( Inc-remental Dynamic
( Ocp)
THURSDAY, JAN 21st
Connectivity
problem
with Edges
vertices sequentially
only components
Start •
with arrive
.
graph
of
connected
as
new edges arrive
.
Keep
track
-Generic
-fer
CONNECTEDCail ==0 then
for each edge
if –
D.C.A
(awl in a sequence of edges doe :
Alg.
\
UNION Carli Tore- ‘–z-
the
-CONNECTED
need FIND :
–
later –
CUH) FindLu)==Find(v1
component
label
of vertex u
findCal
returns
n — m=e
Ala.
,
id of length
:Quick-F
IN ht
belongs to component idsit =k T
Irl
array –
–
if #then
k,
vertex
i
id lil
[ 0417
2345 ooooOO
Find ( it
Initially Cidfiti
e-= 0,1,
=
H constant time
id
0I
7%13/415
tidbit
…,
n-I
Civil
idc ;] ) =
CONNECTED
(Quick – find)
UNION (int : For each element (of id) equal to idf-7 ,
change μ
cost cost
id fit
value to Cnt
:
(
4
operations ) n-
(n- OC ”
ill
n
”
total
l
union
:
(
=O
n
n
-Alg
”
.
2 :
Quick – Union
→ array
slower
fast OCH
Find → onion
be will be
–
still use id each vertex will
will
”
have parent ,
if parent#vertex i is vertex j , then idCi) =j
for root (rootis itsown
:
–
n o w
except
parent)
–
id lil
Initially
=i ,
for
i- O, 1,2,.. . ,
n-I
(Quick –
Union )
5 3
ol2I4567 id 7%10171015171
Find lil
do
–
set
: repeatedly
i a – id fi )
until
i stops changing
of leaf- to- root path
co st
°”II ” id of root of vertex i
: length wors!t-cdsecostfinds.NL#
•
=
id lb )
. …. ..
..-
he comp. the
root of
of j
id —
and
id
…
)
l id a
C idlil ) b–
4%
1
and#b he comp id .
id Cid C
) GI)
Analysis – of
Quick –
Union #
Find :
Icn) 041
Total cost call edges) :
r Cmn)
Thelma)
Ocn ) -:
Union weighted
Quick – do
Onion
trick
:
Quick –
Total cost
:
onion lower weight
but always make tree with ( lower number nodes = vertices)
the
child
( if tie ,
of the
other
ties arbitrarily)
break
ghkd
vertices g, )
claim :
# ology
Quick – union
Find ( it cost
MONDAY, JAN z 5th Proposition :
Proof :
Max height among any
tree in
forest
109241 I-
depth t.
k
Ig Cal depth Cvlelglnl )
consider
arbitrary
node V .
will show that sufficient
.
.
FACT :
Let after
weighted
of =I)
jth
label
change .
quick- union ⇒
Sj ZZSj- I
Sjt,
I 22Sj- ,
2k ⇐ nextstep Z 225k
‘S
2k
2k
..
logan ←nSkI2,zzSo
=
Cyane
ofv is thesame as#oftimesthewind?’
Observe
he the # nodes in tree containing V
depth Sj
changed ( So
v
.- ,
.
.
z
–weighted –
proylem :
( Hmm
union –
whenever a
w orst
find with lath
do find
1
ht
–
direct case time
complexity
doing
( processing
find edges)
w e
, descendant
compression
we make each node in the path
of the root
of
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= mintizl: Ali, En)) A
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m union –
operations .
m
( Hmm
ma(min)) h(mintis inverseAckermannfunction
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ht
= mintizl: Ali, Lan))
A
Ackermann function
Z
login )
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logarithm ‘t ={0ifnel
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