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complexity

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Algorithm time complexity space complexity worst case best case stability

Selection _Sort _o(N2) ₀ _o(N2) o(N') Y

Insertion Sort o(N2) 0 o(N2) o(N)

Bubble Sort o(N') 0 o(N2) o(N') Y

Shell Sort ?L'IE O(N(logN)'z) or O(Nl25) 0 N

:Quick

_Sort _O(N'loeN)

_M-

o(N'z) O(N'loeN) N

Heap Sort 94(o(N'loeN)4?i ⁰ N

|ffrp_Merge _, ^Sort _;. zhz ^O(N'loeN)

--x c

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Distribution Counting o(N) N+M Y

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4z.W tuvnqblAg-+e+

(

g

+u,rl<te.-

t\ar+1

_WlA<t\r\41-

"&z- ^W

^tY?l^3y2+

" 9o&

,

^*Wfix(rlA\\F!+tltl,ll+bhq

vrlj{

't+d"*4t4 4&, &tWrt

tA!,llvri'lfuG)tr

qwilrnu,wbea%,q1]

b

ir,Vryauwt&A|'

tff;Ai],wvtrtw4"t*X;',,ff il,

t;

tPlxl

1u-tlw

/

^"Jc4^t:q{,!'bw^vz^c[4

_"wl ffi ^tf ^'

B-'r'eo-!','p ^->^e^172-w

&

=*L rllllzlzl-tt +,u

l,t* ffi

^N^p.,

+#&r+n'rL ^+'"MJ 4N#- /VPY'"a

_{W P*k}

WA ^lW ^ral

.:\wrtatz Fe,!4a,1rg-^#E:fu4Q

^{t^''))}

ffi

^v,

+at-gq pWf^*

muwlD ⁿ^t

aefJ usvlotavvwfi ltwf{Al"ffi!,'vur-eafi lzl'&4tua,w/t

l+lul,zt

^QY

3 au*o cru@iil t qu@ rye$y@ ^/ M wA@

ffir^r\ e*ugr+fi

v@r4?4ozLltlJzljf+

VTl,aWLt

+e,ia1\t4a

_,95t<,4at

v11Jrt,Y/3i:

+ _^wol EL*rAwL

t

(7)

*

P,adix Tree,

€b€-

^L+E+Lll=

e^4++ / ^ulja4 +€ol +^lE]= E€.4 ,J4+6 /

FlEullg

147>!bV 4,4bLi-4+ 4,4bLi-4+

'lz4+ttz:2321

¹

1l+7

lg,L<.,1 ₀(714|

fual

^{t ?"}^I^F

L " | | r./i+<aD

(+;t*

<4v4

,

---

__y plat/l f?+.-ZAVg

--t

;

ll.

^B-Tree

Radix Trie -E-f *ol=longN+1

/

complexity:

d-E,44,

^{Al^ll}

g+

₌_O(losN)

12. multi-way search tree (m-way search

tree) +

^'i)'rr,\!n

ol4dls

^EFl-76

-1+rlq

.=

fu<ye4.W .ELW

't _{i:.j.)

-

^BalancedTree:

olde4E +^ll€(4++e *^l EqE ^ulq€4)

⁼degenerated caseE 6H€-.J

+-"j4

Eal

/

^AlE,_^l^ll

_^l

E_a_a1g _{^.^._+_}

+-:3 +^l / ^6.lt

^O(loeN)4^4

C*

i' _.e\"-

\'.1 / ++

⁼^AVL^Tree,^B-Tree,^Red-Black^Tree

f,

, ^_

-.B-rree:

ttq

!5_91 qar

at+rl ltlalEl= Eel+^ / Eeg

qlojEl= H=-?

