:{\displaystyle G_{2}}
1414:
1355:
1353:{\displaystyle G_{1}}
1325:
1305:
1285:
1265:
1224:
1204:
1172:
1152:
1132:
1112:
1092:
1072:
1070:{\displaystyle d^{2}}
1045:
1011:
942:
861:
780:
698:
617:
576:
538:
536:{\displaystyle d^{2}}
516:is defined to be the
511:
485:
450:
424:
404:
384:
346:
320:
300:
249:
229:
209:
180:
156:
132:
117:) and a small graph (
112:
88:
62:
2317:, pp. 457–466,
2162:
2158:between vertices in
2142:
2122:
2118:(i.e., the graph on
2102:
2082:
2050:
1982:
1962:
1930:
1904:
1884:
1864:
1838:
1818:
1798:
1775:
1749:
1648:
1549:
1509:
1450:
1423:
1364:
1337:
1314:
1294:
1274:
1236:
1213:
1193:
1161:
1141:
1121:
1101:
1081:
1054:
1034:
954:
873:
792:
711:
627:
585:
547:
520:
494:
459:
433:
413:
393:
358:
329:
309:
289:
238:
218:
192:
169:
145:
121:
101:
71:
45:
2078:is the subgraph of
1745:The zigzag product
1726:
1595:
1232:Formally: Define a
581:whose rotation map
60:{\displaystyle G,H}
2280:Journal of the ACM
2168:
2148:
2128:
2108:
2088:
2068:
2036:
1968:
1948:
1916:
1900:-regular graph on
1890:
1870:
1850:
1834:-regular graph on
1824:
1804:
1781:
1761:
1727:
1712:
1634:
1581:
1535:
1495:
1436:
1409:
1350:
1320:
1300:
1290:-regular graph on
1280:
1260:
1219:
1199:
1167:
1147:
1127:
1107:
1087:
1077:-regular. Thus if
1067:
1040:
1006:
937:
856:
775:
693:
612:
571:
543:-regular graph on
533:
506:
480:
455:with rotation map
445:
429:-regular graph on
419:
399:
379:
341:
325:-regular graph on
315:
295:
267:symmetric logspace
244:
224:
204:
175:
151:
127:
107:
83:
57:
2251:, pp. 3–13,
2171:{\displaystyle S}
2151:{\displaystyle G}
2131:{\displaystyle S}
2111:{\displaystyle S}
2091:{\displaystyle G}
1971:{\displaystyle G}
1893:{\displaystyle d}
1873:{\displaystyle H}
1827:{\displaystyle D}
1807:{\displaystyle G}
1784:{\displaystyle G}
1303:{\displaystyle N}
1283:{\displaystyle D}
1222:{\displaystyle G}
1202:{\displaystyle H}
1170:{\displaystyle H}
1150:{\displaystyle G}
1130:{\displaystyle G}
1110:{\displaystyle H}
1090:{\displaystyle G}
1043:{\displaystyle G}
422:{\displaystyle d}
402:{\displaystyle H}
318:{\displaystyle D}
298:{\displaystyle G}
247:{\displaystyle H}
227:{\displaystyle G}
178:{\displaystyle G}
154:{\displaystyle H}
130:{\displaystyle H}
110:{\displaystyle G}
16:(Redirected from
2347:
2325:
2295:
2269:
2260:
2222:Graph operations
2177:
2175:
2174:
2169:
2157:
2155:
2154:
2149:
2137:
2135:
2134:
2129:
2117:
2115:
2114:
2109:
2097:
2095:
2094:
2089:
2077:
2075:
2074:
2069:
2067:
2066:
2061:
2045:
2043:
2042:
2037:
2035:
2034:
2023:
1999:
1998:
1993:
1977:
1975:
1974:
1969:
1957:
1955:
1954:
1949:
1925:
1923:
1922:
1919:{\displaystyle }
1917:
1899:
1897:
1896:
1891:
1879:
1877:
1876:
1871:
1859:
1857:
1856:
1853:{\displaystyle }
1851:
1833:
1831:
1830:
1825:
1813:
1811:
1810:
1805:
1790:
1788:
1787:
1782:
1770:
1768:
1767:
1762:
1736:
1734:
1733:
1728:
1725:
1720:
1708:
1707:
1695:
1694:
1679:
1678:
1666:
1665:
1643:
1641:
1640:
1635:
1627:
1626:
1614:
1613:
1594:
1589:
1577:
1576:
