Data Structures

1
Data Structures

• ???????????????????????????
• Algorithm ??? ????????????????????????????????????
  ??????

2
Introduction

• ????????????????????????????
• ?????? code ???? ? ???????
• ?????????????????????????????????????????????

3
???????????? ????????????????????? 5 ????? (1 ???
5 ) ???????????????????
Linked ist

• P

1
2
3
5
4
Tree (Binary Search Tree)
4
?????????????????? (???)

• Heap

???????????????????????????? ?????????????????????
?????????????????????????????????????????
???????? Heap ?????????????
5
??????????????????????????

• Big O
• ???
• ????? c ??? N0 ??????????????? NgtN0
• ??????????????? T(N) ?????????????

6
???????????????? BIG O

• ??? T(N) 339N ??? f(N) NN
• ?????? N0 339 ??? C 1
• ?????????????????????????? 339N0 lt 1(N0N0)
• ?????????????????? -gt T(N) lt 1(340N)
  ??????????????

7
???????????????? BIG O (???)

• ??????? T(N)O(N) ??????????????
  ???????????????????
• ?????????????????????????? ?????????????????????
  ????????????????????????????????????????????
  ??????? ?????? O(N) ?????????
• ???? O(N) ??????????????????????????????????
  execute ??? code ???????????????????????

8
????????????????????????? ??????????

• sigmaOfSquare(int n) // calculate
• 1               int tempSum
• 2               tempSum 0
• 3                for (int i1iltni)
• 4               tempSum ii
• 5               return tempSum

1 unit (declare only)
1 unit (assignment)
n1 unit
n unit
1 unit
??? ??? ??? assignment ??????? n ??? 3n unit
1 unit (return)
?????? 5n5 unit
9
??????????????????????????????

• ????????????????????????????????????? running
  time ??????? ??????? ??????????? ?
  ????????????????? ??????? ?????????? running time
  ??????????????????????????????????????????????????
  ????? ? ??????? ?????????????? Big O ???????
• ?????????????? ??????????? loop ????????? run
  ????????????????????????????????????????? loop
  ??????????????? ???????? n ?????
• ?????? ?????????????? loop ?????????? ????Big O
  O(n)
• ???????????? 5n5 ?????? O(n) (????????? 5n5lt
  6n)

10
????? BIG O ??? loop ?????

• For loop ??? Big O ??? ????????????? loop ?????
• Nested loop
• 1      for (i 1 i lt n i)
• 2           for (j 1 j lt n j)
• statements

n ???
n ???
Big O ?????? O(n2)
11
????? BIG O ??? loop ????? (???)

• ?????????????? Nested loop ??????????
• ??? T1(N)O(f(N)) ??? T2(N) O(g(N)) ????????
• T1(N) T2(N) O(f(N)g(N))
• ??????????????? f(n) g(n) n
• ?????????? O(n2)

12
????? BIG O ??? loop ????? (??? 2)

• Statement ????????????????????????
• 1         for (i 0 i lt n i)
• 2             statement1
• 3         for (j 0 j lt n j)
• 4                 for (k 0 k lt n k)
• 5 statement2

O(n)
O(n2)
????????????????? ??????? O(n2)
13
????? BIG O ??? loop ????? (??? 3)

• ????????????? running time ??? Statement
  ??????????????
• ??? T1(N)O(f(N)) ??? T2(N) O(g(N)) ????????
• T1(N) T2(N) max(O(f(N),O(g(N)))
• ??????????? f(n) O(n), g(n) O(n2)
• ?????????? O(n2)

14
????? BIG O ??? loop ????? (??? 4)

• ???????????????????
• 1             if (condition)
• 2                   Statement1
• 3             Else
• 4                   Statement2

O(f(n))
O(g(n))
????????????????? ??????? max(O(f(n),O(g(n)))
15
????? BIG O ??? recursion

• 1mymethod (int n)
• 2 if (n 1)
• 3       return 1
• 4        else
• 5             return 2mymethod(n 1) 1
• 6       
• 7

n ?????, big O O(n)
16
Maximum Subsequence Sum, ???????????????? code
?????? Big O ????????

