afee passe

About This Presentation

Title:

afee passe

Description:

afee passe – PowerPoint PPT presentation

Number of Views:57

Avg rating:3.0/5.0

Slides: 205

Provided by: nikolaossp

Category:

more less

Transcript and Presenter's Notes

Title: afee passe

1
??af??e?e? pa???s??se??
DRAFT
30/03/03

? ???ssa p????aµµat?sµ?? C
?????aµµat?st???? te??????
??µ?? ded?µ????

2
? ?st???a t?? C

1969-1973 ATT Bell Labs, Dennis Ritchie
1978 The C Programming LanguageKR Kernighan
Ritchie
1983 S?stas? ANSI Standardization Committee X3J11
1989-1990 ?p?d??? ANSI/ISO Standard? ANSI C
1990-1999 ??a?e???s? t?? standard ?p? e??????
C9X? C99

3
?a?a?t???st??? t?? C (i)

G??ssa p??sta?t???? p????aµµat?sµ??
G??ssa µet???? ep?p?d??
??????µ?a st?? ??f?as?(??t? ?a? pe??e?t???)
S?et??? ?a?a?? s?st?µa t?p??
F???s?f?a ? p????aµµat?st?? ??e? p???? ??e???
?a? e????eta? ??a ta sf??µat? t??

4
?a?a?t???st??? t?? C (ii)

?d?a?te?a d?µ?f???? st?? p????
??e? ???s?µ?p????e? ??a t?? p????aµµat?sµ? e?????
f?sµat?? s?st?µ?t?? ?a? efa?µ????
??e? ???s?µ?p????e? ?? ß?s? ??a p?????a ?????
???ss?? C, Java

5
Hello world!

include ltstdio.hgt
void main ()
printf("Hello world!\n")
?a?at???se??
? ep????????a µe t?? ??? ??sµ? ???eta? µ?s? t??
ß?ß???????? s??a?t?se??
?? s?µe?? ??a???? t?? p?????µµat?? e??a? ?
????a s????t?s? main.

6
??t??es? p????aµµ?t??
stdio.h
hello.c
???e?aep??efa??da?(header files)
?eta???tt?st??C (compiler)
???a?????d??a?(source)
Objectcode
hello.o
??te??s?µ?p????aµµa(executable)
??ß???????(library)
S??d?t??(linker)
printf
hello
7
??p?? ded?µ???? (ap???)

????a??? a???µ??
int char
?a????sµ?? p??s?µa?s??
signed unsigned
?a????sµ?? µe??????
short long
????µ?? ????t?? ?p?d?ast????
float double

8
???a?a? ap??? t?p?? ded?µ????

char , signed char , unsigned char
signed short int , unsigned short int
signed int , unsigned int
signed long int , unsigned long int
float
double
long double
?e ??????? ???µa ?sa µp????? ?a pa?a?e?f????.

9
???sµ?? µetaß??t??

int x
int x, y, z
double r
unsigned long abc
??????p???s?
int x 1
int x, y 0, z 2
double r 1.87
unsigned long abc 42000000

10
Sta?e??? (i)

????a?e?
42 0 -1 de?ad????
037 ??tad????
0x1f de?ae?ad????
42U 42L 42UL unsigned long
????t?? ?p?d?ast????
42.0 -1.3 de?ad????
2.99e8 µe d??aµ? t?? 10
42.0F 42.0L float long double

11
Sta?e??? (ii)

?a?a?t??a
'a' '0' ''
??d???? ?a?a?t??e?
\n a??a?? ??aµµ??
\' ap?st??f??
\\ ?a?a?t??a? \ (backslash)
\t a??a?? st???? (tab)
\" e?sa??????
\0 ?a?a?t??a? µe ASCII 0 (null)
\037 µe ASCII 37 (??tad???)
\x1f µe ASCII 1f (de?ae?ad???)

12
Sta?e??? (iii)

S?µß???se????
"abc" "Hello world!\n" "a\"51\""
????se?? sta?e???
const int size 10, num 5
const double pi 3.14159
const char newline '\n'

13
S????a

?eta?? / ?a? /
include ltstdio.hgt
/ This simple program greets the
world by saying "hello" /
void main ()
printf( / eh? / "Hello world!\n")
?? pa?a??t? e??a? ?????
/ Nested / comments / are wrong! /

14
??t?p?s? µe t?? printf

?p??? t?p?? ded?µ????
int d
char c
double lf
string s
?a??de??µa
printf("d lf c s\n",42, 1.2, 'a', "hello")
?p?t??esµa
42 1.200000 a hello

15
??sa???? µe t?? scanf

?d??? ??d???? ??a t??? ap???? t?p???
?a??de??µa
int n
double d
char c
scanf("d", n)
scanf("lf", d)
scanf("c", c)

16
??a p?? s???et? pa??de??µa

include ltstdio.hgt
void main ()
int celcius
double farenheit
printf("Give the temperature (C) ")
scanf("d", celcius)
farenheit 9.0 celcius / 5.0 32.0
printf("d degrees Celcius
"is lf degrees Farenheit",
celcius, farenheit)

17
?e?est?? ?a? e?f??se?? (i)

????µ?t???? te?est??
- /
S?es?a??? te?est??
! lt gt lt gt
??????? te?est??
?????? s??e??? (?a?)
?????? d???e??? (?)
! ?????? ????s? (???)
p.?. (x 3 ! 0) !finished

18
?e?est?? ?a? e?f??se?? (ii)

?e?est?? bit p??? bit (bitwise)
s??e??? bit (AND)
d???e??? bit (OR)
ap???e?st??? d???e??? bit (XOR)
????s? (NOT)
ltlt ???s??s? bit a??ste??
gtgt ???s??s? bit de???
?a??de??µa
(0x0101 0xfff0) ltlt 2
? 0x0400

19
?e?est?? ?a? e?f??se?? (iii)

?e?est?? s???????
(a gt b) ? a b
?e?est?? pa???es??
a-1, b5
?e?est?? a???es??
a b1
a x ?s?d??aµ? µe a a x
?e?est?? a???s?? ?a? µe??s??
a a-- t?µ? p??? t? µetaß???
a --a t?µ? µet? t? µetaß???