1564:
1563:
1544:
1542:
1541:
1536:
1534:
1533:
1521:
1520:
1504:
1502:
1501:
1496:
1491:
1490:
1478:
1477:
1465:
1464:
1445:
1443:
1442:
1437:
1435:
1434:
1418:
1416:
1415:
1410:
1405:
1404:
1392:
1391:
1379:
1378:
1359:
1357:
1356:
1351:
1349:
1348:
1329:
1327:
1326:
1321:
1309:
1307:
1306:
1301:
1289:
1287:
1286:
1281:
1269:
1267:
1266:
1261:
1228:
1226:
1225:
1220:
1208:
1206:
1205:
1200:
1176:
1174:
1173:
1168:
1156:
1154:
1153:
1148:
1136:
1134:
1133:
1128:
1116:
1114:
1113:
1108:
1096:
1094:
1093:
1088:
1076:
1074:
1073:
1068:
1066:
1065:
1049:
1047:
1046:
1041:
1015:
1013:
1012:
1007:
999:
988:
946:
944:
943:
938:
927:
916:
915:
910:
892:
865:
863:
862:
857:
852:
835:
834:
829:
811:
784:
782:
781:
776:
759:
758:
753:
735:
724:
702:
700:
699:
694:
653:
652:
641:
621:
619:
618:
613:
611:
610:
599:
580:
578:
577:
572:
542:
540:
539:
534:
532:
531:
515:
513:
512:
507:
489:
487:
486:
481:
479:
478:
473:
454:
452:
451:
448:{\displaystyle }
446:
428:
426:
425:
420:
408:
406:
405:
400:
388:
386:
385:
380:
378:
377:
372:
350:
348:
347:
344:{\displaystyle }
342:
324:
322:
321:
316:
304:
302:
301:
296:
253:
251:
250:
245:
233:
231:
230:
225:
213:
211:
210:
205:
184:
182:
181:
176:
160:
158:
157:
152:
136:
134:
133:
128:
116:
114:
113:
108:
95:binary operation
92:
90:
89:
84:
66:
64:
63:
58:
21:
2355:
2354:
2350:
2349:
2348:
2346:
2345:
2344:
2330:
2329:
2299:
2273:
2233:
2230:
2218:
2206:st-connectivity
2202:
2189:
2184:
2160:
2159:
2140:
2139:
2120:
2119:
2100:
2099:
2080:
2079:
2056:
2048:
2047:
2018:
1988:
1980:
1979:
1960:
1959:
1928:
1927:
1902:
1901:
1882:
1881:
1862:
1861:
1836:
1835:
1816:
1815:
1796:
1795:
1773:
1772:
1747:
1746:
1743:
1699:
1686:
1670:
1657:
1646:
1645:
1618:
1605:
1568:
1555:
1547:
1546:
1525:
1512:
1507:
1506:
1482:
1469:
1456:
1448:
1447:
1426:
1421:
1420:
1396:
1383:
1370:
1362:
1361:
1340:
1335:
1334:
1312:
1311:
1292:
1291:
1272:
1271:
1234:
1233:
1211:
1210:
1191:
1190:
1183:
1159:
1158:
1139:
1138:
1119:
1118:
1099:
1098:
1079:
1078:
1057:
1052:
1051:
1032:
1031:
1028:
1023:
992:
981:
952:
951:
920:
899:
885:
871:
870:
845:
818:
804:
790:
789:
742:
728:
717:
709:
708:
630:
625:
624:
623:
588:
583:
582:
545:
544:
523:
518:
517:
492:
491:
462:
457:
456:
431:
430:
411:
410:
391:
390:
361:
356:
355:
327:
326:
307:
306:
287:
286:
283:
236:
235:
216:
215:
190:
189:
167:
166:
143:
142:
119:
118:
99:
98:
69:
68:
43:
42:
36:zig-zag product
28:
23:
22:
15:
12:
11:
5:
2353:
2351:
2343:
2342:
2340:Graph products
2332:
2331:
2328:
2327:
2297:
2271:
2229:
2226:
2225:
2224:
2217:
2214:
2201:
2198:
2188:
2185:
2183:
2180:
2167:
2147:
2127:
2107:
2087:
2065:
2060:
2055:
2033:
2030:
2027:
2022:
2017:
2014:
2011:
2008:
2005:
2002:
1997:
1992:
1987:
1967:
1947:
1944:
1941:
1938:
1935:
1915:
1912:
1909:
1889:
1869:
1849:
1846:
1843:
1823:
1803:
1780:
1760:
1757:
1754:
1742:
1739:
1724:
1719:
1715:
1711:
1706:
1702:
1698:
1693:
1689:
1685:
1682:
1677:
1673:
1669:
1664:
1660:
1656:
1653:
1644:-graph, where