/

_"J

^LEql

^Mz1l9

_^ie7l

^{un^is]E}

vxi

3-1rLe (-*

r=Eq Eq+ + 9l=

^alrll

xleuli

₌rvrzl;

/

^-d

^r--ql

^Mzllg

_^lE,

^M+17llE

_^l^lg ^rl€ ₊ 9l€. ut

le^+.v*l<_Lt$zu|4)ril(kl+@

auz+zl -

Mxl B-Tree 5.ol ⁼^logy,2N^ola1-

1

^zJ

^:-Ey d4

^M/Z7I1E

_^lRE 7ld (r\ffiot7z<tt)

-

^{,J^4}^,r.l^zJ

uc B

^4lrll^MHE

+^lrJ4

(trr.)

- complexity:t"J,

^{,J^1,}^alxll=

O(logN)

T(N)'<

c *

M

* logr,,r/zN

,,, ).y'iVi'i.(::

- B-Treei- g Gj_+6

=> qlq9__t--E_

rTlE- q!?

_{E_€-l}_?-gGs=+)E

_fZlE

^=.J

/ +^l ^ol€g fl*

^=>^B-TreeL^Batanced^Tree-E_^Worst^caseTl^gi4_,-_--

/ ?€€ 1^l"a+tql

^rI-.J4."J€=jol

€;.ol

+g.

x _B+

Tree:q= alliEsol E49

61619l

ellBql

^-JElq

fljl E4

^Lll+

B+ql=

^715"J

^.1

^,

ULlv'i ", / ,rlol)t,r-trrt 7lo ^HarrHr- ^ E1r-'^rr,-r ;;.-r,^;;r^r ='-;' ,r; '(

\rr.:_

z

,,r,Jrla'J

,Je.

E+7lr* r.Eel^lql^l

^lgqolEjE E--€j ,;4ql

_ftE-"J

.l _a:,1ib ^t\:ilr-1,4i:'{(|il,' _(1,n1,;,t,

ealalrs _+E)E

"J'Jslg

/

⁷⁺

^Lt-trlQ

^r.lH^Eal^mzll

I tr] 7 E]lll,l.lTlol -4-c).1 E

t.;--1i'.'lV.y-,t

j+.,+€;-4

+^1617l gl6ll +Al-J

ttol ^gs-?

€EI

,t€ + 9lg

- m-fl

^EE.l:

_+ol7t *n +ol

H€.

/:* tzll B4q€ c

BUT binary tree:

+ol +n + €g / _"f +41-q4q€

- _14:

,J€- 7ls_+ ajol ullxlalq

q

:-,€81 g]TlEe)

..'--'''-.

%F+"=|]oF -J

LE += €ol= ae. a+ qaf

-.T-?

_! EFI ++(oHrl

+e*TE4

^Batanced^Search^Tre

^, _, _>, . ,,. . . - ^{- _}-_ j

+ _ail,9Nl

13. AVL Tree

-

Aderson-vetskii and LandisTt

^ll?]

/r) ^LEE g+

^ltl_Eqe+

sE+ _^JHqilel ^gopt

^JE +_-dg

ol+=

E?]

-

_-E-F

]-EE

^,\-lH^Eal

Eol

<l-olzl

I

^oldfel^Eal

/ E"l

_^lol7l2 olAlol EIE

r-EE

_{^.^^_+-}xllullxl"lq

+-;?l

AlEll=

"JE=.E4

-

+:":?Ji(!qlanc" fu"to.) ₌

g+

^ltj-E4 +ol - g=+,<lu--e4

_:r-ol

i

¹^?l^qal

-

^balancefactorTl 2o16}_ "J-

3+

^{471^l:}^LL,^RR,^{LR, RL}

"JHg ^E.li

_qlC^lZ

+€€- "+fl+

-

icomplexiLv : O(loeN) _J It'

