Diviele and Conque
Pie
Se
S S
Search
Bina
Maximum and
Algoithm 2 finding Miurn
DAC (P) Megesort
i (Smaii() (4) Qwck sot
Momis Muhipeati
s(P) S) Srassen's
3
else
divid p into
DAC(P), DAC (P).
-)
Combin (DAc(e.), DAC(P),
Search
8inag
4 47 s3 SS 6
2s (29)3) 36
|4
3
S
2
mid
Key 42
mid =
h
int Bin Seaeh (A, n, ke)
2-1,h n
whid (loJ <= high)
mid (24h) 5
=A[mid3)
iF (key vetun mid
min timu - o )
h mid -1 ma hm o(lgn)
3
3
Maimum
2 heignt of tu cormpars iD
Logl6 =(
Recursive Algoithm
TO) Algenthm RBin seavch ( e.b, key)
if (e:h)
if
(i)
(ACJ= = kuy) Singu <lement
ese Smoll P,obemn
3
mid= (2h)
iF (ky Amid))
tetusn mid
if (ky < AÇniaj) tavga
TO) veturn R6in Search (l, mid-1, ky) Arobum
else we ane dind
A
Bin Searchl midt!, h, key) into smatl
3 Subprobkny
3
T(n2) +1 o(eogn)
Merse Sort
Me«ge Merg mose than +wo Usts
A C
2 -K
L’2
4 3 2
I2
4- wy Merging
Algorithm Merg (A,B, m, n) Not a 900 d M
thod
i=!, j= i, k1
So, use 2 wayMegiá
if (AC)<8C3))
c[K++] Alit+]
c[K++] = B[j++]
(Ans
3
element
A B D
wkia (ism)
L1:
3
wa(in)
c[k+*]= Bj+ L3
3
ThY atv Process
2 lway Merge Sost
Considex +his as 8 G'sts
array
A 3|7 S 4 8 lisb qre alyeady Sostd
2 4
jst Pass 3 9
4
of Pas5es = oq n
4
2 Pass S 2
3d Pa5s 2 3 4
Merge Sost
mid h
ir (ieh) Aoben s (arse Di'da into ubprbus.
mid +))
T(b)mergesot (a, mid) n
T(Q) rnerge so4 (mid 41, hgh) 21()+n:
merg( e, mid,h)
2,8
Merging is do ni
12 Postord e trav(is al.
3,4
mesg
9|3|3
Pros of aey Sost
Lary Siu st
(2) linked i'st RAM
ExKYna So« hng
4) S+abu
S
duplcate
4
Lsost Merged Arraregmert
not d'sturbe
3 4 S
Stable
uo)/ Me* Sos
Cons of Merg Sot On) fet a space ) Tnseon sot
in placa Sost) 2) Bubbu so
Space (not
Smatl Problem
2No Slow
Fox small si U'sts, it is
J beHey*
wase hme in seussion
Recursive ’ Stoack (Coqn) stack <paeo it neds .
spae taken
o(n+ tog")
Quk Sost
Studert -7eache
ate Snallc than
posihon
than
is in ngnt should be (argeY
Ideq An ee ment eleme t on
ment and
that ele
that ele mn
|2 I3
6
|6
3 4
Snalles
han
follos DACG stratgy of Ust
Y s e r tend
4 S
3
A
Pivot = 10 of Pivot
find soyt poHon
Procedue than pivot
find an ele ment q7cate
until sou |(ssn than Pivot
elemet
jou ind a n
Swap Parthoning procedurt
(arsu than )o
Smalley th an|o
| '
Porhion (l,h)
Pvot A[4]
i=R, j=h
whia (i<i)
Qicksost (e.h)
do
perfosm quickint on
wki'a(aliÜ< pivot)
and
3
do
Quicklor(,j)
(AL')>pivot)
Quicklort (j t1, h) 3 whia
if (i<i)
Swap(Ali),ALi))
3
j’ Posinon
whre paihony Suap (ALW,A[))
is lo
3
1,1S
Pabhon at
9, Is
1,?
s. (9,1
o(n togn)
Pasihon at every mdd e eu ment
Best time’ 0 (ncoan)
wosst Case ’ 0 ( n ) List is atrea dy Sod
2,) ]
3,2]
S,
Impronng Ruek Sot
midda elemennt as Pivot
Selcct
Rand om
eleme t as pivot
Select
2
size ’ o(begn ) to o (n)
Stack
Strassen's Max Mulhplcation
b b
A=
be
for (izo, ikn, it+)
for(=o; j<n',j*1)
fos (ko; K<n,+)
C[i,3I += Ali, k] * B[*,i)
Small Problcm
If Si2e of matix is greatY than 2.
