Seminar ppt of a maxflow problem

8/16/2019 Seminar ppt of a maxflow problem

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An efcient distributed max-ow

algorithm or Wireless SensorNetworks

Niveditha Zacaria

R7042 1


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Wireless Sensor Networks

(WSNs):Highl distri!"ted networks o# s$all%lightweight wireless nodes%&e'loed in large n"$!ers%onitors the environ$ent or sste$ !$eas"ring 'hsical 'ara$eters s"ch as

te$'erat"re% 'ress"re% h"$idit

*ach node:

sensing + 'rocessing + co$$"nication 2


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,or$al -. ,/W ro!le$

ra'h G=(V,E) 3 a ow network ○ &irected% each edge has capacity c(u,v ) 0

○ wo s'ecial vertices: source s, and sink t ○ ,or an other verte5 v, there is a 'ath s→…→v →…→t

Flow 3 a #"nction f : V × V → R○ Capacity constraint : ,or all u, v ∈ V : f (u,v ) c(u,v )

○ Skew symmetry : ,or all u, v ∈ V : f (u,v ) 6 3f (v,u)○ Flow conservation: ,or all u ∈ V 3 s, t 8:

, or ( , ) ( , ) 0

( , ) ( , ) 0

v V

v V

f u v f u V

f v u f V u

∈

∈

= =

= =

∑

∑2/5

2/15

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s t 0/9

a

b

9


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4

a5 ,low (contd)

We want to nd a ;ow o# $a5i$"$val"e #ro$ the so"rce to the sink

&enoted !


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N!A!"N # ASS$%&!"NS

@ :3 sensorsWireless Sensor Network 6 (@%*)

* :3 links (directed and #"ll d"'le5)

/ink 'resent:3 neigh!oring nodes eg:(i%A)* *

1 *ach sensor can !e "niB"el identied Chas atleast 1 trans$itter and 1 receiver

2 - $echanis$ e5ists #or neigh!ordiscovering

9 here is a 'ro'er deliver o# 'ackets #ro$ a

node to its oneDho' neigh!or E


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'esidual Network() *

Network consisting o# links that have'ositive resid"al ca'acities

Resid"al ca'acit D $a5i$"$additional ;ow that can !e sent #ro$node i to node A

Resid"al F"%vG 6 c F"%vG 3 # F"%vG


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+eight unction(h*

h : @ DIN is a height #"nctionhFsG 6 n

hFt G 6 0hF"G J6 hFvG + 1

-lgorith$ '"shes ;ow downhill% whichis #ro$ the higher node to the lowernode

7


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)eneric push-relabel

algorithm )&'

"nput:- network F@G 6 * stored as an adAacen

listutput: a5D;ow # : * DI NK

# Fi% AG 6 c F i% A G 3 resid"al F i% A G

L


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Algorithm ,

1 hFso"rceG 6 nK2 hFsinkG 6 0K9 ,or ( all " * @ 3 s%t8 )4 hFvG 6 0KE ,or ( all ("%v) * * ) resid"alF"%vG 6 cF"%vGK7 ,or ( all links e o"tgoing #ro$ so"rce )L sat"rate e ;= While (v is the active node% v * @) 10. i# ( (v%") is down hill )11. '"sh $in e5cess FvG% resid"al Fv%"G 8 "nits #ro

v to "K12. "'date resid"al F"%vG and resid"al Fv%"G =


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Asynchronous Algorithm

he $ain !od o# the 'ro'osedalgorith$ which e5ec"tes in sensor

nodesn'"t: /ocal in#or$ation o# callernode% s"ch as role and e5cess as wellas oneDho' in#or$ation s"ch as ;owval"es and resid"al ca'acities #oreach neigh!or

"t'"t: a5D;ow or nearDo'ti$al ;ow

10


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Algorithm

1 # : * DI N% # Fi% AG 6 cFi% AG 3resid"alFi% AG

2 nitialiOe()K9 i#( role v 6 SPRQ*)

4 e5cess v 6 s"$ v (resid"al vF"G)K

E "sh()K while (tr"e)7 listen()K

11


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Algorithm .

n'"t: he local in#or$ation o# node v which are

the ca'acities o# inco$ing links cFv% "G% role v

* SPRQ*K SNK R*/-8 and the totaln"$!er o# the nodes (n 6


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"nitiali/e(*

1 *5cess v 6 0K

2 h v 6 init T height(role v)K

9 #or (all " * neigh!ors)4 i# (("% v)is an inco$ing link into v in

) resid"al v F"G 6 0K

E else i# ((v% ") is an o"tgoing link#ro$ v in ) resid"al v F"G 6 cFv% "GK

7 h vF"G 6 init T height(role ")K19


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init 0 height(k*

"nput 1 role k * (SPRQ*% SN% R*/-) andthe total n"$!er o# the nodes (n 6


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Algorithm 2

"nput1 /ocal varia!les s"ch as the resid"al

ca'acities o# the neigh!ors% role%height and e5cess We ass"$e thatthe algorith$ r"ns at node v

utput1 - PSH $essage carring a ;ow val"e

1E


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3 !he push operation 4 push(*

1 i# (role v U6 SN)

2 while (e5cess v I0) CC (" * neigh!ors v s"ch that (resid"al v F"G

I0) CC (h vF"G J h v))

9 V 6 $in(e5cess v %resid"al vF"G)K

4 e5cess v 6 e5cess v D VKE resid"al vF"G 6 resid"al vF"G D VK

send (PSH% V) to "K

1


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Algorithm 5

"nput1 /ocal varia!les s"ch as theresid"al ca'acities o# the neigh!ors%

role and height

utput1 So$e H*H $essagescarring the new height val"e o# thecaller node

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!he relabel operation 4relabel(*

