[e-mail]

1
????????????

• ???
• ywdeng_at_mail.knu.edu.tw
• http//w3.im.knu.edu.tw/joseph/

2
??(Sorting)

• ????(Internal Sort)
• ???????????,????
• ????(External Sort)
• ???????,??????????,????????(?????)????

3
??(Sorting)

• ????(Comparative Sort)
• ??????(Key Value)
• ????(Distributive Sort)
• ????????????

4
??(Sorting)

• ???(Stability)
• ?????????R1, R2
• ?????R1??R2??,??????R1???R2??,????????????(Stable)
  ?????????????(Unstable)????

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????(Bubble Sort)

• ????
• ?N???????,??????????,??????????,????????
• N?????N-1???
• ?????O(N2)
• Stable
• Comparative Sort

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????(Bubble Sort)
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????(Bubble Sort)
void bubble_sort(int data, int size) int
base, compare, temp for (base 0 base lt
size - 1 base) / ???????,??????
/ for (compare base 1 compare lt
size compare) if (database gt
datacompare) temp
datacompare datacompare
database database temp

8
????(Bubble Sort)JAVA
public BubbleSortDemo(int A) boolean done
false for (int i A.length (i gt 0)
(!done) i--) done true for
(int j 0 j lt i-1 j) if (Aj
gt Aj1) int t Aj
Aj Aj1 Aj1 t
done false
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?????(Insertion Sort)

• ?????K???????????L??
• ????,??????,????(????)
• ?????O(N2)
• Best Case????????????,???????????
• Stable
• Comparative Sort

10
?????(Insertion Sort)
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?????(Insertion Sort)
void insertion_sort(int data, int size)
int base, compare, temp for (base 1 base
lt size base) /
????????,?????,??????????????? / temp
database compare base while
(compare gt 0 datacompare - 1 gt temp)
datacompare datacompare - 1
compare-- datacompare
temp
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????(Shell Sort)

• ????(Insertion Sort)????
• ????????,?????????????,????????????????
• ???????????(?????)??,????????????????????
• ???????????????
• ??????????????????,???????????????
• ?????
• ???N/2
• ???N/4
• ????1

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????(Shell Sort)

• ???????O(N2),??O(N1.5)
• Unstable
• Comparative Sort

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????(Shell Sort)
0
1
2
3
4
5
6
7
8
9
10
11
12
13/26
13/43
13/81
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????(Selection Sort)

• ??????????????????
• ???????????????????
• ?N-1?????N-1???????N-1???
• ?????O(N2),??????????????O(N)
• Stable
• Comparative Sort

16
????(Selection Sort)
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????(Selection Sort)
void select_sort(int data, int size) int
base, compare, min, temp for (base 0 base
lt size - 1 base) /
???????????????? / min base
for (compare base 1 compare lt size
compare) if (datacompare lt
datamin) min compare
if (min ! base) temp datamin
datamin database
database temp
18
????(Merge Sort)

• ????????????L1?L2????????????L
• ?????p?p1?p2????L?L1?L2???
• ??p1??????p2?????,??????p?????
• ??????????L1?L2?????
• ????
• ?????????????????,????????????
• ????????????
• ?????O(N log N),???????
• Stable
• Comparative Sort
• ??????

19
????????????L1?L2????????????L
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????????
void merge(int data1, int data2, int data3,
int size1, int size2) int arg1 0, arg2
0, arg3 0 data1size1 32767
data2size2 32767 for ( arg3 lt size1
size2 arg3) / ?????,?????????????
/ if (data1arg1 lt data2arg2)
data3arg3 data1arg1
arg1 else data3arg3
data2arg2 arg2

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????(Quicksort)

• Divide-and-Conquer
• Partition-Exchange
• ???
• ?????????K1???,?N?????????,??????????K1?,?????????
  ??K1
• ???????????????Quicksort??
• ???????O(N log N),??O(N2)
• Unstable
• Comparative sort

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????(Quicksort)
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void quick_sort(int data, int left, int right,
int size) / left?right?????????? / int
lbase, rbase, temp if (left lt right)
lbase left 1 while (datalbase lt
dataleft) lbase rbase right
while (datarbase gt dataleft) rbase--
while (lbase lt rbase) /
?lbase??rbase,?????? / temp
datalbase datalbase
datarbase datarbase temp
lbase while (datalbase lt
dataleft) lbase rbase--
while (datarbase gt dataleft) rbase--
temp dataleft
dataleft datarbase datarbase
temp quick_sort(data, left, rbase - 1,
size) quick_sort(data, rbase 1, right,
size)
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????(Heap Sort)