14. Red-Black Tree

,<ttol Red or Blackgj ^er_.1 Black

/

_lIagk

r-Eq

^l^lg

^Red^or^Black

,

_5"J

-

root nodegel

ttt _U

^{,eE7/l^lE}

_44zl ,lt +ili 4E l^l4

^r1_qel

+

^Hll^Hr+

^-?tl

Air+

=>

+€"1

Al'.1

E4 /

-ES ^worstcaseoll.r.l

go4rle+r!yrlr5{9_F_9Lqll.r}+,a€Else

_-H-5:EBST-H-I+

Eg4 -

^rotation$AVL tree$

zl$

operation

/

AIE

e+

^{371^l}^ol-a-

1. f;r{1LE Red ^:$-E-tE,

SeL-

^->^Black,^.Jolulxl

LE

-> Red, --jofulxlE

SEL-e

_^J46tr+Ezl^l

tE q€

^,$+

2. frg-5c ^Black,

t"JL-7l- +sE s=+^l4

^:

+E

^7l=

^teld,

+sH.J L-zl

d."J-*

l=e

^J461_l 3H

!t€

a-"J

3. ar€-rc Biack, aloJ-u=7l-

+gE E+

_^|

l

^:

^ge-Lr

^->^Black,^-Jolulxl^tLE^->^Red,^-golulxi

t-E +qc

-

complexity : ^g11et5;

(8)

H,

^r_

³ bl6v$"l E?l=_g:% +

^q,F(

fi|trry

r'bffiwzutil

%WEt .t"d4l+rt

_wn++w

-

- gg

w1 ^q4iy zrlnau*Ftt'+7+

rtt441

44 ^g4

"

Vff-

^?r1l

hli(d1-&Bll'<fulrl ;[eg.

^ftl

++

e"ley

JzlezetE

^ynq

*\L

."-

+LWw

.

€4+'+th47,!+441W"$uuE - lnwb *[Nw

^.

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lli.

Graph

€C(vertex=node)4 td(edge)_+.

ol+qAl

^le+-r. /

^{,4^llrJE}

$rJ€ q.JB6l=

_{^lg+d_}

,de(path)

:

^-.J

LEoll^] 4E ^{L-zzl^le]}

^,J

/ t$(cycle) :^lg €€4 E €€ol ,Jg eE

xff(degree) :

at49 €dql Bolgl= ,Jdg _i=>

d"J^lf(in-degree)/d€^l+(out-degree) (uJ-Jra11ng;

uJ.J(directed)re11n'

=gr-Ee+

^-E+tLE7l

€6llC/

*EJ-T(undirected)rfllE:

€g]-Ee+ E+t -E --* ^-_#hP _+ESg

nzll

+)i€

^TlTl+"J-?

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⁼^n(n-

^I)/z

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eraph)

_-a*L-)ro

ol^'l*lrl/o.ti^^^^+\ - E ttxtl]/xtxt\^l Fdo=

^l .ttT-t otL ,\lEr t0 (' ,t --A

oJd6}+(adjacent) ₌

+ +1]€(€E)ol Hse ^{olq^l}

^g1=Algl

,a'q$'flKY''oeYi(O

^LTJC

-

Spanning Tree

: C[}E4 = rzll-r$ _+ts r?llE +

^_q=

€€4 Eeg +ts

r4lE(cycleol

S= ^-ibE

eGf

^{f-q-1_)j: ^{AlSE4ql^l} 7}e^l(td ul+E

-.J) -"Jol

al^"J E4 _+**Wgvtr*,

alftj?^; e E

ltrlETi

=\ /

^olTl

^9ar

Diikcirno-r;rFr^ 6^rr^r^t^r or'"oY#t€FEHil

cD Kruful

,' " rlf _'

O Prtm

slful

tr!4L ^,A,ro W.\);V,VW*S, 4{

^4M

),.11

dpe{al+l.( t4+l!''l

rL

Nr Vwsr^lw4-

^'Q'f'}

'yL-al?

: bltlt"l

^Ul++.,*Ys\^bsel-L* W+Vr=-W ^P'FoweL)

!r! ww9+a

,iV;ffW{Y,qaw"'+latw)

tff,rn_ _[6,rul

0

^sollin

, _sl'U.q^)

fl ory1rg:4epd7.W:fOv)

(10)

1l; o n/\

i. LJ- ₀

-

^Thread:

^{Ee,rll^}

qlql,<lpl Are

+g]

^z

^l^Hg ^qal ^tgg ^gB+ol de6l= ^re.reJE

E.gl

rH

_€-B

^lt ^E+E / _^t?