• Maximum Subsequence Sum ???
• ?????????????? A1,A2, ??? An
• Maximum Subsequence Sum ??????
  ????????????????? (????????????????
  ????????????????????)
• ???????????? -2, 11, -6, 16, -5, 7
• ???????? 11, -6, 16 ??? 21 (????? ?????????????)
  ??????????????? ????? ??????? 11, -6, 16, -5, 7
  ?????????????? 23 ?????????? Maximum Subsequence
  Sum

??????
17
?????????? max sub sum ?????? 1

• 1 int maxSubSum01 ( int a)
• 2 int maxSum 0
• 3         for (int i 0 i lt a.length i)
• 4          for (int j i j lt a.length j)
• 5             int theSum 0
• 6                   for (int k i k lt j
  k)
• 7             theSum ak
• 8                  
• 9                   if (theSum gt maxSum)
• 10         maxSum theSum
• 11               
• 12               
• 13         return maxSum
• 14      
• 15

index ???
index ???????
???????????????????????
???????????????????????
18
?????????? max sub sum ?????? 1 (???)

• ??????? 1 ????? big O O(n3) ????????????
• ??????????????????? ??????????????????????????????
  ????
• ???????????????? 0 ???2 ?????? ???????????????????
  ??? 0 ??? 3 ???????????????????????
  ???????????????????????????????? 0 ???2 ???

19
?????????? max sub sum ?????? 2

• 1             int maxSubSum02 (int a)
• 2             int maxSum 0
• 3                   for (int i 0 i lt
  a.length i)
• 4             int theSum 0
• 5                   for (int j i j lt
  a.length j)
• 6             theSum aj
• 7                   if (theSum gt
  maxSum)
• 8             maxSum theSum
  
• 9            
• 10               
• 11               
• 12         return maxSum
• 13             

????????
???????????????????????????????
BIG O O(n2)
20
?????????? max sub sum ?????? 2 (???)
-2
11
-6
4

• ????? i0, j0 theSum -2
  maxSum 0
•  
• ????? i0, j1 theSum -2 11
  9 maxSum ???????? 9
•  
• ????? i0, j2 theSum 9 (-6)
  3 maxSum ????????? 9
•  
• ????? i0, j3 theSum 3 4
  maxSum ????????? 9

21
?????????? max sub sum ?????? 3

• ??? divide and conquer
• ????????????????
• ???????????????????????? ????
• ??????????????????????? ????
• ??????????????? (?????????????????????????????????
  ???????????????)

22
?????????? max sub sum ?????? 3 (???)
1 -2 7 -6 2 8 -5 4
Max sub sum ?????????? 7
Max sub sum ?????????? 10
??????????????????????????? (-6) ??? 1
?????????????????????? (2) ??? 10
Max sub sum ?????????????????????????? ??? 1 10
11 ???????????
23
?????????? max sub sum ?????? 3 (??? 2)

• 1int maxSumDivideConquer (int array, int
  leftindex, int rightindex
• 2 // ????????????????????????????????????????????
  ??????????????? ???????????????????????????
• 3 if (leftindex rightindex) // Base
  Case
• 5        if (arrayleftindex gt 0 )
• 6             return arrayleftindex
• 7             else
• 8                   return 0 // maxSubSum
  ?????????????? 0 ????????
• 9       
• 10     int centerindex (leftindex
  rightindex)/2
• 12     int maxsumleft maxSumDivideConquer(array
  , leftindex, centerindex)
• 13 int maxsumright maxSumDivideConquer (
  array, centerindex 1, right)

T(n)
T(n/2)
T(n/2)
24
?????????? max sub sum ?????? 3 (??? 3)

• 14     int maxlefthalfSum 0, lefthalfSum 0
• 15        // ????????????????????????????????????
  ??????????????????????????
• 16     for (int i center i gt leftindex i--)
  
• 17             lefthalfSum lefthalfSum
  arrayi
• 18             if (lefthalfSum gt maxlefthalfSum)
  
• 19             maxlefthalfSum lefthalfSum
• 20             
• 21     

O(n/2)
25
?????????? max sub sum ?????? 3 (??? 4)

• 22 int maxrighthalfSum 0, righthalfSum 0
• 23    // ?????????????????????
  ??????????????????????????????????????
• 24   for (int i centerindex 1 i lt
  rightindex i)
• 25    righthalfSum righthalfSum array i
• 26             if (righthalfSum gt
  maxrighthalfSum)
• 27             maxrighthalfSum righthalfSum
• 28              
• 29   

O(n/2)
26
?????????? max sub sum ?????? 3 (??? 5)

• 30 // ????????? max ??????? 3 ???? ???
  ????????????, ???????????, ????????????????? 2
  ?????
• 31 return max3 (maxsumleft, maxsumright,
  maxlefthalfSum maxrighthalfSum)
• ??????? ????????????????????
• T(n) 2T(n/2) 2O(n/2)

????????? max ????????? ????????????
??????????????
27
?????????? max sub sum ?????? 3 (??? 6)

• ?????????? unit time ?????? ?????????????? BIG O
  ???
• T(n) 2T(n/2) 2O(n/2)
• 2T(n/2) O(n)
• 2T(n/2) cn
• ??? n ????????????