• ???1
• ?N?????Heap????????,Heap?O(log N)??,??O(N log N)
• ?Heap??N???????????????O(log N),??N???O(N log N)
• ??????? O(N log N)

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????(Heap Sort)

• ????????,??N???
• ???2
• ????????Heap
• ??N??????Complete Binary Tree,?????????????
• Ai?????A2i,????A2i1,????Ai/2
• ?AN/2????Heap???????(AN/2?????Internal
  Node),?????AN/2 -1,,??A1
• ?????O(N)
• ?N?????O(N log N)

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????????Heap
27
????????Heap 1
28
????????Heap 2
29
????????Heap 3
30
????????Heap 4
31
????????Heap 5
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?????(Binary Tree Sort)

• ???????????(Binary Search Tree)
• ????(In-order Traversal)

33
????(Radix Sort)

• ???Bucket Sort?Bin Sort
• ??????LSD(Least Significant Digit)?MSD(Most
  Significant Digit)??
• d????d???
• ??Distribution Sort
• Stable
• ?????O(N)
• ??????????dN,d????

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????(Radix Sort)1radix10
35
????(Radix Sort)2radix10
118
326,228
032
0
1
2
3
4
056
076,879
882
291,199
5
6
7
8
9
36
????(Radix Sort)3radix10
032,056,076
118,199
228,291
326
0
1
2
3
4
879,882
5
6
7
8
9
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Radix Sort ???
Algorithm radix_sort(A, first, last,
maxDigits) // Sorts the array of positive decimal
integers Afirst, last // into ascending order
// maxDigits is the number of digits in the
longest integer for (i 1 to maxDigits)
Clear bucket0, bucket1, ..., bucket9
for (index first to last) digit
i-th digit from the right of Aindex
Place Aindex at end of bucketdigit
Place contents of bucket0, bucket1, ...,
bucket9 into the array A
38
????????????
39
??(Search)

• ?????????????????
• ????(Internal Search)
• ????(External Search)

40
Sequential Search

• ???????
• Sequential Search ?????????
• ??? Linear Search ????
• ?????O(N)
• ??
• ?????????????????????
• ????????????,???????????????

41
Sequential Search
int sequential_search(int data, int target, int
n) int i for (i 0 i lt n i)
if (target datai) return
i return -1
42
Binary Search

• ??????????
• ?????????
• ???????????,??????????????,???????????????????????
  ????,??????????????,???????
• ?????O(log N)

43
Binary Search
int binary_search(int data, int n, int target)
int left 0, right n-1, mid while
(left lt right) mid (left right) /
2 if (target datamid)
return mid else if (target gt
datamid) left mid 1
else right mid - 1
return -1
44
Interpolation Search ???

• ???????(Uniform Distribution)
• ??s?????,x?????,low??????????????????(left),high
  ??????????????????(right)

45
????

• Pattern Matching
• ??????????????????
• ?????????? ?? ????
• ??????????,????????
• ??? Brute Force Method
• ????????????????,????
• ?????O(NM),N?????????,M?????(Pattern)???

46
????-???
int strfind(char string, char pattern)
int pos_s, pos_p, len_s, len_p len_s
strlen(string) len_p strlen(pattern)
/ ???string?????????????(len_s-len_p) / for
(pos_s 0 pos_s lt len_s - len_p pos_s)
for (pos_p 0 pos_p lt len_p pos_p)
/ ???????,???????? / if
(patternpos_p ! stringpos_s pos_p)
break / ?pos_p?pattern???????,??????
/ if (pos_p len_p - 1) return
pos_s return -1
/ ???? /
47
????-KMP?

• Knuth-Morris-Pratt
• ?????????????Pattern??? ,???????????
• Failure function
• ?Pattern??????????(Prefix)??