20
E?t???? ?a? ??e???? ???? (i)

?e?? e?t???
??t??? ??f?as??
a b5
a
??t??? if
if (a gt b)
max a
else
max b

21
E?t???? ?a? ??e???? ???? (ii)

S???et? e?t???
if (a gt b)
min b
max a
else
max b
min a

????e? ??a eµß??e?a
if (x lt y)
int temp x
x y
y temp

22
E?t???? ?a? ??e???? ???? (iii)
while (s??????) e?t???

??t??? while
int i 1, s 0
while (i lt 10)
s i
i

23
E?t???? ?a? ??e???? ???? (iv)
do e?t??? while (s??????)

??t??? do-while
int i 1, s 0
do
s i
while (i lt 10)

24
E?t???? ?a? ??e???? ???? (v)
for (a?????p???s? s?????? ß?µa) e?t???

??t??? for
int i, s
for (i1, s0 i lt 10 i)
s i

25
E?t???? ?a? ??e???? ???? (vi)

??t??? break
int s
for (i0, s0 i lt 10 i)
int x
scanf("d", x)
s x
printf("Sum is d\n", s)

if (x lt 0)
break

26
E?t???? ?a? ??e???? ???? (vii)

??t??? continue
int s
for (i0, s0 i lt 10 i)
int x
scanf("d", x)
s x
printf("Sum is d\n", s)

if (x lt 0)
continue

27
E?t???? ?a? ??e???? ???? (viii)

??t??? switch
switch (ch)
case 'a'
printf("alpha\n")
break
case 'b'
case 'c'
printf("beta or c\n")
break
default
printf("other\n")

28
E?t???? ?a? ??e???? ???? (ix)

?t???te? ?a? e?t??? goto
int i 1, s 0
loop
s i
if (i lt 10)
goto loop
printf("The sum is d\n", s)

??? goto d?µ?µ???? p????aµµat?sµ??!

29
??µ? t?? p?????µµat??

?? p????aµµa ap?te?e?ta? ap?
s??a?t?se??
?a??????? µetaß??t??
??af???? ap? t?? Pascal
????? p????aµµa, d?ad??as?e? ?a? s??a?t?se?? de
d?af???p?????ta?
t? ????? p????aµµa ???µ??eta? main
?? d?ad??as?e? ????? ap?t??esµa void
??e? ?? s??a?t?se?? st? ?d?? ep?ped?
de? ep?t??p??ta? f???asµ??e? s??a?t?se??
µ??? p??asµa ?at? t?µ? (call by value)

30
?a??de??µa

int f (int x, int y)
return x y
int a
void p (int x)
a f(x, x1)
void main ()
p(6)

31
???a?e? (arrays) (i)

????????a µetaß??t??
µe t? ?d?? ???µa,
µe t?? ?d?? t?p? ded?µ????,
se s??e??µe?e? ??se?? µ??µ??.
?a??de??µa
int a50
double d15, d210
? a???µ?s? a????e? ap? t? 0 !
for (i 0 i lt 50 i)
ai i

32
???a?e? (arrays) (ii)

??????p???s? p??????
int a3 6, 7, 42
char c 'a', 'b', 'c', 'd', 'e'
S?µß???se????
???a? p??a?e? ?a?a?t????
char s6 "abcde"
?e?µat????ta? µe t?? ?e?? ?a?a?t??a '\0'
char s6 'a', 'b', 'c',
'd', 'e', '\0'

33
???a?e? (arrays) (iii)

?a??de??µa
int a3 5, 6, 7
int b3 2, 3, 1
int i
for (i 0 i lt 3 i)
printf("d times d is d\n",
ai, bi, aibi)

5 times 2 is 10 6 times 3 is 18 7 times 1 is 7
34
????d??stat?? p??a?e? (i)

?a??de??µa
int a35
char c5104
??????p???s?
int i33 1, 0, 0,
0, 1, 0,
0, 0, 1
char s10
"my", "name", "is", "joe"

35
????d??stat?? p??a?e? (ii)

?a??de??µa p???ap?as?asµ?? p??????
double a34 ... ,
b45 ... ,
c35
int i, j, k
for (i0 ilt3 i)
for (j0 jlt5 j)
cij 0.0
for (k0 klt4 k)
cij aik bkj

36
??sa???? st?? d?µ?? ded?µ???? (i)

???????µ??
?epe?asµ??? s????? e?t???? p??,?ta?
e?te?est???,ep?t???????? ??p??? ep???µ?t?
ap?t??esµa
?ed?µ??a e?s?d?? ?a? e??d??
???e e?t??? p??pe? ?a e??a?
?a?? ???sµ???
ap??
? e?t??es? p??pe? ?a staµat?

37
??sa???? st?? d?µ?? ded?µ???? (ii)

????p????t?ta
??t?? e?t?µ?s?? t?? ap?d?s?? a??????µ????
s????t?s? t?? µe??????t?? p??ß??µat?? p??
ep??????
?????? e?t??es??
????t???t?ta µ??µ?? p?? apa?te?ta?
????ß?? µ?t??s? p???p????t?ta?
t f(n)
???? µe??????, s?µß???sµ?? ?, O, T
t O(f(n))

38
??sa???? st?? d?µ?? ded?µ???? (iii)

???sµ?? t?? ???se?? ?, O, T
??? ????
g O(f) ? ?c. ?n0. ?n gt n0. g(n) lt c f(n)
??t? ????
g O(f) ? ?c. ?n0. ?n gt n0. g(n) gt c f(n)
???? µe??????
g T(f) ? ?c1, c2. ?n0. ?n gt n0. c1 f(n) lt
g(n) lt c2 f(n)
???ta?? µe????? ???se?? p???p????t?ta?
O(1) lt O(logn) lt O(?n) lt O(n) lt O(nlogn)lt O(n2)
lt O(n3) lt O(2n) lt O(n!) lt O(nn)

39
??sa???? st?? d?µ?? ded?µ???? (iv)

?fa??es? ded?µ???? (data abstraction)
??a????sµ?? ?d??t?t?? ?a? ???p???s??
?d??t?te? e??? t?p?? ded?µ????? t??p?? µe t??
?p??? d?µ?????e? ?a?e?? ?a? ???s?µ?p??e? ded?µ??a
a?t?? t?? t?p??
???p???s? e??? t?p?? ded?µ????? t??p?? µe t??
?p??? a?apa??sta?ta? ta ded?µ??a st? µ??µ? t??
?p?????st? ?a? p????aµµat????ta? ?? d?a??s?µe?
p???e??

