Streaming Algorithms

Streams Si S Sm

S number

Questions

distinct elementsitem frequenciesheavy hitters

averagemedian

Frequencies Heavy hitters

Find countIX sto x Em EcountME x

Find every item thatappears I 2 Em timesreturn noitemthat appears Emtimes

XI times X appears in stream

AlgorithmSet 5 of item count pairsFor each item Xj C stream

if Xy C is in set increment Celse add Xj 1 to setif ISl K decrementcount foreach itemRemove all itemswithcount 0

Output countyif XES return countelse return 0

Ex K 2

Stream 2 5 7 2 2 5 5 5 5 7 2

is pity I iNote if S is full no need to add delete new item

claim output XI E x

why Increment count for at most c times

claim output1 12 Ix

Why in total court of X incremented xtotal increments m I xwhen count x decremented

K items are decrementedE m total decrementsC Mlk decrements of

Conclusion if output1 1 0 thenIx E Mlk

else if output41 0 thenC Z x E

ChooseK Y

Space 0 E tog m t tog nError Cm

qNole Countevery

bit

Heavy hittersreturn X if count x z 2 EM

Graphisstream of edges

Goal 0 n or Oln polylog Intl space

Connectivityis it connectedconnected components

ALLf forest initiallyfor each edge e

if F U e has no cyclesF FU e

idea maintain spanning forest

check for cycles Union Find

Space Oln log n

MST

Bipartite Graphs

Shortest Paths

Goal answer queries dlunt

Approximation return

dla.ME answers c d n v

TT3 C logn

Ideamaintain subgraph Hno small cycles in It allowed

AlgyH 0for each edge e U V

if d la v 2K 1 in Hadd e to pf

return It

Claim dath ul I da thVIda 4,4 E KK 1 do In v

why Every edge i It is in Gand

Let u x X Xe V beshortest path in G Then eitherXy Xp E H or d Xi XpHE 2k I

TH is a 12k it spanner

How big is It

Every cycle in It is size Z 2kt

t.es

girth 1422kt Isize of minimum cycle

Every graph with girth 72K has

01h edges

Proof Let It be graph with girth 72Kand Z 10 n edges

Repeat until none leftif u EH has degree C 2N thendelete u and adjacent edges

At most 2N n edges deletedI 8h remain

graph not empty

Now It has min degree 2hIt has no cycles E 2K

KEEFE

AhhhFix u in HLet T be all nodes at distance EkT is a tree

Leaves may haveedges between the

nodes 2n n

contradiction

Conclusion The graph we constructhas 0 n l edges

Two extremes

15 2Space 0 n tog m

3 spanner

K log n

Space 0 n logmI O n togmO login spanner

Better

Erdos Girth ConjectureFor KZ 2 n sufficiently large D n nodegraphwith girth 72k and I n Jno better spanner possible

why If It is such a graph andIt CH is spanner any edge inHl H haspath 22K in

It so H is not a 211 1 spanner

Matching

f V E M C E is a matching ifn nodes V eye CM e hem edges

streamingM 0for each edseetu.ir in stream

feuded intimeif U and V not matched in M

add e to M

Claim It is a matchingNever add conflicting edges

et M't max matchingclaim IM'tf E z lmy

M is Yz approximation

PI Let e EM'tIf e EM charge 1 to eElse

F e EM adjacent to e

charge 1 to e

total charges MY

Each e EM is charged E 2one for each endpoint or one for e

To eEM'tteem't

total charges E 21Mt

In IE 21Mt

Weighted Matching

each edge has weight to let

u 10 V

Greedy

to 44144

M 0for eachedge e uit

let C edges U and V in Mif ulet WIC then

M MICm Mule

Does it work

L o eo o o eo o o o

L Max weightmatch 217greedy weight 2not a good approximation

8 pp

J GreedyM 0for eachedge e uit

let C edges U and V in Mif wle Itf WIC then

M MICm Mule

ttt Hr its City Itsootoo o eo o o am.no o_O

Max weight It It8 t t l fn E IRL

zGreedia E Ith in this example1Mt't not in general

Terminologyedge e is borne if whenadded to Medge e killed by E if e removedfrom M when e bornedge e is survivor if bornandnever killed

T e edgeskilled by ET e edgeskilledby E ET e

Tg e edges killed by e ETg le

Tree ofthe Dead T e Tj e

ofa T Cel

f NT let

A Ariesclaim W Tj et It 8 w T let

each edge eu let Its T Ie

ttt w Tle I 8 II V Tle jhile t ult le

u let ultle

sublet cute

ultlells WII

charging Scheme

For each e EM't1 if e E T le't for Survivor e thencharge wle toe

2 if e H Tle then it was never bornwhen e arrived wld E htt U ccase 1 C e I

charge wle to ew let C its w e

case 21 C e e Icharge wle1h to e

u leftwleyCharge Wleth to ez

wle.ltbledNole wle E Its u left Ute

charge to e E Its u letcharge to e E Its U e

Total charge w Mt

Each EE Tle UM is charged E 2 Its Uce2 unborn in MOR f killed in M't

Betterif U v kills e u wand e charged fur unborn h gthen move charge for 4 y to es tu v

movecharge

f sey g EM't

Note Only charge u v for unborn Medges adjacent to U or V

no edge has more than 2 chargesno killed edge has more than I charge

iv then su killed

fo Moves I charge

I w1h41 I 2 Its wcml t its E.mu Tle

C 2 Its Ulm t Its fam

C 2 Its Ulm t WIM

E w m 3 28 t j

choose 8 7 approx0.17

What if more passes

E passes for ITT approx

Let M be stir approx I pass

RepeatMold Mfor eachedge e uit

let C edges U and V in Mif wle Itf WIC then

M MICm Mule

Until wlmwlm.la E l t k

each iteration increases ulmby factor Hk

After l iterations 2 T

ulm't WIN 4M It Kl

l C log 16

fix K later

k01

Let My M after iteration jc unchanged in iteration j

B My h Mg 1

HK w IMAI y

4M

when 4Mt it w by Itu Bgdone 2 wlMI

wlmf.twtulBdZLkmwlMDt8ulBg

SolvefrBjWIMg truly 2 ltHwlM

It 74

WHIZ lt klM211 tk

sumoygif.in

OPT fly 3 2 ulna UNFIT

no tree of dead

2 Its W Bg2 Its f s 28 E p I

Efftst2Dulm Itt U Ba

Ef's 3 2Dump lit ftp JulMD

Set K y

E 2 38 WINDset 8 21

E 2 2 E U ng 2 24 Ulm