Dancing Links

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Dancing Links

• by Donald E.Knuth (Stanford University)
• 2000.11.15

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????

• ??????
• ????????????
• ??
• ???????????
• ?????????,?????
• ???????????????
• ??????????????????????????????????????????????????
  ?????
• ??????????

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????

• ????
• 1960??,??LR(k)??? ,?????????
• 1968?,???????????????,?????????(????,??????,????
  ?)
• 1968?,??Knuth-Morris-Pratt (KMP)???
• 1974?,????????????????????,??ACM????????(?????????
  ?)?
• 1977?-1982? ?????????????????????TeX?METAFONT
  ?(???????????????????)
• 1994???Concrete Mathematics(????)?
• 2000?11?,??Dancing Links????????NP?????
• 2006?7?,?Dancing Links???????????????
• ??
• http//www-cs-faculty.stanford.edu/knuth/

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????

• ??????
• ??
• ??????,??????L?R??,???????????????????????????
• ???,?????R??????,?????L??????,?????????head,Lhead
  Rheadhead,???????
• ??????x,??
• x R L x
• x L R x

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????

• ??????
• ????
• LRxLx
• RLxRx
• ????
• Lzx
• Ryx
• Rxz
• Lxy

LX
RX
x
y
z
x
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Dancing Links

• ?????????????
• ????????x , L R x L x , R L
  x R x ??x????????,????(1)?
• ????????x????????
• ?????x???,????x???L?R????,???x????????????(1)?????
  ?L R X x , R L x x,????(2)?

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Dancing Links

• ??
• ???x????,L?R??????????????????(?????)
• ????,??????,???????????????????
• ??(2)??????????????x,???????(1)?

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Dancing Links

• ????????(2)???
• idea 1????????????,??????????????????
• idea 2??????????(??N????)?????-??,??????????????
• idea 1??????,???(2)????????????idea
  2????????(1)???(2)????,????????-???,??????????????
  ????????(Dancing)?
• ??????????(Links)???(1)???(2)???????????,?Knuth???
  Dancing Links?

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Exact Cover Problem

• Exact Cover Problem(??????)
• ??????MN?0-1??,????????,????????1?
• ???,??????1?4?5?,?????????????1?1?
• ????NPC??,??????????????,?????NPC??,???????,??????
  ??????

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Exact Cover Problem

• ???????????
• Knuth????X??,????0-1??A???????????????

??,??2,3??,???????,???5??????
???? ?????1,4,5
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Exact Cover Problem

• X???????
• ??????????,??????????????,??????????????????O(n)??
  ?
• X?????-???,???????????,?????????????????,?????O(n
  2)?
• ??????, Knuth???DLX??,?Dancing Links X???

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DLX

• ??????????,??????????????,?????,L,R,D,U???????????
  ???????,??C?,????????????,?????1??????????
• ?????(??????)??????,????S????????????,????????????
  ??(????),????head??????,?????

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DLX

• DLX???????,DLX?????????????????search(k),??????k0
  ,?????k????,?????,???????X?????????

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DLX

• ?????
• ???Ok??????????????,?????????????????
• ???????????,???????????CEF,????????????????,??????
  ?????????

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DLX

• ???c
• ???c?,???????????????,???????????,????????????????
  ????,???????????????,???????

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DLX

• ?c???(???)
• ?c????????????????c??????,?????,??????(1)?????????
  ????,???????????2?????????????

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DLX

• ?c???
• ???????(2)???????,????????????,??????????c????,???
  ??????????

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????

• ?????A???

19
????

• A????,D??G???????
• ???????

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????

• ???????,??????,??????,?????A,D?,????????
• A D
• B G
• C E F

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????

• ?????????,????????????,??Knuth????????,?????Scott?
  ????????,???????O(1)??????,??????????????????
• Scott??,??12????????,??88????,?????????????

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????

• ?????????F I L P N T U V W X Y Z,?????????????????
  
• ????72????,??12????????,?60?????????
• ?????????????????????????,??1568??????
• ??i????????,????????r1,?(i,r1)?1,????????x,?(i,x)?
  1,?????????????(?),?????????(?)????(??)???

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????

• ?????????(???????????)???????103005,DLX????????10?
  ???,???????????,???????????????(72
  1568),??1011?,?CPU 108/sec???,????103s,??10????
  ?,?DLX,??1????????300??????

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????

• ????N??,Y??,???
• ?????16?????312,512,659?????5,801,583,789????
• DLX?????
• ?????

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??

• ??Dancing Links ???????????????DLX??,???????????
  ????????
• ????????????????????(?N????,??????)???????????????
  ????,?????????????,???????????????????????????,???
  ??????????????
• Knuth??????????????????Dancing
  Links??????,?????????????????????????

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???

• ????
• ??????????????9x9????,??????1-9???????,??????????
  ???????1-9????????????,?????????9?3x3????????1-9??
  ????????????,??????????,?????????,????3x3??????1-9
  ??????

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???

• ????
• ??????NPC??????,???????????????,??????????????
• ???????,????????,???????
• ??????????(r,c,k)????r?c?????k?
• ???????????,??????r???????k,c???????k,(r,c)???????
  k???b(?????)?????k?
• ??????NNN999729?,????994324?,????????????
  (?),?????????????????(?)?????(1?1),?????729324?
  ????????
• ???,(r,c)??????????????????NN?

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???

• ?PKU???????(TLE10s),????????(???,?????????????)?D
  LX(??)?????????
• ????4????????
• DLX(31ms)
• ????(531ms)

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???

• ???DLX????,????????(???????????),???????????,?????
  ?,???????????1?
• ??????????????,????????

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???

• ??????
• ??n???,m???,???????k???,??????????????(???????????
  ,????????)
• ????
• ?n??????,????????(????????????????,???????????????
  ???)????
• ?2????????????????????high,0??????low,????????????
  mid,????????r??????c,?(r,c)1?
• ?????????,???,????????1?1,?????????????k(?????),
  ??DLX?????????

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Thank You!Q A
??,??????????? ??,?????????? Knuth???????????????
?????, ??,???????????Dancing Links ?????,???????
??????,????????????? ?????,???????????
????Knuth????,??Dancing Links??? ???,???????,?????
?!