1633:
1630:
1625:
1621:
1617:
1612:
1608:
1604:
1601:
1598:
1593:
1588:
1584:
1580:
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1571:
1567:
1562:
1558:
1554:
1532:
1528:
1524:
1519:
1515:
1494:
1489:
1485:
1481:
1476:
1472:
1468:
1463:
1459:
1455:
1433:
1429:
1408:
1403:
1399:
1395:
1390:
1386:
1382:
1377:
1373:
1369:
1347:
1343:
1319:
1299:
1279:
1270:-graph as any
1259:
1256:
1253:
1250:
1247:
1244:
1241:
1218:
1198:
1182:
1179:
1166:
1146:
1126:
1106:
1086:
1064:
1060:
1039:
1027:
1024:
1022:
1019:
1018:
1017:
1005:
1002:
998:
995:
991:
987:
984:
980:
977:
974:
971:
968:
965:
962:
959:
948:
936:
933:
930:
926:
923:
919:
914:
909:
906:
903:
898:
895:
891:
888:
884:
881:
878:
867:
855:
851:
848:
844:
841:
838:
833:
828:
825:
822:
817:
814:
810:
807:
803:
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797:
786:
774:
771:
768:
765:
762:
757:
752:
749:
746:
741:
738:
734:
731:
727:
723:
720:
716:
692:
689:
686:
683:
680:
677:
674:
671:
668:
665:
662:
659:
656:
651:
648:
645:
640:
637:
634:
622:is as follows:
609:
606:
603:
598:
595:
592:
570:
567:
564:
561:
558:
555:
552:
530:
526:
505:
502:
499:
477:
472:
469:
466:
444:
441:
438:
418:
398:
376:
371:
368:
365:
340:
337:
334:
314:
294:
282:
279:
265:to prove that
243:
223:
203:
200:
197:
174:
150:
126:
106:
82:
79:
76:
56:
53:
50:
40:regular graphs
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2352:
2341:
2338:
2337:
2335:
2324:
2320:
2316:
2315:
2310:
2306:
2302:
2298:
2294:
2290:
2286:
2282:
2281:
2276:
2272:
2268:
2264:
2259:
2254:
2250:
2249:
2244:
2243:Wigderson, A.
2240:
2236:
2232:
2231:
2227:
2223:
2220:
2219:
2215:
2213:
2209:
2207:
2199:
2197:
2193:
2186:
2181:
2179:
2165:
2145:
2125:
2105:
2085:
2063:
2053:
2031:
2028:
2025:
2015:
2012:
2009:
2006:
2003:
2000:
1995:
1985:
1965:
1942:
1936:
1933:
1910:
1887:
1867:
1844:
1821:
1801:
1792:
1778:
1758:
1755:
1752:
1740:
1738:
1722:
1717:
1713:
1709:
1704:
1700:
1696:
1691:
1687:
1683:
1675:
1671:
1667:
1662:
1658:
1651:
1623:
1619:
1615:
1610:
1606:
1599:
1596:
1591:
1586:
1582:
1578:
1573:
1569:
1565:
1560:
1556:
1530:
1526:
1522:
1517:
1513:
1505:-graph, then
1487:
1483:
1479:
1474:
1470:
1466:
1461:
1457:
1431:
1427:
1401:
1397:
1393:
1388:
1384:
1380:
1375:
1371:
1345:
1341:
1331:
1317:
1297:
1277:
1254:
1251:
1248:
1245:
1242:
1230:
1216:
1196:
1188:
1180:
1178:
1164:
1144:
1124:
1104:
1084:
1062:
1058:
1037:
1025:
1020:
996:
993:
989:
985:
982:
975:
969:
966:
963:
949:
931:
928:
924:
921:
912:
896:
889:
886:
882:
879:
868:
849:
846:
842:
839:
831:
815:
808:
805:
801:
798:
787:
769:
766:
763:
755:
739:
732:
729:
725:
721:
718:
706:
705:
704:
684:
681:
678:
672:
666:
663:
660:
649:
646:
643:
607:
604:
601:
565:
559:
553:
528:
524:
503:
500:
497:
475:
439:
416:
396:
374:
354:
335:
312:
292:
280:
278:
276:
275:Reingold 2008
272:
268:
264:
260:
255:
241:
221:
201:
198:
195:
186:
172:
164:
148:
140:
124:
104:
96:
80:
77:
74:
67:, denoted by
54:
51:
48:
41:
37:
33:
19:
2312:
2305:Trevisan, L.