40
??sa???? st?? d?µ?? ded?µ???? (v)

?? ?ept?µ??e?e? ???p???s?? ep??e?????t??
p???p????t?ta t?? a??????µ??
???a? s???? ep?µ???? ? a??a?? t??p?? ???p???s??
??p???? t?p?? ded?µ????
? afa??es? ded?µ???? e?a??st?p??e? t?? a??a???
p?? apa?t???ta? st? p????aµµa p?? ???s?µ?p??e?
??a? t?p? ded?µ????, ?ta? a????e? ? t??p??
???p???s??

41
?f???µ???? t?p?? ded?µ????

?f???µ???? t?p?? ded?µ???? ???(abstract data
type)
?a?????e? t?? ?d??t?te? t?? t?p?? ded?µ????
de? ?a?????e? t?? ???p???s? t??
???eß????? ???sµ?? ???
s??ta?? ???µa t?? t?p?? ?a? ep??efa??de? t??
p???e??
s?µas??????a ?a???e? p?? pe?????f??? t?
?e?t?????a t?? p???e??

42
?a??de??µa ??? S????a (i)

S??ta??
???µa t?p?? set (s????a a?e?a???)
?p??efa??de? p???e??
const set empty
set add (int x, set s)
boolean member (int x, set s)
set union (set s1, set s2)
boolean subset (set s1, set s2)
boolean equal (set s1, set s2)
S?µe??s? ?e????µe ?t? ??e? ???ste? ?at?????a ?
t?p?? boolean.

43
?a??de??µa ??? S????a (ii)

S?µas??????a
????µat???? ???sµ?? member
member(x, empty) false
member(x, add(x, s)) true
member(x, add(y, s)) member(x, s) a? x ! y
????µat???? ???sµ?? union
union(empty, s) s
union(add(x, s1), s2) add(x, union(s1, s2))

44
?a??de??µa ??? S????a (iii)

S?µas??????a
????µat???? ???sµ?? subset
subset(empty, s) true
subset(add(x, s1), s2) band(member(x, s2),
subset(s1, s2))
????µat???? ???sµ?? equal
equal(s1, s2) band(subset(s1, s2),
subset(s2, s1))

45
S???e???µ???? t?p?? ded?µ????

S???e???µ???? t?p?? ded?µ???? S??(concrete
data type)
?a?????e? t?? ?d??t?te? t?? t?p?? ded?µ????
?a?????e? epa???ß?? t?? ???p???s? t??
???e ???ssa p????aµµat?sµ?? ?p?st????e?
???sµ????? S??, p.?.
ap??? t?p?? a???a??? a???µ??, p?a?µat????
a???µ??, ?a?a?t??e?, ??????? t?µ??
s???et?? t?p?? p??a?e? (arrays), e???af??
(records), de??te? (pointers), s????a (sets)

46
S??s? ??? ?a? S?? (i)

??? set s????a a?e?a??? s 1, 3, 7, 9
???p???s? 1 p??a?a? ap? boolean

???p???s? 2 bits e??? p??a?a ap? int

Stat???? t??p?? ???p???s?? S??

47
S??s? ??? ?a? S?? (ii)

???p???s? 3 ap?? s??dedeµ??? ??sta

???aµ???? t??p?? ???p???s?? S??
???p???s? 4 s?µß???se??? p?? pe????e? t?
µa??µat??? ???sµ? t?? s??????
"x x1 \/ x3 \/ x7 \/ x9"
"x 1ltxlt10 /\ x21 /\ xltgt5"
??s???? ? ???p???s? t?? p???e??
?pe???s????a "x x gt 100"

48
???a?e? ?? ??? (i)

?as??? p???? p??sp??as? st???e???
ai
S?????? ???p?????ta? µe ??p??? S?? p??????
(arrays)
??st?? p??sp??as?? O(1)
? S?? t?? µ???d??stat?? p??a?a epa??e? ??a t??
???p???s? ???e ??? p??a?a
S????t?s? loc ?p??????e? t? ??s? e??? st???e???
t?? ??? p??a?a st? µ???d??stat? S?? p??a?a t??
???p???s??

49
???a?e? ?? ??? (ii)

??? p??a?a d?? d?ast?se?? n ? m

loc (n, m, i, j) m (i 1) j 1
????µ?s? ?at? st??e?
loc (n, m, i, j) n (j 1) i 1

50
???a?e? ?? ??? (iii)

??? ??t? t????????? p??a?a n ? n

loc (n, i, j) i (i 1) / 2 j 1
?µ???? ??a s?µµet?????? p??a?e?

51
???a?e? ?? ??? (iv)

??? t??d?a?????? p??a?a n ? n

loc (n, i, j) 2 i j 3
52
???a?e? ?? ??? (v)

??? a?a??? p??a?a n ? m

???p???s? µe d?ad??? p??a?a
???p???s? µe t?e?? p??a?e?
row 1, 2, 3, 3, 4 col 1, 3, 2, 3, 5
val a1, a2, a3, a4, a5

53
??a??t?s? se p??a?e? (i)

Se???a?? a?a??t?s?
?a st???e?a d?at?????ta? ?at? se???
??st?? O(n)

?e?t??s? st???e?? f?????? (sentinel)

7
54
??a??t?s? se p??a?e? (ii)

???p???s? se C
int ssearch (int a, int n, int x)
int i
for (i 0 i lt n i)
if (ai x)
return i
return -1

55
??a??t?s? se p??a?e? (iii)

???p???s? se C µe f?????
int ssearch_s (int a, int n, int x)
int i
an x
for (i 0 ai ! x i)
return (i lt n) ? i -1

56
??a??t?s? se p??a?e? (iv)

??ad??? a?a??t?s?
? p??a?a? p??pe? ?a e??a? ta????µ?µ????
??st?? O(logn)

???e? µ???d?? a?a??t?s??
?????te?? ??st?? ? pe??ss?te??? ?????
???a?e? ?ata?e?µat?sµ?? (hash tables)

57
?a????µ?s? p?????? (i)

?a????µ?s? ep?????? (selection sort)
?d?a
??a ???e i ?at a????sa se???
ß?e? t? µ????te?? t?? st???e??? µet? t? ai
a?t?µet??es? t? µe t? ai
??st?? ?(n2)
?e?t??s? st???e?? f??????