⁺⁺⁷¹

^gol:*

+41

rg,rll,rlg

_l-Er= 6lL+__

/aIdH=> lock =

al*ql q4

-.J /.zllE7t

^lg+?l ^lgg _=€Algol ^EH

^{trllrl^l}

+=

^allEql^l € + szl

_"JF

synchronized =

I'J

^DIIAEE^="J

Hql

^-.J

^BllEtrJ

€+7ib 6lE+ 6l= a

Context-swtich(Eq4-,r-$\

:

^ne,rll

:-rl _+gql

€-"J-f ^{ng^'ll}

^e

^dq1El=

aqlEel EE^ll^

Hlra _I

7td(EE^1],.

^J€^ltrtr+ ^{pCB^t€)}

4€. olC Efl^,ll^q

_r{6J

_€H4^ll

-+Ed:sFW+=e

- ^lol€ :

^{nE^llr.}

^t^l6lr flg

-"E,#-;*

aqlE

=

os\Z

Lll qls4EGi+=.n

^lg=G^lla

ul6ll _H.€.dll

F9oI

^:g

€H7l Alql4se 4q ^ltol

^6l€.)

ArNt tl t f? ! 1) tl P; tql'!4 _|₌L,l ( ( k

EerEIs Eordl =4E +t tgl$ trllol^l4,al$

ol trllol^l=

g+qe

^I€-6t=

7lH

trdro\

^ =ir J

:----=:.:-_ rd

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-

=> 5"J-.J

=tlE

^L+Hd

r-ar+€. _/ _q+ _EtrE _g^J _| ^'d

2. ^llrEEllolC _^llr\rLll

ult /lE 7lB i I _WV

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f 7l+ 4t-

^Eg=q,

_^llrHE-+

'\lliLj-:-

^atq

"l"l ^ol'l z\llr-L.l:E

_^llrHEE E+qg

E-':rL _^f€-6t=,.l6efE

7lH

/lE

,

_.2'

i t

=\ Tlol 'r7l xlOE*L7l ol;ll rlQ / .)1 s f,l-,d*L HlrU I \y

=> Tl9rofl_AojdtTl _9.16ll

^l+ /

^E\+.€Sq+"J^J

:--- _ \r--

19 +iil 6I : g-:Isl ^oCos=7] ^>/6lrlEl n a rej\ =

]V il

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->

^lgEl^l ?Jr Hg,il *iJ _L:,1 rz("\-rl _Cfrt

'l | - -- '€==+a-+*

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el+

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³⁷¹^<

-f$g

^jr_+_re₌₇₁->

^leElxl Saln Hg

^uJ

*rJ ^r.- : t -/

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^l

^a7l:

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nfolxl FiglE

+ts

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trl^lb

g:.J

----_r:, r ,_, \.

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lIU+EJ(EllolE aTll

),ltrllllemporat), ₎

/ ^E€-J a€J(^l+€*, ^lt1)/idgl_glrJJ_+

=-^l

nllolxl(spatial) 414

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qlol^l _sjaJe+1

/ trn€4Etrq"Jg aTlI) / ^E€-J ^{Egt(q^3} d=+1, ^lt1;7 ^{a^x?ffiz}

EEs-d

^{Lll€.zl^l}

4xll€ + ^gle

- Localitv(l+c): re^ll.\71 _d-€El=

_€'"J

+7lq6l^l= El6g

^trll

_"J+

Dllol^ltrJ

d+4qs a}6ol= €€

-

Working Set

: rilelllr.Zl l _^l+ d56lb

^{qlol^lEE} ^{t of}^llE

/ #zlq6}^lql

Al+^14^l

ggl

7 X

/ ^{qol^l}

^ql

ttl+ +7lqg^l

=

^{+llol^l}^qalloJ

⁺

replacement: 414

i^ll) ^: ?}se

^716}

g*€i?I

^l€-61^l

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_qlol^l_ra^ll

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: 4^llg

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7lq6l E^l $qg +^lqle

^qlolxl

lxll

7f xl

=. ^ci

/

trl4ql+=7li'€el"+

/

^{trllol^l}

Eall"J+t

^la^l"J

trllol^l +xll

t

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g€

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B^J LRU(Least Recently _Used)

