in general tcn a divide and of at each level ne no 10610
TRANSCRIPT
Solving Recurrences
mergesort : Tcn) = Z - T ( Iz) + och) ⇒ Tcn) - ocnlogn )M"
Pei'
- covering : Tcn) = 4. TCF) t OCD ⇒ ?
(nxn tile)
In general : Tcn) = a - TCF) + Nc runtime of"
divide.
"
and"
merge"
at each level :
④ ne
- no⑤ . . . . . . b- a.IE = F. ne
h - loan € . £10610 aint = i. ne
C
'
,
9 :
:'
ah - (⇒ = #Y - no
④ . . ..
. - -
Tcn) = a. TCF) t na sum -_ nc - ( ltqtq't . . - + qh)= a. (a.TCF ) + CET ) + n
'
g-
= a?TCFz)ta + n'
'
y q= #Claim .
.
Let 9>0 be a fixed constant .-
For any integer h> o.
Oh) if get
It qtq -t . .. + 9h = { htt if f- I
⑦Cgh) if 9>1 .
Proof : One can verify the 9=1 case.
When 9 , htt (Fix this to 0.1)the largest
-1+9+92-1? ! -19k = < Iq -
- OG).top
term1
When q > I beitqtqzt . . . tqh=9"g÷=qh(9÷h)cqh .# =ocqh) .
tthe largestterm
Master Theorem-
:
The solution to the recurrence Tonka - TCF) thc is
(where a ,b, c > o are constants
① (nc ) logba - c ⇒ acb'⇒ gel .
{ ⑦ Cnc - login) l°9ba=cnc.qhnc.fzjosbn-nc.ba?:!n
⑦ (n'Bba ) logb A > C =nc .E aiogbn.at#o=n'IT
- c na
logyA = C nclogn
Applications of Master Theorem.
> c n'09bar
Example l : Tcn) = 2T (Z) t ⑦ (n),a- 2. b=z,c=l
Tcn) = ⑦ ( nlogh) logba - I = C .
Example 2 : Tcn) = 4. TCF) t ⑦ ( l ) , a=4 b=2.
⇐ o
Tcn) = ( NZ ) logba -_ 2 - c
Example 3 : binary search .
Tch) = TCI) t ⑦ (1),
a=l b=2 c=o
logbA=0=cTcn) = ⑦ ( login)
Example 4 : Tcn) = 2 Tern ) t ⑦ ( 21092T )Answer : out of the scope of this course .
one common approach is to prove by induction.
Example 5 : Tcn) = 2T ( Nz ) -1 ②( NZ ) a=z 6=2 c=2
Tcn) = ⑦ (m2)logba =/ cc
Example 6 : matrix multiplication : C = AXB. Cij - I Aik - Big .
nxnnaive implementation : A ,
B , C E R
runtime = ① (as )
a- III. I:] BY::B.:] ⇐ ⇒Cu = An B" t AIZBZI Tch) = 8 TCI) t Oth )Gz = All Biz + Are 1322
9=8 ,b - 2
,
C= z
Cz, = Az, Bil t Azz Bylogba=3 > cCz, = Az, Biz + A-221322
f-(n) = ⑦ C? )Strassen Algorithm
(source : Wikipedia) TGt-7.IE ) -1%4J
time to split the matricesand perform additions
and subtrations.
9=7 6=2,C= 2
Tch) = n'%7
= m2-81 ⇐ n3,