58
?a????µ?s? p?????? (ii)

?a????µ?s? ep??????, ???p???s? se C
void ssort (int a, int n)
int i, j
for (i 0 i lt n-1 i)
int min ai, minj i
for (j i1 j lt n j)
if (aj lt min)
min aminj j
aminj ai
ai min

59
?a????µ?s? p?????? (iii)

?a????µ?s? e?sa????? (insertion sort)
???s?µ?p????µe a?t? t?? t??p? ?ta? ta????µ??µe ta
?a?t?? µ?a? t??p???a?
?d?a
??a ???e i ap? t? de?te?? ?a? ?at a????sa
se???
t?p???t?se t? ai st? s?st? t?? ??s?µeta?? t??
st???e??? p?? e??a? p??? ap? a?t?
??st?? ?(n2)
?e?t??se??
st???e?? f??????
d?ad??? e?sa????

60
?a????µ?s? p?????? (iv)

?a????µ?s? e?sa?????, ???p???s? se C
void isort (int a, int n)
int i, j
for (i 1 i lt n i)
int x ai
for (j i-1 j gt 0 j--)
if (x lt aj)
aj1 aj
else
break
aj1 x

61
?a????µ?s? p?????? (v)

?a????µ?s? f?sa??da? (bubble sort)
?d?a
??a ???e i ?at a????sa se???
??a ???e j gt i ?at? f?????sa se???
a? aj-1 gt aj a?t?µet??ese ta aj-1 ?a?
aj
??st?? ?(n2)
?e?t??se??
staµat? a? se ??a p??asµa de? ???e? a?t?µet??es?
se p??? s?µe?? ????e ? te?e?ta?a a?t?µet??es?
a??a?? ?ate????s?? µeta?? d?ad?????? pe?asµ?t??
(shake sort)

62
?a????µ?s? p?????? (vi)

?a????µ?s? f?sa??da?, ???p???s? se C
void bsort (int a, int n)
int i, j
for (i 0 i lt n i)
for (j n-1 j gt i j--)
if (aj-1 gt aj)
int temp aj-1
aj-1 aj
aj temp

63
?a????µ?s? p?????? (vii)

?a????µ?s? µe d?aµ???s? (quick sort)
?d?a
d???e?e ??a t??a?? st???e?? x t?? p??a?a
d?aµ???se t?? p??a?a, µetaf????ta?
ta st???e?a µ????te?a t?? x st?? a???
ta st???e?a µe?a??te?a t?? x st? t????
a?ad??µ???, ta????µ?se ta d?? µ???
??st?? ?(n2) st? ?e???te?? pe??pt?s? ?(n logn)
?at? µ?s? ???

64
?a????µ?s? p?????? (viii)

Quick sort, ???p???s? se C
void qsort (int a, int n)
qsort_auxil(a, 0, n-1)
void qsort_auxil (int a, int lower,
int upper)
if (lower lt upper)
int x a(lower upper) / 2
int i, j
for (i lower, j upper
i lt j i, j--)
while (ai lt x) i
while (aj gt x) j--

65
?a????µ?s? p?????? (ix)

Quick sort, ???p???s? se C (s????e?a)
if (i lt j)
int temp ai
ai aj
aj temp
qsort_auxil(a, lower, j)
qsort_auxil(a, i, upper)

66
?a????µ?s? p?????? (x)

????? t??p?? ta????µ?s??
?a????µ?s? t?? Shell (shell sort),??st?? ?(n2)
?a????µ?s? se s??? (heap sort),??st?? ?(n
logn)
?a????µ?s? µe s?????e?s? (merge sort),??st??
?(n logn)

67
???sµ?? t?p??

S?????µa ap??? t?p??
typedef double real
real x, y
S?????µa s???et?? t?p??
typedef double vector 10
vector v
for (i 0 i lt 10 i)
vi i

68
?pa???µ?se?? (enumerations)

? t?p?? enum
enum color_tag
GREEN, RED, BLUE, WHITE, BLACK
enum color_tag c
c GREEN
typedef enum
GREEN, RED, BLUE, WHITE, BLACK
color
color c WHITE

69
??µ?? (structures) (i)

? t?p?? struct
struct complex_tag
double re, im
struct complex_tag z

z.re 1.0 z.im -1.0 typedef struct
complex_tag complex complex z complex zs 100
70
??µ?? (structures) (ii)

?a??de??µa
complex complement (complex z)
complex result
result.re z.re
result.im -z.im
return result

71
??µ?? (structures) (iii)

?a??de??µa
typedef struct
char firstName 30
char lastName 50
char phone 10
int age
double salary
record
record r 1000
r423.age 32

72
???se?? (unions)

? t?p?? union
union number_tag
int i
double d
union number_tag n

n.d 1.2 printf("lf\n", n.d) n.i
42 printf("d\n", n.i) printf("lf\n", n.d)
/ wrong! /
73
?e??te? (pointers) (i)

?etaß??t??
p?? de? pe??????? µ?a t?µ? ??p???? ap??? t?p??
ded?µ????
a??? t? d?e????s? µ?a? ????? µetaß??t??
?a??de??µa
int x, p

42
74
?e??te? (pointers) (ii)

?e??t?d?t?s?
? d?e????s? µ?a? µetaß??t??
p x
?p?de??t?d?t?s?
t? pe??e??µe?? µ?a? d?e????s??
printf("d", p)
p 7

42
75
???a?e? ?a? de??te? (i)

????µ?t??? de??t??
int a3 7, 6, 42
int p

p (a0) p a p a
printf("d\n", p) printf("d\n", (p1)) p
p2 printf("d\n", p)
76
???a?e? ?a? de??te? (ii)

?s?d??aµ?a p?????? ?a? de??t??
??a? p??a?a? e??a? ??a? de??t?? st? p??t?
st???e??.
ai ?s?d??aµ? µe (ai)
?? p??a?e? ?µ?? e??a? sta?e??? de??te?, d??ad?
de? µp????? ?a a??????? t?µ?
int a3 7, 6, 42
int p a
p / correct /
a / wrong! /

77
???a?e? ?a? de??te? (iii)

?a??de??µa a?t???af? p??????
int a10, b10
int p a, q b, i
/ assume b initialized /
for (i0 ilt10 i)
p q

78
???a?e? ?a? de??te? (iv)

?a??de??µa e?t?p?s? s?µß???se????
void putchar (char c)
char s "Hello world!\n"
char p
for (p s p ! '\0' p)
putchar(p)

79
?e??? de??t?? ?a? de??t?? se ?e??

? ?e??? de??t?? NULL
?pa???e?eta? ?a ap?de??t?d?t??e?!
int p NULL
p 42 / wrong! /
? µ???? de??t?? p?? a?t?st???e? st?? ?e?d? ??????
t?µ?, ?ta? ???s?µ?p??e?ta? se s??????

