Union-Find

1
???? ??????

• Union-Find

????? ?? ???? ?? ????? ?????
2
???? Union-Find

• ???? ?????? ?????? ?? ??????? ?????
• ????? make-set(x) ????? ????? ??? ???? x
• ????? union(x, y) ????? ??????? ?? x ?-y
• ????? find(x) ????? ?????? ???? x ????, ??????
  ????? ?????? (representative) ?? ??????

5
5
11
7,11
7
7
????? ?? 3 ?????
???? ?? 11
????? ?? ???' ?? 7 ?-11
3
????? Union-Find

• ????? ?"? ?????? ???????
• ?? ????? ???? ????? ??????
• ????? ??? ?????? ???? ???? ????? ???????
• ????? ???? ??? ??????
• ????? ?"? ????
• ?? ????? ?????? ??? (???? ??????? ???? ???)
• ????? ?????? ?? ????? ??? ?? ?????? ???
• ????? ?"? ????? ??? ???? (????? ??? ?? ????)
• ????? ?"? ????? ??? ?? ?????

4
????? Union-Find

• ?? ?????? ????? ?????? ?"? ?????
• ????? ????? ??? ?????? ???????. ????
• ??????? ????? ?????
• Union By Rank

Rank0
Rankx
Rankx
Rankx1
5
????? Union-Find

• ?? ?????? ????? ?????? ?"? ?????
• ????? ????? ??? ?????? ???????. ????
• ??????? ????? ?????
• Union By Rank ???? ?? ??????? ?? ????? ????
  O(logn)
• ????? ??????? (Path Compression)

??? (a)
6
????? Union-Find

• ?? ?????? ????? ?????? ?"? ?????
• ????? ????? ??? ?????? ???????. ????
• ??????? ????? ?????
• Union By Rank ???? ?? ??????? ?? ????? ????
  O(log n)
• ????? ??????? (Path Compression) ???? ??
  ??????? ?? ??????? ?? ???? n ?????? ????? (?????
  ??) ?-f ?????? ????? ???? ??????
• ?? ?? ????? ?? ??? ?????? ?????????? ?? ???? m
  ?????? ????

??' ??????? ??' ?????, ?? ?' ????? ??? ??? ???...
???? ??? ???????? ?? union-find ???? ??? ?-4
?????? ?????? http//mathworld.wolfram.com/Ackerm
annFunction.html
7
????? 1

• ?????? ???? ?????? ?? union-find (????? ??
  ???????? ?? union by rank path compression),
  ????? ???? ?? m ?????? (make-set, union, find)
  ??? ?? ?????? ?-union ????? ???? ?????? ?-find.
  ???? ??? ???? ?????? ???? ?? ??? ??????, ?????
  O(m)
• ?????
• ??? ???? ???????
• ????? ????? ????

????? ?????
????? ????
???? ???????
8
????? 1

• ????? ????? ????
• ??? ????? ?? m ?????? ???? ??? O(m) ?? ?????
  ????? ????? ???? ????
• ??????? ?????? ????? ?? ??????
• ?? ?? ??????? ???????
• make-set
• union
• find

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????? 1

• ????? ????? ???? (????)
• ???? ???? ?? ?-path compression ???????? find
  ????? ??????? ??????
• ??? ????? make-set ???? ???? ????? ??????? ?????
• ?????? find ???? ????? path compression
• ?? ???? ????? ??? ??? ????, ???? ????? ???? ????
  ?? ??????

Find(a)
??"? m ?????? ??? ??? ???? O(1) amortized
10
????? (-) 2

• ?????? ??? G(V,E), ???? ?? ????? ??????? ?? ????
• ???? ???? ????? ???? ?????? ????? ???? ??? ????
  union-find
• ??????
• ???? ?-make set ?? ?? ????? ?? ????
• ????? ?? ?? ?????? ???? ??? ???? ?-
• union(find(u),find(v)) ???? u ?-v ???? ????
• ?? ?? ????? ??????? ???? ??? ???? ?????? ????
• ????????

11
????? 3

• ????? ???? HEX ?? ??? ??????, ?? ???? ???? ?????
  ????? ????/???? (???? ?????). ???? ????? ??????
  ???? ????? ?? ??? ?????? ???
• ???? ????? ??????? ?? ???? ???? ?? ???? ?????? ??
  ??? ???? O(log n) ????? ???? ?? ??? ??????? ?????

12
????? 3 - ?????

• ????? ???? union-find ??? ???? ????? ????????,
  ???? ???? ????? ??? (??? ?? ????)
• White-right
• White-left
• Black-top
• Black-bottom
• ??? ??? ????
• make-set(X) ?? ?-X ????? ???? ?? ?????
• union ?? X ?? ?? ??? ????? ?? ????? ??? (??? ??
  ?? 6 ????? ??? ?????)
• ????? ?? ???????? ???? ?????
• ?? ???? ??????
• find(W-R)find(W-L) ??? ???? ????
• find(B-T)find(B-B) ??? ????? ????
• ??"? ??? ???? O(log n) ????, ?????

13
????? 4

• ???? ???? ?????? ?-union-find
• ?? ????? ????? ?????? ??????? ??-???????, ??
  ????? ????? ????? ????? (????) ?????? ?? ??????
• Find ????? ????? ??????? ????? ?????
• Union ?????? ??' ???? ?????? ??????? ?? ?????
• ??? ??? ???? ???? m ?????? ???? ????? ????? ?? n
  ?????? ????? ??? ??? (?? ???)? (?????? mgtn)

14
????? 4

• ?????? find, make-set ?????? ??? ????, ??? ?????
• ???? ????? ?? ????? ??? ?? ?? ??? ?????? ?????
  ???? ?????? ????????
• ???, ??"? ????? ???? ?????? O(n log n) ?????
• ???? ???, ?????? ?-union ?????? ????? ????? ?????
  ??????
• ??"? ?????? O(m n log n)