:

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?jg otl"ll=,txll

/-.-stsl-"llE g^J

/

^+:"J

^qelg

LFU(Least Frequently Used) :

.<|$rullzl zltt4e

^nllolxlg

i^ll /

_^t€.El^l

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^{nllol^l7l}^EelloJ

4l+

^l^l

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Thrashing(-xl-llC)

: Iie^ilr.g

=> "J^lH: ^working^set+^1, ^nllolxl^.]4

^l,J

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+^ll

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dill ^I

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r-l r-l .^. x o Tl E - o-rE -rr"l 6

(11)

ilffold(Demand Paging)

04 se]+E

trllsalql 4^H6l= "JH

/

^{7l^J}

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^l^EJql^J "Jol

rlg

-:,"J'J4oe

SwappingzlH(6ll+E r_E-r+

d^ilE +7lq6l^lql

^-*B6ll

Es^l *E ^oErH4

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:

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uils;l

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a8 _^leEl^l ^gg

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-t ^{i^1l^ltJ,} zlg*tl, ++selql"J €Et

j^t=* _';l

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xl?g E€4"-e

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Hl +l c ^(Non-preemptive) a4l

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^l aEJ

oltrI€Flfl

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uul]",tol

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+ s=

541€="r

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+llrl-g.rcl_4 /,

*-4C +^l t

T f\-fJ

l-l- ^sJF(s,

L O

^14

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but -o.€

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SJF(Shortest Job First) ;

zl rQAll.lTl

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rg^1r.

e-t tt.tuu-lo--1le.o

ifegfo.iority) :+d+97l rlxJ+e ^{Ee^il^ql}

cPUU^l

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=\$einEZlH:

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_^lg:*B

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Al d (Preemptive) A4l €ed ⁼> ^rIolal,r.l _E-.J,'.]z\eJl

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q _+c+gl7l

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_cpu=^zJ-xll-+_uql",t"I

^lg7tb

=> 6}C:

+d+gl 59 ^nfl,rila

^uul=

^.]4

7l:. / \lE:

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qulail-

_aail

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fl=J)el=

tgB €ol4l ^ns.rll5 Es g+ _+q6lq

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/r: ggigl,

qelqil_E_1,

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dd6lts

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I

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^l

^L++L+t-

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=>

6ll€H :

ql"J

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^LllE

- io.

_oTF!

t

_|_I

Lll€.olq

,JI+=

_€-AItrJg0l+6ll

tlq ^r+t

,llg

^ul+

_I

^lE

^+Ale+

c4 _uu-!l'

,\latr t

^.r.l_u-irI'el

c t

rI-.': ^rJo}^t 6*"

I

zuul ll d

rl^l

-H,

(12)

tliEg=

1. OSI(Open System Interconnection)

74lG

^D

:5Col

"JqL+=

nl€g TtzlE

_+B:"J

4 ^(zllgg al6s'J, re.EgE E+4 ^qql

^xll+)

-

1 Physical layer(E4 4l+)

=

+

^z+EJ

e €.i5-6i€gGe)tr=

\/.V/\/+-.:-s-=.+ ^4l+

/

^tilolEr€-E4trir^Jl

tlss t+ / ^g'l= tleith+

^.-_{l.-_-..i-}"

-

2 Data iink layer(rllolEJ€r 416)

=

+q4 4l€g

F-"J ^cllolEJ

d+ql

Altrl€

^ll+ / ^grl4 +a

_^I€(MAC^address)

/L{ia*A

"Jdo}41

d+

-

3 Network

layer(t-llegf

Zl+)