? de??t?? se ?e?? void
Ge???? µ??f? de??t?
?pa???e?eta? ?a ap?de??t?d?t??e?!
?pa???e?eta? ? a???µ?t??? de??t??!

80
?e??te? a?t? pe??sµat?? µe a?af???

??d??a? Pascal
procedure inc (var x integer)
begin
x x1
end
... inc(a) ...
?s?d??aµ?? ??d??a? C
void inc (int px)
(px)
... inc(a) ...

81
?e??te? se e???af??

S??t?µ???af?a
p-gtx e??a? ?s?d??aµ? µe (p).x
?a??de??µa
struct
int x, y
coordinates, p
coordinates.x 1
coordinates.y 3
p coordinates
printf("d\n", p-gtx)
printf("d\n", p-gty)

82
?etat??p?? t?p?? (i)

?µµese? (coercions)
double d 3 (µe a???es?)
int x 3.14
int f (int x) (µe p??asµa pa?aµ?t???)
f(3.14)
?µese? (type casting)
( t?p?? ) ??f?as?
(double) 3
(int) 3.14
(int ) NULL

83
?etat??p?? t?p?? (ii)

???ß??µa p?? µp??? ?a t?p?s? t? ap?t??esµa t??
p?a?µat???? d?a??es?? d?? a?e?a??? a???µ??, ?????
??e? µetaß??t??
int x 5, y 3
????? ??s? 1 printf("d", x/y)
????? ??s? 2 printf("lf", x/y)
S?st?? ??se??
printf("lf", 1.0 x / y)
printf("lf", (double) x / (double) y)
printf("lf", (double) x / y)

84
???aµ??? pa?a????s? µ??µ?? (i)

S??a?t?se?? ß?ß???????? ltstdlib.hgt
void malloc (size_t n)
???aµ??? pa?a????s? µ??µ?? µ????? n bytes.??
ap?t??esµa p??pe? ?a µetat?ape? st? s?st?
t?p?.?p?st??feta? NULL a? e?a?t???e? ? µ??µ?.
void free (void p)
?p?d?sµe?s? t?? µ??µ?? st?? ?p??a de???e? t?
p.?? p p??pe? ?a ??e? d?µ???????e? µe
p??????µe?? ???s? st? malloc.
??sa bytes ??e?????ta?
sizeof(type) p.?. sizeof(int)
sizeof(variable) p.?. sizeof(x)

85
???aµ??? pa?a????s? µ??µ?? (ii)

?a??de??µa
int p
int i
p (int ) malloc(sizeof(int))
p 42
free(p)
p (int ) malloc(10 sizeof(int))
for (i 0 i lt 10 i)
pi 42
free(p)

86
???aµ???? S?? (i)

???s?µ?p?????ta? ??a t?? ???p???s? ???
?a?ade??µata
s??dedeµ??e? ??ste?
d??t?a
???f??
??? ???p?????ta? st? C
µe ?at?????? s??d?asµ? d?µ?? (struct)?a? de??t??
(pointers), ?a?
µe d??aµ??? pa?a????s? µ??µ??

87
???aµ???? S?? (ii)

??e??e?t?µata ??a?t? stat???? S??
?e? ep?ß?????? pe?????sµ??? st? µ???st? p?????
t?? ded?µ????
? µ??µ? p?? ???s?µ?p??e?ta? e??a? a?????? t??
p?a?µat???? p?????? t?? ded?µ????
??p??e? p???e?? ???p?????ta? ap?d?t???te?a
?e???e?t?µata ??a?t? stat???? S??
G?a sta?e?? ?a? ???st? p????? ded?µ????,
???s?µ?p????? s?????? pe??ss?te?? µ??µ?
??p??e? p???e?? ???p?????ta? ????te?? ap?d?t???

88
???aµ???? S?? (iii)

?d?a
?atas?e???eta? ??a s????? ??µß??
???e ??µß?? pe????e? p????f???e? ?a? s??d?sµ???
p??? ?????? ??µß???

89
?p?? s??dedeµ??e? ??ste? (i)

???a? ??aµµ???? d?at??e??
???e ??µß?? pe????e? ??a s??desµ? st?? ep?µe??
??µß?
? te?e?ta??? ??µß?? ??e? ?e?? s??desµ?

90
?p?? s??dedeµ??e? ??ste? (ii)

?a??de??µa ??sta a?e?a???
struct node_tag
int data
struct node_tag next
typedef struct node_tag Node,
LinkedList
?e?? ??sta

91
?p?? s??dedeµ??e? ??ste? (iii)

???s???? st???e???
ap?fas? p???a p??ste?e?
d?µ??????a???? ??µß??
a?t???af?p????f???a?
s??des????? ??µß??

i
92
?p?? s??dedeµ??e? ??ste? (iv)

?fa??es? st???e???
ap?fas? p???st???e???a afa??e?e?
?atast??f???µß??
s??des??p????p????µß??

93
????? (i)

First In First Out (FIFO)?,t? µpa??e? p??t?,
ß?a??e? p??t?
???? a?e?a???
??? queue
const queue queueEmpty
void queueInsert (queue qp, int t)
int queueRemove (queue qp)
int queueHead (queue q)

94
????? (ii)

???p???s? µe ap?? s??dedeµ??? ??sta

95
???p???s? ????? se C (i)

???p???s? µe ap?? s??dedeµ??? ??sta
typedef struct list_tag
int data
struct list_tag next
ListNode
??p?? queue
typedef struct
ListNode first
ListNode last
queue

96
???p???s? ????? se C (ii)

?de?a ????
const queue queueEmpty NULL, NULL
??sa???? st???e???
void queueInsert (queue qp, int t)
ListNode n (ListNode )
malloc(sizeof(ListNode))
if (n NULL)
printf("Out of memory\n")
exit(1)

97
???p???s? ????? se C (iii)

??sa???? st???e??? (s????e?a)
n-gtdata t
n-gtnext NULL
if (qp-gtlast NULL)
qp-gtfirst qp-gtlast n
else
qp-gtlast-gtnext n
qp-gtlast n

98
???p???s? ????? (iv)

?fa??es? st???e???
int queueRemove (queue qp)
ListNode n
int result
if (qp-gtfirst NULL)
printf("Nothing to remove"
" from an empty queue\n")
exit(1)

99
???p???s? ????? (v)