2301:Reingold, O.
2284:
2278:
2258:math/0406038
2246:
2235:Reingold, O.
2210:
2203:
2194:
2190:
2182:Applications
1793:
1744:
1332:
1231:
1187:spectral gap
1184:
1029:
353:rotation map
284:
256:
187:
35:
32:graph theory
29:
2275:Reingold, O
2098:induced by
1419:-graph and
273:are equal (
2309:Vadhan, S.
2239:Vadhan, S.
2228:References
1021:Properties
281:Definition
161:is a good
2029:×
2013:∘
2001:∘
1937:⊆
1756:∘
1714:λ
1701:λ
1688:λ
1672:λ
1659:λ
1620:λ
1607:λ
1566:⋅
1523:∘
1484:λ
1398:λ
1318:λ
1255:λ
647:∘
605:∘
560:×
501:∘
199:∘
78:∘
2334:Category
2216:See also
2046:, where
997:′
986:′
925:′
890:′
850:′
809:′
733:′
722:′
389:and let
271:logspace
163:expander
950:Output
93:, is a
139:degree
34:, the
2253:arXiv
1978:then
1926:- if
1545:is a
1446:be a
1360:be a
409:be a
351:with
305:be a
1880:, a
1860:and
1814:, a
1684:<
1333:Let
869:Let
788:Let
707:Let
285:Let
269:and
2319:doi
2289:doi
2263:doi
2178:).
1229:.
277:).
38:of
30:In
2336::
2307:;
2303:;
2285:55
2283:,
2261:,
2241:;
2237:;
1791:.
1737:.
1330:.
703::
185:.
2326:.
2321::
2296:.
2291::
2270:.
2265::
2255::
2166:S
2146:G
2126:S
2106:S
2086:G
2064:S
2059:|
2054:G
2032:D
2026:S
2021:|
2016:H
2010:G
2007:=
2004:H
1996:S
1991:|
1986:G
1966:G
1946:]
1943:N
1940:[
1934:S
1914:]
1911:D
1908:[
1888:d
1868:H
1848:]
1845:N
1842:[
1822:D
1802:G
1779:G
1759:H
1753:G
1723:2
1718:2
1710:+
1705:2
1697:+
1692:1
1681:)
1676:2
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1663:1
1655:(
1652:f
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1629:)
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1611:1
1603:(
1600:f
1597:,
1592:2
1587:2
1583:D
1579:,
1574:1
1570:D
1561:1
1557:N
1553:(
1531:2
1527:G
1518:1
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1493:)
1488:2
1480:,
1475:2
1471:D
1467:,
1462:1
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1454:(
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1407:)
1402:1
1394:,
1389:1
1385:D
1381:,
1376:1
1372:N
1368:(
1346:1
1342:G
1298:N
1278:D
1258:)
1252:,
1249:D
1246:,
1243:N
1240:(
1217:G
1197:H
1165:H
1145:G
1125:G
1105:H
1085:G
1063:2
1059:d
1038:G
1016:.
1004:)
1001:)
994:i
990:,
983:j
979:(
976:,
973:)
970:b
967:,
964:w
961:(
958:(
947:.
935:)
932:j
929:,
922:b
918:(
913:H
908:t
905:o
902:R
897:=
894:)
887:j
883:,
880:b
877:(
866:.
854:)
847:a
843:,
840:v
837:(
832:G
827:t
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821:R
816:=
813:)
806:b
802:,
799:w
796:(
785:.
773:)
770:i
767:,
764:a
761:(
756:H
751:t
748:o
745:R
740:=
737:)
730:i
726:,
719:a
715:(
691:)
688:)
685:j
682:,
679:i
676:(
673:,
670:)
667:a
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661:v
658:(
655:(
650:H
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569:]
566:D
563:[
557:]
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551:[
529:2
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440:D
437:[
417:d
397:H
375:G
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364:R
339:]
336:N
333:[
313:D
293:G
242:H
222:G
202:H
196:G
173:G
149:H
125:H
105:G
81:H
75:G
55:H
52:,
49:G
20:)
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