????? O(n) lt cn ????????
(1)
28
?????????? max sub sum ?????? 3 (??? 7)

• ???????????????? ?????

(2)
(3)
(X)
29
?????????? max sub sum ?????? 3 (??? 8)

• ???????? (1) (2) (3) .. (x) ?????
• ??????????????????????????????????? ???? c
  ???????????????????? log2 n ?????????????
• ??? n ?????? ?????
• ???????? T(1) ???????????? ????????????????????
  T(n) ?? Big O O(n log n)

30
?????????? max sub sum ?????? 4

• ???????????????? 2 ?????????? ????????????????????
  ?????
• ????????? maximum subsequence sum
  ????????????????? ????????? 3, -5, 1, 4, 7, -4
• ??? -5 ??????????????? sequence ??????????????
  ?????????????????????????????? (?????????????? 1)
  ??????????????????????????? (?????????????????????
  ?????? maximum subsequence sum ??????????????????
  0)

31
?????????? max sub sum ?????? 4 (???)

• ??????????????????
• subsequence ???????????????? ?????????????????????
  ??????? maximum subsequence sum ???
• ????????????????????????????????? ?????? i ????
  index ???????????? subsequence ??? j ???? index
  ???????????????? subsequence (????????????????????
  ?? subsequence ???????????) ??? p ???? index ?? ?
  ??????? i1 ??? j

3 4 1 -3 -9 1 5
i
j
p
32
?????????? max sub sum ?????? 4 (??? 2)

• ??????????????? (???)
• ??????????????????????????????????? j ????? ????
• ??? aj ?????? ??????????? maximum subsequence
  sum ????????? ?????? maximum subsequence sum
  ???????????????????
• ??? aj ??????? ????????????? aiaj
  ?????????????? aiaj ??????? ????????
  aiaj ??????? ?????? ???????????????????????
  ????? maximum subsequence sum ????????????????
  ?????????????????? aj ??????? ??????????????
  ???????????????????? ????? subsequence
  ?????????????? ????????????????????? j
  ??????????? ?????????????? i ???

33
?????????? max sub sum ?????? 4 (??? 3)

• ?????????????? i ???????????????????????????
• ??????????? ????????? aj ????????????????
  aiaj ??????????? ??????????? ????????? i
  ????????????????????????? i ??? p
  ??????????????????????? ai ??? ap (??? p
  ????????????????????? i ??? j) ???????????????????
  ???????? ??????? ????????? j ??? i ???????????
  ???????????????????? max sub sum ???
• ??????????????????? max sub sum
  ?????????????????? ??????????????????????????????
  ??????????? j1 ????????????? i ????? j1 ???

3 4 1 -3 -9 1 5
i
j
p
34
?????????? max sub sum ?????? 4 (??? 4)

• 1    int maxsubsumOptimum (int array)
• 2     int maxSum 0, theSum 0
• 3         for (int j 0 j lt a.length j)
• 4          theSum theSum array j
• 5                   if ( theSum gt maxSum)
• 6             maxSum theSum
• 7                   else if (theSum lt 0) //
  ?????? aj ????????????
• 8                  
  //sequence ??????
• 9                     theSum 0 //
  ?????????????????????????
• 10 // j1
• 11               
• 12               
• 13       return maxSum
• 14  

35
????? log ?? BIG O

• ???????????????????????? (O(1))
  ??????????????????????????????????? ? ???? ? ???
  (??????? 3 ??? maximum subsequence sum)
  ????????????? Big O ???? O(log n)
• ??????????????????????????????????????????????
  ?????????????????????????????????? O(n) ????

36
?????????????????????? O(log n)

• ?? 5 ?????????????????????????????????????????????
  ????????????
• ????????????????????????????????????? (Big O ???
  O(n)) ?????????????
• ??????????????????????????????? (Binary Search)

37

• int binarySearch (int a, int x)
• int left 0, right a.length 1
• while (left ltright)
• int mid (left right)/2
• if (amid lt x )
• left mid 1
• else if (amid gt x)
• right mid 1
• else
• return mid
• return -1 // ?????????????????????????

Big O O(log2 n)
38
???????? O(log n) (???)