?fa??es? st???e??? (s????e?a)
n qp-gtfirst
result qp-gtfirst-gtdata
qp-gtfirst qp-gtfirst-gtnext
free(n)
if (qp-gtfirst NULL)
qp-gtlast NULL
return result

100
???p???s? ????? (vi)

???tas? st???e???
int queueHead (queue q)
if (q.first NULL)
fprintf(stderr, "Nothing to see" " in
an empty queue\n")
exit(1)
return q.first-gtdata

101
St??ße? (i)

Last In First Out (LIFO)?,t? µpa??e? te?e?ta??,
ß?a??e? p??t?
St??ßa a?e?a???
??? stack
const stack stackEmpty
void stackPush (stack sp, int t)
int stackPop (stack sp)
int stackTop (stack s)

102
St??ße? (ii)

???p???s? µe ap?? s??dedeµ??? ??sta

103
???p???s? st??ßa? se C (i)

??p?? stack
typedef ListNode stack
?de?a st??ßa
const stack stackEmpty NULL
??sa???? st???e???
void stackPush (stack sp, int t)
ListNode n (ListNode )
malloc(sizeof(ListNode))
if (n NULL)
printf("Out of memory\n")
exit(1)

104
???p???s? st??ßa? se C (ii)

??sa???? st???e??? (s????e?a)
n-gtdata t
n-gtnext sp
sp n
?fa??es? st???e???
int stackPop (stack sp)
ListNode n
int result
if (sp NULL)
printf("Nothing to remove"
" from an empty stack\n")
exit(1)

105
???p???s? st??ßa? se C (iii)

?fa??es? st???e??? (s????e?a)
n sp
result (sp)-gtdata
sp (sp)-gtnext
free(n)
return result

106
???p???s? st??ßa? se C (iv)

???tas? st???e???
int stackTop (stack s)
if (s NULL)
printf("Nothing to see"
" in an empty stack\n")
exit(1)
return s-gtdata

107
??p?? s??dedeµ??e? ??ste? (i)

?p?s?? ??aµµ???? d?at??e??
??? s??desµ?? se ???e ??µß?, p??? t?? ep?µe?? ?a?
p??? t?? p??????µe??
Ge???? µ??f?, p.?. ??a ???p???s? ?????

108
??p?? s??dedeµ??e? ??ste? (ii)

??p?? ??µß?? DListNode
typedef struct DListNode_tag
int data
struct DListNode_tag next
struct DListNode_tag prev
DListNode
??p?? dlist
typedef struct
DListNode first
DListNode last
dlist
?de?a ??sta
const dlist dlistEmpty NULL, NULL

109
??p?? s??dedeµ??e? ??ste? (iii)

??sa???? st???e??? st?? a???
void dlistInsert (dlist lp, int t)
DListNode n (DListNode )
malloc(sizeof(DListNode))
if (n NULL)
fprintf(stderr, "Out of memory\n")
exit(1)
n-gtdata t

110
??p?? s??dedeµ??e? ??ste? (iv)

??sa???? st???e??? st?? a??? (s????e?a)
if (lp-gtfirst NULL)
n-gtprev n-gtnext NULL
lp-gtfirst lp-gtlast n
else
n-gtprev NULL
n-gtnext lp-gtfirst
lp-gtfirst-gtprev n
lp-gtfirst n

111
???????? ??ste? (i)

??p?? clist
typedef struct
ListNode first
ListNode last
clist
?de?a ??sta
const clist clistEmpty NULL, NULL

112
???????? ??ste? (ii)

??sa???? st???e???
void clistInsert (clist lp, int t)
ListNode n (ListNode )
malloc(sizeof(ListNode))
if (n NULL)
fprintf(stderr, "Out of memory\n")
exit(1)
n-gtdata t

113
???????? ??ste? (iii)

??sa???? st???e??? (s????e?a)
if (lp-gtfirst NULL)
lp-gtfirst lp-gtlast n
n-gtnext n
else
n-gtnext lp-gtfirst
lp-gtlast-gtnext n
lp-gtlast n

114
???????? ??ste? (iv)

?fa??es? st???e???
int clistRemove (clist lp)
int result
if (lp-gtfirst NULL)
fprintf(stderr, "Nothing to remove" "
from empty list\n")
exit(1)
result lp-gtfirst-gtdata

115
???????? ??ste? (v)

?fa??es? st???e??? (s????e?a)
if (lp-gtfirst lp-gtlast)
free(lp-gtfirst)
lp-gtfirst lp-gtlast NULL
else
lp-gtfirst lp-gtfirst-gtnext
free(lp-gtlast-gtnext)
lp-gtlast-gtnext lp-gtfirst
return result

116
???????? ??ste? (vi)

??t?p?s? st???e???
void clistPrint (clist l)
ListNode n
for (n l.first n ! NULL n
n-gtnext)
printf("d\n", n-gtdata)
if (n-gtnext l.first)
break

117
?a????µ?µ??e? ??ste? (i)

??p?? slist
typedef ListNode slist
?de?a ??sta
const slist slistEmpty NULL
??sa???? st???e???
void slistInsert (slist lp, int t)
ListNode n (ListNode )
malloc(sizeof(ListNode))
if (n NULL)
printf("Out of memory\n")
exit(1)

118
?a????µ?µ??e? ??ste? (ii)

??sa???? st???e??? (s????e?a)
n-gtdata t
while (lp ! NULL (lp)-gtdata lt t)
lp ((lp)-gtnext)
n-gtnext lp
lp n

119
?a????µ?µ??e? ??ste? (iii)

?fa??es? st???e???
void slistRemove (slist lp, int t)
ListNode n
while (lp ! NULL (lp)-gtdata lt t)
lp ((lp)-gtnext)
if (lp NULL)
printf(d was not found\n, t)
exit(1)
n lp
lp (lp)-gtnext
free(n)

120
?a??µet??? t?? p?????µµat?? (i)

?p??efa??da t?? p?????µµat??
int main (int argc, char argv)
?a??µet???
argc ? a???µ?? t?? pa?aµ?t???
argvi ? i-?st? pa??µet???
argv0 t? ???µa t?? p?????µµat??
?p?t??esµa
a???a??? a???µ?? p?? ep?st??feta? st? ?e?t???????
s?st?µa
s?????? 0 ??a ep?t??? te?µat?sµ?