Table of next possible matching positions
48
????-KMP?
49
Knuth-Morris-Pratt Algorithm

• Pre-compute table of next possible matching
  positions
• Try to match the characters
• If there is a mismatch, use the look-up table to
  find where to start matching next
• If the relevant part of the table is -1,
  increment the position on the main string
• If any other number, begin comparison at this
  location on the pattern but at the current
  position on the main string
• If there is a complete match, you can find the
  start position by taking the pattern index of the
  last matched character away from the current
  position
• After a complete match, check the last position
  in the next array this will tell you if you can
  attempt another match starting in this string
  again
• Return matches

50
Pre-compute table of next possible matching
positions
void preKmp(char x, int m, int kmpNext)
int i, j i 0 j kmpNext0 -1
while (i lt m) while (j gt -1 xi !
xj) j kmpNextj i
j if (xi xj) kmpNexti
kmpNextj else kmpNexti
j

• x the search pattern
• m length of the search pattern

51
KMP Pattern Matching
void KMP(char x, int m, char y, int n) int
i, j, kmpNextXSIZE / Preprocessing /
preKmp(x, m, kmpNext) / Searching / i
j 0 while (j lt n) while (i gt -1
xi ! yj) i kmpNexti
i j if (i gt m)
OUTPUT(j - i) i kmpNexti

52
???(Hashing)

• ????
• O(1) ??
• ??
• ??????
• ????????????????
• ??
• ????
• ??(Collision)????????????
• ???????? O(N) ??

53
???????

• Hash function h(x)
• Map keys into positions in a table
• Hash table
• Bucket(?)
• Each position of the hash table
• Home bucket
• The position h(x)
• Slot(?)
• The number of elements that a bucket may hold
  equals the number of slots in the bucket

54
???????(Hash Table)
55
???????

• Collision(??)
• ?h(x1)h(x2),x1?x2????????bucket?
• Overflow(??)
• ????bucket?collision????slot????
• Loading Factor ????
• Loading Density ????
• ?N/(SB) i.e.????/(????)

56
???????

• ?????yf(x), x,y????
• ???xgty
• ??????????
• Avoid collision
• ????????key?????,??collision????
• Spread keys evenly in the array
• Uniform distribution
• Easy to compute
• The running time of the hash function should be
  O(1)

57
???????

• ??(Division)
• ????(Mid-Square)
• ?????(Digit Analysis)

58
The Division Hash Function

• ??????? h(x)xM
• Many-to-one mapping
• ??
• ????
• M ???????????
• ??
• ???key????????bucket

59
The Division Hash Function
public class DivisionMethod static int M
1031 //?? public static int h(int x)
return Math.abs(x) M
M ???? 20 ?????!
60
The Division Hash Function
????8864, 100, 2, 77, 729
h(x) x 11
Hash Table
77
100
2
729
8864
0
1
2
3
4
5
6
7
8
9
10
61
Middle-Square Method

• ??????
• ???????????????
• ???????????????W???,W2w(w?word size)
• Hash function

62
Middle-Square Method
public class MiddleSquareMethod static int k
10 // M1024 static int w 32
public static int h(int x) return (x x)
gtgtgt (w - k)
??????LSB? (w-k)?????
63
Middle-Square Method
5
1
0
3
2
4
X
5
8
4
9
7
6
2
6
0
4
3
0
X2
????,????
0
0
0
0
5
8
h(X)
gtgtgt??4?
64
Middle-Square Method

• ??
• ????key????(Scatter)
• ??
• Keys with a large number of leading/trailing
  zeros will collide
• Middle-square method only consider a subset of
  the bits in the middle

65
?????(Digit Analysis)

• ????????
• ????? Digit ??????,??????? Digit ??,????????????
  Digit ???

66
???????(Overflow Handling)

• Linear probing ????
• ???????(?????)
• Rehashing ????
• ????????h1(x),h2(x),h3(x),,hm(x),?h1(x)????,???h2
  (x),?????????h3(x),

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???????(Overflow Handling)

• Quadratic probing ????
• ? h(x) ??,????? ( h(x) i2 ) b ? ( h(x) i2 )
  b, 1? i ? (b-1)/2
• Chaining ????
• ? Linked-List ????????????????
• Slot ????????

68
Linear Probing ????

• Linear open addressing ??????
• When home bucket is full, collision item is
  placed in the next available bucket
• Regarding the table as circular
• Clustering problem
• ??????

69
Clustering Problem
????88, 108, 48, 58, 228, 888
h(x) x 10
Hash Table
48
58
228
888
88
108
0
1
2
3
4
5
6
7
8
9
70
Linear Probing ????

• ??
• Start from its home bucket
• If it is not in its home bucket, Looking forward
  (circular) until
• Found!
• An empty bucket is reachedgtelement does not
  exist!
• Return to the home bucketgtelement does not
  exist!
• Worst case insert and search time ?(n)

71
Rehashing ????

• Use 2 distinct hash functions
• h K?0,1,,M-1
• hK?1,2,,M-1
• hi(x)(h(x)ih(x)) mod M