2x 2
}t not
A= Au A1 bi bi3
bay b
434
bg bet
A
ba3
b4 b43 b4
Algonthm MM (A, &, n)
if (ns 2)
C- + formulas
3
else Mabie adeiha
mid- n /
MM (An, B1 )Mm(Ain, 6a1, nla)
B22, nz)
B2t, nh)
MM (A,6.,, nl-) + MM(A21,
n/)
MM(A,B,s, nh ) + MM(An2, 6a2,
ns2
T(n) =
8T(nlz) +n?
6
(ase I )
Shassen'has done ?
what mulhp¯ cations
C) = A, Bn t An B2)
+ Ba)
Pe (A +Az) (61 C2| A2, B t Azz B2
Q (A| + A22) B
R= Au (62 -B22) ng2
S= A22 (Bz1 - B)
T (A) + An) B22
Bi)
U= (A2 - An) (6+
V= (Aj2 - Aaz) (B21 t Ba) o(ntog?)- o(n)
C) P+s-T+ V
Max- Min Using Dind and Conquer
4
|4 Ma|+
6
6
mid: 4
Max 9 |4
+
4
|lo 4
9 6
Mar 1|
Max Min lo Min 8
Ma
T(n) = mar
MihM,n
:. 0 3n2)= o()
Algorithm MaxMin Ai, . ma*, Min)
man mìn ALi)
else if (==j-i)
if (ACi) <ACS1)
max mas
max AS
max maX 1
else
max A[i
mine A(i) min =min
3 el6e mih min
mid = (+3))
pmar Min (A, , mid, max, min)
Man Min(A, midt, , max, min)
T(n) =
n2
l2r (nl3) + n>
hreedy Metho
used for Solng opimzation Probems
requse eitheY min oy may
P: A B Cons aunt 12 hs min cost
(CondiKon)
waue Bike
mad
Solution Satis ying Condihon The decision is
without wosying about
min feasib Solution
the effect of the
Curent cdesion jn
oph'mal Soluton not be opimat)
future
(nly 1)
Soling ophmization Probltms
St agies for
O Greedy Method
2 Dynamic Progran ming
and Bound
Braneh
Method
Grcedy
Poblen Shold be Sold in &tges
ns
Algorithmn Geedy (a,n)
9 4
L1 to n do
for
Slect (a)
if feasiba(2) then
Solution =
3
3
3
20
0 20 D
30
Knap cack
Problem (fractionay Knapsa)
Objcts (0): 2 3 4 S n7
1S 18
Profit (P): 4
3
wights(w): 2
(faction)
Mavimize tha
ophnation problem
Consraint
3 4
4'S 3
Plw: objectie
1 ma
2|3
M3
I 5 - 2 - 4-S-|=2- 2 -0
Profits = II6 t 213 S + Jxs +
with DeadUnes
Tob Sequening
n=S
Jobs
Profits 20 IS S
deadints 2
Machin
Highest job pr ofit shoul be do fivst
factional Knapsack cweights, values, w)
n= len (wights)
ims e enptg ist
iems
APPend ratio, w) V[iy, ') 4o
Sost lescending ordet of Yatio
items in descending
V_may e-o, x[]+ array of Os of siun, vemaining Capacity<w
fos each item in items!
let (atioo oight, value, inder) - tem
If (remainingcapaity weignt)
nlinde]<1
Vmay Vmax 4 valwe
Yemainingcapa city<renaining cepa'y- wight
else
Raction emaing-capau ty wg ht
[inde]e fraction
Vmay Vna t faction x valw
Breat the (oo pP
Retun
Vmas,x)
Q’ Mazimum mechngs
Activity Selection Proble n
A'go'
0) Sort the actirihes as per finishng time in ascending ordey
2 Seteet fiyst actint is ale s than
time
(3) Seleet ne w achrty if its star hng
time.
poeousy seleched actity's Fir'sh
aYe checked.
actiihes
(4) Repeat slep 3 till all
Acoity 4 2 3 4
Start 1
2 8
4
Fini'sh to fish hn
sort aceosding
ag 4 S
Achiital 4
Stat
4
KN'sh
icishjXst4rt1 Start ; > fnish
Actiity
3 S
Stat
S
Finish
Pseudocod
Ago
etor (s,f)
Greeedy Aetiritysll
ne length [s]
fos i 2 to n
do if si fi
+hen A EAU{is
Tetun A
Dptinal Mesge patten
|List A C
2
Sies ’
2 2
Total cos t t)3 t6= 40
(04)
TC I|+5+I6 =3 2
A D
Lists’ 30
30
20
(Go
30
20
