????? (Analyzing Algorithms)

1
?????(Analyzing Algorithms)

• ??
• ???????? ??Java??, ??? ?, ??????, 2005.
• Introduction to the Design and Analysis of
  Algorithms -- A strategic approach, 2E, R.C.T.
  Lee et. al., NcGraw Hill, 2005.
• Introduction to Algorithms, Cormen et. al., MIT
  Press.

2
?????

• ?????????????????????(resources)?????
• ??(Resources)???(memory),??(communication),??(ban
  dwidth),??? (logic gate),??(time)

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?????

• ???????(running time)???????????????????????
• ????????????(machine-independent) ???

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???(Complexity)

• ??????
• ?????(time complexity)
• ?????(space complexity)
• ????????????????????,
• ???????,?????????
• ??????time complexity!

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????

• ?????????????????
• ????(best case)?????????????????????????
• ????(worst case)?????????????????????????
• ????(average case)??????????????????????????

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????????????

• ??,????????????????(worst-case running time)
• ??
• ????????(upper bound)?
• ??????????
• ?????????????????????(insertion sort)
  ??????(quadratic function)?

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????????????
Algorithm Prime1(n)Input????2????nOutputtrue?false(??n????????)for i?2 to n-1 do if (ni)0 then return falsereturn true

• ??????,????2??????n,?n???,????Prime1???????????(n
  i)???????((ni)0)??n-2???,?????n??????,?n????,???
  ?Prime1??????????????????1?,?????n??????

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?????????????

• ??????,????2??????n,?n???,????Prime2???????????(n
  i)???????((ni)0)?? -2???,?????n??????,?n????,?
  ???Prime2??????????????????1?,?????n??????

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?????Prime1

• ??,????????,?????(worst case)?,???Prime1??????????
  ????n???????????(best case)?,???Prime1???????????
  ????n????????????,???Prime1?????(linear)
  ?????????(??????????)????(constant) ??????????

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?????Prime2

• ?????????,?????(worst case)?,???Prime2????????????
  ????n???????????(best case)?,???Prime2???????????
  ????n????????????,???Prime2??????(square root)
  ?????????????(constant) ??????????

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????(Asymptotic Notation)

• ????,???????????(asymptotic notation)??????????,??
  ??????????????????????,???????(problem
  size)???????????
• ???????????(??????)???,???????(???????)???????(???
  ????)???????????????,??????,????????????????????
• ??????????(??????)????,??????????????????????????
  ??????????????????????????????,???????????????(???
  ???)??????,??????????????????????????

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??(Order)

• ????????????????????,?????????????????(order)?????
  ,???????????????,?????????????????,???????????????
  ?????,??????????????,??????????????????
• ??,?????????????n-2,??n???(??????)?,?????????????n
  ,?????????(???)??(??????????????????????????????)
  ?????????????35n212n11,??n???(??????)?,?????????
  ????n2,????????????????????28n31245n2162n321,?
  ?n????(???????),?????????????n3(??????)?

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?O??(Big-O Notation)

• ???????O??(Big-O notation)??????????,?O???????????
  ???????????(order)???,??????????O?????????

????,?????????????????????,????????????????????????(order),????????O????(O ??order??)?
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???O??

• ?????????O??
• ?? ?O?? (Big-O notation)
• ?f(n)?g(n)????????????????,????????cgt0??????n0???
  ???n?n0??,f(n)?cg(n)??,????f(n)O(g(n))?(??
  ?f(n)???Big-O of g(n)?,???????????f of n is of
  Big-O of g of n?)?
• ??,??35n212n11??,??c58?n01(58?351211??),???n
  ?n01?,35n212n11?cn2(58n2)??,??,???35n212n11
  O(n2)?

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?????(Asymptotic Upper Bound)

• Def f(n) O(g(n)) ??(upper bound)"
• iff ? c, n0 ? f(n) ? cg(n) ? n ? n0
• e.g. f(n) 3n2 2
• g(n) n2
• ? n02, c4
• ? f(n) O(n2)
• e.g. f(n) n3 n O(n3)
• e. g. f(n) 3n2 2 O(n3) or O(n100 )

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?????(Asymptotic Upper Bound)
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?????(Asymptotic Lower Bound )

• Def f(n) ?(g(n)) ??(lower bound)
• iff ? c, and n0, ? f(n) ? cg(n) ? n ?
  n0
• e. g. f(n) 3n2 2 ?(n2) or ? (n)

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?????(Asymptotic Lower Bound )
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Theta??

• Def f(n) ?(g(n))
• iff ? c1, c2, and n0, ? c1g(n) ? f(n) ?
  c2g(n) ? n ? n0
• e. g. f(n) 3n2 2 ? (n2)

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Theta??

• f(n) ?(g(n))

21
Theorem

• ??????f(n) ? g(n),
• ?

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????????

• ?????????????????????????(?????,problem
  size)??????????,??????????????????(?????,execution
  time)??????????,?????????????????,???????????????
  ??,?????????????????????????????????????

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?????

• ????????????????????,??????????????,??????????????
  ??
• O(1) lt O(log n) lt O( ) lt O(n) lt O(n log n) lt
  O(n2) lt O(n3) lt O(2n)O(n!) ? O(nn)

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??????????O????????
????? ??O???? ??
162 O(1) ?????(constant)??
63log n4 O(log n) ???(??)(sub-linear, logarithmic)??
37 52 O( ) ???(square root)??
n-2 O(n) ??(linear)??
156n81 O(n) ??(linear)??
35n212n11 O(n2) ??(quadratic)??
28n31245n2162n321 O(n3) ??(cubic)??
14?2n457n3248n2-45n81 O(2n) ??(exponential)??
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???????????????????
log n n n logn n2 n3 2n
0 1.00 1 0 1 1 2
1 1.41 2 2 4 8 4
2 2.00 4 8 16 64 16
3 2.83 8 24 64 512 256
4 4.00 16 64 256 4,096 65,536
5 5.66 32 160 1,024 32,768 4,294,967,296
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???????????????
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???????????????(???)
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???????? vs. ???????

• ??????????O(p(n))????,???p(n) ????????,???????????
  ???(polynomial time algorithm)?????,?p(n)???????,?
  ?????????????(exponential time algorithm)

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????? (ceiling) ?????? (floor)
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????(modular operation)

• ?????a ? ?????n, ??a mod n ? a/n ???
• a mod n a -?a/n?n.
• ?(a mod n) (b mod n), ?????? a ? b (mod n)???a
  ?b ?mod n (modulo n)????(equivalent)??
• ????a ? b (mod n) ?a ?b ?mod n?????????