121
?a??µet??? t?? p?????µµat?? (ii)

?a??de??µa
int main (int argc, char argv)
int i
printf("Program s called with "
"d parameters\n", argv0,
argc-1)
for (i 1 i lt argc i)
printf(" s", argvi)
printf("\nand will return 0\n")
return 0

122
S??a?t?se?? e?s?d??-e??d?? (i)

?as???? s??a?t?se?? e?s?d??-e??d??
int printf (const char format, ...)
int scanf (const char format, ...)
??d???? ?a?a?t??e? st? format
????a??? a???µ??
d st? de?ad??? s?st?µa
u ????? p??s?µ? st? de?ad??? s?st?µa
o ????? p??s?µ? st? ??tad??? s?st?µa
x ????? p??s?µ? st? de?ae?ad??? s?st?µa

123
S??a?t?se?? e?s?d??-e??d?? (ii)

??d???? ?a?a?t??e? st? format
????µ?? ????t?? ?p?d?ast????f se µ??f?
-ddd.dddddde se µ??f? -ddd.dddddd e
/-dddg se µ??f? f ? e
????? t?p??c ?a?a?t??e?s s?µß???se????p de??
te?

124
S??a?t?se?? e?s?d??-e??d?? (iii)

?a?a??a??? st? format
???e??? a???µ??h a???µ?? short p.?. hd,
hxl a???µ?? long ? double p.?. ld,
lfL a???µ?? long double p.?. Lf
????? ap?te??sµat??8d a???µ?? se µ???? 8
?a?a?t????20s s?µß???se??? se µ???? 20
?a?a?t????8d a???µ?? se µ???? 8 ?a?a?t???? µe
08d a???µ?? se µ???? 8 ?a?a?t????, ta p??ta
0-8d ?p?? t? 8d µe st????s? a??ste??

125
S??a?t?se?? e?s?d??-e??d?? (iv)

??s?d??-???d?? ?a?a?t????
int putchar (int c)
int getchar ()
??s?d??-???d?? s?µß???se????
int puts (const char s)
char gets (char s)
??e???? t????? ded?µ????
int eof ()
? sta?e?? EOF pa??st??e? t? t???? t?? ded?µ????
?a? ??e? t?p? int.

126
?a??de??µa

??t???af? ded?µ????
?? ?a?a?t??e? p?? d?aß????ta? e?t?p????ta?, µ????
?a pa???s?aste? t???? ded?µ????
void main ()
int c
while ((c getchar()) ! EOF)
putchar(c)

127
S??a?t?se?? d?a?e???s?? a??e??? (i)

??p?? a??e???
FILE fp
?????µa a??e???
FILE fopen (const char filename,
const char mode)
?a??µet??? moder a?????s? (read)w e???af?
(write)a p??s???? (append)t ?e?µe??
(text)b d?ad??? ded?µ??a (binary)

128
S??a?t?se?? d?a?e???s?? a??e??? (ii)

??e?s?µ? a??e???
int fclose (FILE fp)
??s?d??-???d?? ?a?a?t????
int fputc (int c, FILE fp)
int fgetc (FILE fp)
??s?d??-???d?? s?µß???se????
int fputs (const char s, FILE fp)
char fgets (char s, int n, FILE
fp)

129
S??a?t?se?? d?a?e???s?? a??e??? (iii)

?as???? s??a?t?se?? e?s?d??-e??d??
int fprintf (FILE fp, const char
format, ...)
int fscanf (FILE fp, const char
format, ...)
??e???? t????? a??e???
int feof (FILE fp)

130
S??a?t?se?? d?a?e???s?? a??e??? (iv)

??s?d??-???d?? p????? ded?µ????
size_t fwrite (const void p,size_t size,
size_t num, FILE fp)
size_t fread (void p,size_t size, size_t num,
FILE fp)
? a???a??? t?p?? size_t ???s?µ?p??e?ta? ??a t?
µ?t??s? ????? µ??µ?? se bytes.

131
?a??de??µa (i)

??t???af? d?ad???? a??e???
int main (int argc, char argv)
FILE fin, fout
unsigned char buffer1000
size_t count
fin fopen(argv1, "rb")
if (fin NULL)
return 1
fout fopen(argv2, "wb")
if (fout NULL)
return 2

132
?a??de??µa (ii)

(s??e???eta?)
while (!feof(fin))
count fread(buffer, 1,
1000, fin)
fwrite(buffer, 1, count, fout)
fclose(fin)
fclose(fout)
return 0

133
S??a?t?se?? ß?ß???????? (i)

??s?d?? ?a? ???d?? ltstdio.hgt
?e????e? ??e? t?? s??a?t?se?? e?s?d??-e??d??
????a????sµ??a a??e?a
FILE stdin t?p??? e?s?d??FILE
stdout t?p??? ???d??FILE stderr t?p???
???d?? sfa?µ?t??
?s?d??aµ?e?
printf(...) ? fprintf(stdout, ...)scanf(...) ?
fscanf(stdin, ...)
?.?p.
S??a?t?se?? d?a?e???s?? a??e??? µe t??a?a
p??sßas? (random access)

134
S??a?t?se?? ß?ß???????? (ii)

??a?e???s? s?µß???se???? ltstring.hgt
size_t strlen (const char s)
??t??s? a???µ?? ?a?a?t???? t?? s?µß???se???? s.
char strcpy (char s1, const
char s2)
??t???af? t?? s?µß???se???? s2 st?? s1.
char strcat (char s1, const
char s2)
???s???? t?? s?µß???se???? s2 st? t???? t?? s1.
int strcmp (const char s1,
const char s2)
S?????s? t?? s?µß???se???? s1 ?a? s2.

135
S??a?t?se?? ß?ß???????? (iii)

?etat??p? s?µß???se???? ltstdlib.hgt
int atoi (const char s)
?etat??p? t?? s?µß???se???? s se int.
long int atol (const char s)
?etat??p? t?? s?µß???se???? s se long int.
double atof (const char s)
?etat??p? t?? s?µß???se???? s se double.

136
?a??de??µa

??a d??at? ???p???s? t?? strcmp
int strcmp (const char s1,
const char s2)
while (s1 s2 s1 ! '\0')
s1 s2
return (s1)-(s2)
?p??es? ?? t?µ?? t?? t?p?? char e??a? st?
d??st?µa 0-255 (??? a???t????)