=

4E dq _"t ^r++€,

=44 ^{+^(IP)} ^€el6l= ^4l+ /

^4lz,l

.lEdE €€ /ip;"kaisE

+-.Jdtq

d+

-

4 Transport

layer({+

_4l+)

=

t/at9

4l+?J ?lEJnllola

7lb q

ull^l^l

Fg / ^{EE^llq} q q+^ilq /

^{TCP, UDP}

rtrE=

//s_esi[_""tJ

-

5 Session layer(r.llf;

zll5';

^{"

=

FCalb F Exe t ^6l:-ErJ q _E€+^l g.J / _^lldg gr gn t4 /

dialogueg4(= ^rlle+_{= FAI)}

-

6 Presentation layer(+E ₄₁₆₎

=

d*

^rllolel

EE e€ /

^LllolEl

^EE

1}s_ H€t, "J+(compression) "i.Jsol(encryption)fa3 /ex. ^jpes,avi

-

⁷Application layer(g8 4l+)

=,\|$xlz|

ol+dlb

r-JlEf_=

*Eoer+ /Sgoerllr- ^Jnt ^€H,as

t7ltelnet(23), ^http(80),ftp(2i) protocol

2. TCPIe

erg

? ^t

:oSr74l6=€'Hl Tllg^l7l

q"J4l E+g Alg^l 44 / rcp/ip=d€4es ^t+q= ^nsE=ql

^ql-il

^a(AlBlEt) - r,2

= Network Access layer(t-11e9;f

d+ 4l+), / - 3

= Internet layer(olEi'l 4l€)

- 4

=Jrun"port

laver(d*416)

_Iclrnp.

pe./ep*\ot / -

s, 6,

i

= Application layer(g+ _4l+)

3.

Tldf+a E4+a. ^x1"1

a,, I

-

MAC address:

4 $ul$ol €q"J Fdg

^?l6ilLjlE?JfNtql^l

^le ^+Bdl=

^q-.Jg

^{dll+= a}

/

^,+r+ELllE fl_=-

gElg

?lEJtrlloja=

n+E

MAC +aE^ ^7td

i-rcP/lP

=is--=- ,rl-8alb r4E91:qj,r1 _rp

+n _^i+dlq F4l. nrt, 4+4se

^MAC

+a _^l+6tq ^qojEl dB -

ARP(Address Resolution Protocol) : _IP->

MAC e>

RARp(reverse) : 1,q4g

-j 1o

(.JEJ'Jl4l+3)

-

_SEl4oJ

Tttl+A= 844.J €zlq+ae

E+61-b 4. ⁼

+^

^r,ld|⁼rllg(Mapping)

4. _{TCP E6J}

/

^IP

Eg

-

TCP(Transmission Control Protocol)

:l;t,',i

^:

: _E7j

^l-:J€ ^jae_tr+

/

^Nelzj,

/ ^g: sgg

_{^llrfl_E}

-

^UDP(UserDatagram Protocol)

:

ulE,3€ EeEE /

Hlllel4(rllolEl

E+q+

5-E)

/ -

IP(lnternet _Protocol)

: Hlge6j

EEE€-

/

^LllolEl

C+ /

"J4l^l(+C^l)+El

Fg /

^Et:

S'JS

9.1-"J

a7l €€ ^rl'J le

Htsl+

/ ^ilel€Hr+= -z+d s+El= *+ql _^lg

+4^l(+41^l)zl^lel ++B eE=

^,J€dl=tJl_^l€:

"

.::t _I

t,,"

r.t.',;\';'l ^:;.'2t, :.

l/

(13)

LftVltt*r4g

(14)

eiilig

xoll^lYgg-+"Jf :y -y

-

€elq(Domain) :

6Ji9

,Jql

€9€ ^d--J / ^x4 ^{d^llg}

^d--J

-

*9(Codomain)

:

^-.J+E,

tJE"l 4":lY d-? /

^vrJ

^{t^llal}

^d-.J

-

xl$(Rance)

:€Eqql ql$"l= _+qel ga$ _{EolE d-.J} /

^CF+el

^s= +"J ,JE d-.J /

_^14

^c +q

1.

q-.J+

_C?