• ???????????????

long gcd (long m , long n) while (n!0)
long rem mn m n n
rem return m
???? Big O ??????????
????????????????????? Big O (????????) ??????????
????????????????????????????????????
39
????? big O ????????????????????

• ???????????????????????
• ??? M gt N ???? M mod N lt M/2 ???????????????????
• ??? N lt M/2 ???????????????????????? N
  ??????????? ?????????????????? M/2 ?????????
• ??? N gt M/2 M ??? N ????? 1 ?????? M-N ??????? 1
  ?????? M (gt M/2) ??????????? ????????????? M/2
  ?????????
• ??????????? ???????????? n ???????????????? m
  ????????????????? (n2) ???????????? (n2)
  ???????????????????????????????????????????? n
  ???????????????????????????????? ???????
  ??????????????????????????????????? ?

40
???????? O(log n) (???2)

• ????????????? xn ????????????????????????
• long power (long x, int n)
• if (n0)
• return 1
• if (isEven (n))
• return power (xx, n/2)
• else
• return power (xx, n/2)x

Big O ??? O (log2 n) ???????????????? 2 ?????
(??????) ????????????????????????? method
41
?????????????? O(log n)

• logk n O(n) ????? k ????????????
• (????????????????? logarithm function
  ?????????????????????)
• ????????? f(n) loga n ????? big O ???? O(logb
  n) (a ??? b ????????????????????? 1)
• (??????? log function ?? growth rate ???????)

42
?????????????????? log ?? growth rate ???????

• ??? ???

43
Runtime -????(??)?????(????)

• c
• log n
• logk n
• n
• n log n
• n2
• n3
• 2n

44
????????? ? ??????????? Big O

• Big Omega ( )
• T(N) (g(N)) ????????????? C ??? N0
  ????
• T(N) gt C g(N) ??? NgtN0
• ???????? ??? f(N) (N2) ????????? f(N)
  (N) (N1/2) ????????????????????????????????
  ?????????

45
????????? ? ??????????? Big O (???)

• Big Theta ( )
• T(N) (h(N)) ?????????? T(N) O(h(N)) ???
  T(N) (h(N))
• (????? c1, c2, N0 ??? c1h(N) lt T(N) lt c2h(N)
  ?????? N gt N0

46
????????? ? ??????????? Big O (??? 2)

• small O
• T(N) o(p(N)) ??? T(N) O(p(N)) ??? T(N)
  (p(N))

47
?????????????????????????

• T(N) O(f(N)) ??????????????????? f(N)
  (T(N))
• ???????????????????? f(N) ???? upper bound ???
  T(N) ??? T(N) ??????????????? lower bound ???
  f(N)
• f(N) N2 ??? g(N) 2N2 ????? Big O ??? Big
  ??????? ??????? f(N) (g(N))
• f(N) N2 ?? Big O ?????????? (???? O(N3), O(N4))
  ???????????????????? O(N2)
• ????????????????? f(N) (N2) ?????????????
  ??? Big O -??? N2 ??????? Big O ???????????

48
???????????? ???????????????????????

• ??? T(N) ??? Polynomial degree k ????
• T(N) (Nk)
• ??????????? ??? T(N) 5N4 4N3 N ???????????
  T(N) (N4)

49
Best case Worst case ??? Average case

• worst case ??? ???????????? run
  ????????????????????????????
• ???? best case ??? ????????? run
  ?????????????????????????
• average case (??????????) ????
• ?????????? input (??????????) ???????????
• ??????????? input ????????????????????????? (unit
  time) ????? run
• ????????? (average case running time)
  ????????????? unit time ?????? input / ?????
  input

50
Average case (???)

• ????????????????????????????? ????????????????????
  ?
• input ?????????????????????????????????
• ????????????????????????????
• ?????????????????????????????????????? input
  ??????? ? ????????????????????
• Average case ????????????????
  inputi ??? unit time ???????? inputi

51
????????????? Average case

• ??????????????? x ?????????????? n
• Best case ???????????????? x ?????????
• Worst case ????????????? x ???????????????????????
  ?? (?????????????????)
• Average case
• ??????????????????????????????????????? x ???????
• ??????????? ???????? x ????????????????????? ???
  1/n

52
????????????? Average case (???)

• Average Case running time 1/n
  (?????????????????????????????? x ??????????? 1)
  1/n (?????????????????????????????? x
  ??????????? 2) ... 1/n (???????????????????
  ??????????? x ??????????? n ?????????????????)
• (1 2 n) / n (n1)/2
• O(n) big O ??? worst
  case