? while (s1 s1 s2)
137
???epe?e??ast?? (preprocessor) (i)

??t??? include
include ltstdio.hgt
include "myheader.h"
??t??? define
define MAX_CHARS 1000
char sMAX_CHARS
define INC(x) ((x))
INC(a)
INC(p)

138
???epe?e??ast?? (preprocessor) (ii)

??t???? ifdef, ifndef, else ?a? endif
define DEBUG
ifdef DEBUG
printf("debugging is on\n")
else
printf("debugging is off\n")
endif
ifndef DEBUG
printf("optimizations allowed\n")
endif

139
??ad??? d??d?a (i)

Binary trees
??? s??desµ?? se ???e ??µß?
a??ste?? ?a? de?? pa?d?

???e ??µß?? ??e?0, 1 ? 2 pa?d??
???a ? a?????? ??µß?? t?? d??d???
F???a ??µß?? ????? pa?d??
????? ??µß?? a???µ?? s??d?sµ?? ap? t? ???a

140
??ad??? d??d?a (ii)

??p?? ??µß?? TreeNode
typedef struct TreeNode_tag
int data
struct TreeNode_tag left
struct TreeNode_tag right
TreeNode
??p?? tree
typedef TreeNode tree
?e?? d??d??
const tree treeEmpty NULL

141
??ad??? d??d?a (iii)

??sa???? se d??d?a
?a????st??? ap?fas? se p??? s?µe?? t?? d??d???
?a e?sa??e? ? ???? ??µß??
?s?????sµ??a d??d?a (balanced trees) t? ß????
d?? f????? d?af??e? t? p??? ?at? 1
S????t?s? µ???st?? ß?????
int treeDepth (tree t)
if (t NULL) return 0
return 1 max(treeDepth(t-gtleft),
treeDepth(t-gtright))

142
??ad??? d??d?a (iv)

??sa???? se ?s?????sµ??a d??d?a
void treeBalancedInsert (tree t, int d)
if (t NULL)
t (TreeNode ) malloc(sizeof(TreeN
ode))
if (t NULL)
printf("Out of memory\n")
exit(1)
(t)-gtdata d
(t)-gtleft (t)-gtright NULL

143
??ad??? d??d?a (v)

??sa???? se ?s?????sµ??a d??d?a (s????e?a)
else
int d1 treeDepth((t)-gtleft)
int d2 treeDepth((t)-gtright)
if (d1 lt d2)
treeBalancedInsert(
((t)-gtleft), d)
else
treeBalancedInsert(
((t)-gtright), d)

144
???d?a ?e????? µ??f??

???e ??µß?? k ??e? nkap???????, ?p?? nk??

??d???p???s? µe d?ad??? d??d?a

145
???s??s? d?ad???? d??d??? (i)

Se??? µe t?? ?p??a d?as?????ta? ?? ??µß??
?at? ß???? (depth first)
?at? p??t?? (breadth first)

?at? ß????
42, 10, 7, 14, 12, 30, 50, 55, 52
?at? p??t??
42, 10, 50, 7, 14, 55, 12, 30, 52

146
???s??s? d?ad???? d??d??? (ii)

??t?p?s? ?at? ß????
p??? ap??, µe ???s? a?ad??µ??
???p???s?
void treePrintDF (tree t)
if (t ! NULL)
printf("d ", t-gtdata)
treePrintDF(t-gtleft)
treePrintDF(t-gtright)

147
???s??s? d?ad???? d??d??? (iii)

??t?p?s? ?at? p??t??
µe t? ß???e?a ????? ??a t?? ap????e?s? de??t??
p??? t??? ??µß??? p?? de? ????µe ep?s?ef?e?
???p???s?
void treePrintBF (tree t)
queue q queueEmpty
if (t ! NULL)
queueInsert(q, t)

148
???s??s? d?ad???? d??d??? (iv)

??t?p?s? ?at? p??t??, ???p???s? (s????e?a)
while (!queueIsEmpty(q))
TreeNode n queueRemove(q)
printf("d ", n-gtdata)
if (n-gtleft ! NULL)
queueInsert(q, n-gtleft)
if (n-gtright ! NULL)
queueInsert(q, n-gtright)

149
????µ?t???? e?f??se?? (i)

?a??de??µa
3 (24 / 2) (9 - 3)
?a??stas? se µ??f?d?ad???? d??d???
?? a???µ??sta f???a
?? te?est?? st????p????p??? ??µß???

???s??s? ?at? ß????

150
????µ?t???? e?f??se?? (ii)

???eµat??? pa??stas?
infix notation
? te?est?? a??µesa state???µe?a
d?f????µe????e?????ta?pa?e???se??
? s?????? µ??f???a t?? ?????p?
?p?t??esµa
(3 (24 / 2)) (9 - 3)

151
????µ?t???? e?f??se?? (iii)

????eµat??? pa??stas?
prefix notation
? te?est?? p??? tate???µe?a
??? d?f????µe??,de ??e?????ta?pa?e???se??
ap?? µ??a????a?????s?
?p?t??esµa
3 / 24 2 - 9 3

152
????µ?t???? e?f??se?? (iv)

?p??eµat??? pa??stas?
postfix notation
? te?est?? µet? tate???µe?a
??? d?f????µe??,de ??e?????ta?pa?e???se??
ap?? µ??a????ap?t?µ?s?
?p?t??esµa
3 24 2 / 9 3 -

153
??ad??? d??d?a a?a??t?s?? (i)

Binary search trees
??ad??? d??d?a µe t??pa?a??t? ?d??t?te???a ???e
??µß?
???? ?? ??µß?? t??a??ste??? pa?d??? ?????t?µ??
µ????te?e? ? ?se?t?? t?µ?? t?? ??µß??
???? ?? ??µß?? t??de???? pa?d??? ?????t?µ??
µe?a??te?e? ? ?se?t?? t?µ?? t?? ??µß??

154
??ad??? d??d?a a?a??t?s?? (ii)

?a d?ad??? d??d?a a?a??t?s??d?e????????? t??
a?a??t?s? st???e???
??ad??µ??? a?a??t?s?
a? ? t?µ? p?? ??te?ta? e??a? st? ???a, ß?????e
a? e??a? µ????te?? ap? t?? t?µ? t?? ???a?,a??e?
?a a?a??t??e? st? a??ste?? pa?d?
a? e??a? µe?a??te?? ap? t?? t?µ? t?? ???a?,a??e?
?a a?a??t??e? st? de?? pa?d?
??st?? a?a??t?s?? O(log n)
?p? t?? p???p??es? t? d??d?? ?a e??a? ?s?????sµ???

155
??ad??? d??d?a a?a??t?s?? (iii)