-

-.J+ f:Xui Yoll

qdl

"Jrllgqia"J

"l ga

xoll

rllSatl y= _dejq,

*ffl=v

"Jtll"JqlS

/

€zl6Ji,:;-.J.

2. E€-J(Power

Set)

^Ot)

5. ExlS4l(equivalence

relation)

_ri

- qtr €Bl

^Aql^rll-.J

olcg4lzl ,t^14, rlld4,

"lds4l(d-* ^AE

^{+^16}}^d-"J)

'*^f4(s=vzl xltl

^l4lt+

$41€) /

{ol!(xzi

^yel

Sfllr ^y4 ^zt

^S4lElE

^xE

^zq+

trll, -"J+

Yq 4 ?A

voll Qlal f(x) = ^yoJ

gA _x:. t

dll+ €^ll:"Jqq.

XE

9oe ^r= ^-l+E ^fE

^EEI+et

^6|I ^x

⁼^f-l(y)B^L+EIH

+ ^gl+qq.

t

->-Jff(x)t

₎or zJl-"Jf(xrgi t ->"Jif(x)J )

5q7i=

e+e+

rg^l _{ge e+}

^{271^lE}

^Llbrl.

zj

,r,1zjo1

:-g4it_le ^,J gA _+"J+

_+-.Ji+.

q-.-.-.---..---.- --:;::l ,-,.J . ,:: :,a: : ': _-:.. -+:E;4*::-1|,?

:il \

- ^q'J

d-.J Aql

ql6lq AE s= _++d-JEe _"l+qd _d-Jg Aq Ed:*ol4l ^-rltl+.

AE gAE 7\+)l

nTll+E

E€-*E ^7ll+= 2'7lEqq.

^p(A)= 2o

E

^L++H

+ ^gl+q+

*€€=>Ael

qE +Bd-"Jg _"JE

^trll

4 ^{gAql qj6l} r _flaz|

ol$4l ga,tr_ €7lr

cf.

$aif:

^'y=3

3.

+-+4 flBH _f-\

- _^19+

ngl _{Ql-* g^1}

'*+

^p(n)91^{QlaI p(1)}

tr"ln

^oJEE

_^l€+

^kql^rll6ll^p(k)71

tol4 ^zt€*44;,'

ol

rl

k+191 trll6lq _n(t+t;o1Al "J ^trl]

s= _^lg+ ng *i

^p(n)Qltll6ll Cl"Jg

6B6lt. "JHEqq.

x{zl=1 x}E+

cf. fl.$H(proof by contradiction)

.: ((U'Ji!,'itt)

4.

g4otd ^'|r,J' ' ,i-*',

= _E-= 73+71

€Hdl^l s€-

^trI

^l+

^l€-61=

+€H

(pzl

tlllolE

qz|

zJ]lolrl. 7l€ol €*qHo

^o_+_^.'.p71alolgotEt)

111,

ts

;#iE.=;'['f;'ii;-i.

*-tr61g=> TlEellrloJ _+-(AxB)g

++d:.J

€rtr1]

4€s. trE / ⁺ / _"J-J€ ^rrllE Fes

^+i-J

^7l;-

(directed graph=> ^oJd ^;JaJl4 4E-) olal

s"J €EI=g 'd-*

^aq1,<1

d:*

^e-aq

S4l'+n

"J-Jq+.

dol4ol$

^€:sr-^Aql

^q"J 5^l

Sz]laln :clll+.

rlld4(x4 v4 ⁹⁴¹⁹⁸ vi

^xP]^941€)

szlE /

^Hlqd4(xs+^y7l^{?J3 trll$}

^rld s4lrt dB)

6

d-JE

8*(Equality of

Sets) i''

^lE ee + ^d-.Jg "J-clq+. +

^d--J

^A, ^Bzl 9lg

^trli

^Ael gaTl s+

^Bql

+61r., BE gaE _E+

Aql

+"lE

d-'J ^e+

d-?

= ^le eqn all ^ol5

d-clE 11f,."14

g-slqtl.

(15)

)' aplvur,l LNLaI

: _ft)t _[<pl zA4l^l+,.]w+q

ww 4 ^q6tg, _{%w{_.}pux} .trqr+& _6z)

)ll J , r--

I;f;bv/ffi7"WilYFh til^7dv/

i€wiqr)6|A"{,ry+*;+t

MS). %fu-,ru+ ^@l9vtl,lE ^4*le,l hAl^l-

^ryrg _4Wt!

u*

At4 ^aV* 4 ^\.&EW.

tV _U1-I0tO

cgtl-El'1ay,p'r,,J+u?Z v6tul,W 7\n12

Alrrgq

W1-4 ^rf9F,fr ?+m,ry1,W4 A ^bTb+ ^LV

^s^tw

^Kbv.

T eQp\Q(y44fuf,

g,11tlr1rt!t!/'L

f"/',fv

($4rLoWq-)

..-_E_

r1tffiz,T4tbLWz

_:::^ctlnlE

_+!-,H + ^qo)iltu W

i p:.4,i.11,,

r .:1-1-3 , *.)L \":-' Yz1-

"pvq\qt4ffi)t Wf-;:,^rrr #,,i#,

,.

"t1wt-lftiti.,+'4,'F'42,4-,;4rb

'ablelffi-c2arn

t'1pr

^=-

^v@vllolb"l @+:f'A'Eryill ^cII+ ^$Rl+ rl+,+T,

(w1v1.<twlxt)

ol,tt,1,<14

('

{1a41-wz2-) ^6t

l ^rtrbZ

o il&rftEnaysulrr'n),!Ir*l,#,f

;yffy,t^I#,4+r+;;,W

^---^'

:a 194^lttuYu4e+)t :W+a.##") ^rl*{

^|

_etfxt M+W

-J...

e4 _{4V_t1}

H\TT',Y,1,:,Y*s='i+'tr

⁺

;"!'L*n

^v^t'rr+r-

^>wt+ _v,txl

I A'7-'"d--v

-*prtt--._

1W{g+W

-eExl+ve+w .>;azuk1. ) t!/LltL4tttefttr\\tllr U,[Nt^il|-*b,yT^rutq\ ?]VI+ tar\i\g4vl .inM\t 4/;,t,./,")

9 ?!+(Abstr'zurk'h) .. _/ _eEyZ

(ct+

g 4Plaxerltqnce)r

,/ri

'" _,t&l)

^olvl_Kjrlqrl

Ww- _thq /

*yz;ttL__>Lqs,pt4b,s>1

DiryU?rlyr'rrrflisril t,+tZW" *dat 42\ ^s"lru

^v4)o+,

^rtl+F

^

^zt,-),^,-,,,

^{^t}

^{^,}

/ ', / ' +lst

*4rJlz a +vq , ^t'144L

fr

flntvlgLPl(w

tir,u*j,r

I rnlrJ-\t) r" 2;'l

,,,r\wtrJt,y,b\244 ,c4'4'tlt4"

o'L

tft +

daqble'

+47i

^eW't+fr

_W^Wte

^v*Etl{efSatc4lb6l,l

E+hl.r) _^tiJLW,W

l'"iiiir^ii,""ua ^c*f

^{al^1}^cl^wP;J^-+hi*J^z<W

ex.

_,^u&pay_,

_LV?g

₊

"..

af.urtw,lql 7ttr4.#au. ,r)4,treqtqrk-ztd,il

|/o t"?.q4L"fl &-l(+._vWw?u@ill?g4++A+

( /"h'l'+/+'Ft'

'XL!%4Qwr,"l,rg)

^,^,t+W4t'11_il.v-

+"jwb-'a'J;'tF- _+tb4,. tl* tll,Hel+pysJ+

4t4ry1u1E *r-s-;.+tLq 4

^vdutkr4w

€H= Elr gl= tllolEl / ?€a: ]--tel tsrJg

- rLE :