1 i# ( role v * SPRQ* )CC ( role v * SN )2 check Tcanceling()K

9 h 6 $in(" * neigh!ors v %resid"al v F"GI0) hvF"G8+1K4 h v 6 hK

E #or (all " * neigh!ors v)

Send(H*H% h v) to "K7 '"sh ( )K

1L


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Algorithm 6

shows the reaction o# nodes "'on receivingPSH and H*H $essages

When a node v receives a H*H $essage

#ro$ node w% "'dates the height o# w in itsneigh!ors ta!le (h vFwG)n the other hand% when one node receives a

PSH $essage% acce'ts the e5cess ;ow i# it can%otherwise% it

sends !ack a R*,PS* $essage to the sender-#ter that% the sender roll!acks the '"sh i$'act

and recovers the 'revio"s state

1=


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Nodes reaction upon recei7ing messages rom the network 4listen(*

"nput1 -n inco$ing $essage #ro$ the

channel with w as its sender and V as itsdata oreover% the node "ses localvaria!les s"ch as resid"al ca'acities%e5cess val"e and the height

utput1 &e'ends on the inco$ing

$essage and the state o# the node t$ight !e an "'date in local varia!les or

calling other o'erations to 'ro'agate datathro" h new $essa es 20


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n receiving PSH $essage :

1 i#(hvFwG Ihv)

2 resid"alvFwG 6 resid"alvFwG + VK9 e5cessv 6 e5cessv + VK

4 i#(rolev U6SN)

E '"sh()K i#(e5cessv I 0)

7 rela!el()K

L else send (R*,PS*% V) to wK 21


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n receiving R*,PS* $essage :

1 e5cessv 6 e5cessv + VK

2 resid"alvFw G 6 resid"alvFwG + VK9 "sh()K4 i# (e5cessv I 0)

rela!el()

n receiving H*H $essage :

E hvFwG 6 VK 22


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Algorithm 9

e$'oral Qanceling

- techniB"e to discover tra''ed ;ows and redirect the$!ack into the so"rce

o i$'le$ent s"ch a canceling% each node $"st !e eB"i''ed! a ti$er and each PSH $essage $"st !e a"g$ented witha ti$esta$'

Qonves an idDseB"ence that indicates the loo'D#ree 'ath #ro$

the so"rce to the c"rrent node

"nput1 /ocal varia!les s"ch as the resid"al ca'acities o# theneigh!ors% role% height and e5cess We ass"$e that thealgorith$ r"ns at node v

29

: li ti h k


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:anceling operation - check0canceling(*

1 isra''ed 6 e5a$inera''ed,low(PSHX)K2 i# (isra''ed CC rolev U6SPRQ*)

9 while (there is an "nhandled PSH

$essage containing tra''ed ;ow)4 V 6 the #low val"e o# the PSH $essageKE " 6 re$ove the last id #ro$ the idT seBK e5cessv 6 e5cessv D VK

7 send (Q-NQ*/% V% idT seB) to "K

24


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Algorithm ;

Nodes reaction "'on receiving the Q-NQ*/ $essage

"nput1 -n inco$ing $essage #ro$ the channel with

w as its sender and V as its data oreover% thenode "ses local varia!les s"ch as resid"al

ca'acities% e5cess val"e and the height utput: So$e changes in local varia!les and

another Q-NQ*/ $essage

1 i# (rolev U6SPRQ*)2 resid"alvFwG 6 resid"alvFwG + VK

9 " 6 re$ove the last id #ro$ the idT seBK4 send (Q-NQ*/% V% idT seB) to "K

2E


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!he adapti7e algorithm

n this section% two new o'erations%

'"ll and '"sh!ack are introd"ced%which are "sed to ada't the $a5D;ow in the

network when ca'acities change

Whenever the ca'acit o# the link ("% v)increases% " invokes '"ll o'eration to

trans$it $ore ;ow% i# 'ossi!le

Si$ilarl% once the ca'acit o# link ("% v)decreases% it triggers the '"sh!acko'eration 2


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Algorithm


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Algorithm ,=

n'"t: /ocal varia!les and dv" which is the

a$o"nt o# red"ction in cFv% "G

"t'"t:

- PSH>-Q $essage carring twoval"es% one is #or ;ow diYerence andthe other is #or the height val"e o#the node

2L


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!he pushback operation 4pushback( *

1 i# (resid"alvF"G I6 dv%")

2 resid"alvF"G 6 resid"alvF"G 3 dv%"K

9 else4 5 6 dv%" D resid"alvF"GK

E hv 6 init T height(v)K

send (PSH>-Q% 5% hv) to "K

7 resid"alvF"G 6 0KL e5cessv 6 5K

= '"sh( )K

2=


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>rawbacks

1 ost a''lications o# WSN are sensitive to long delas !"t

still the 'ro'osed algorith$ is slow in so$e sit"ation

2 he algorith$ does not have an a''ro'riate ter$inationdetection $echanis$

ie the rela nodes and es'eciall the so"rce do not know

when the algorith$ ter$inates and what are the ;ow'aths

,ort"natel% the sink node has this in#or$ation and itcan in#or$ other nodes ! "sing the idDseB"ence

90


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:onclusion

n this 'a'er the $a5D;ow 'ro!le$ has!een restated in a wa that the o!Aectiveis not onl nding the o'ti$al $a5D;ow

!"t also red"cing $essage and ti$eoverheads and !eing ada'tive to networkchanges

he algorith$ consists o# severalhe"ristics which $ake it asnchrono"s%lightDweight% ada'tive and realDti$e

91


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'eerences

-n ecient distri!"ted $a5D;ow algorith$ #orWireless Sensor Networks :Sa$an Ho$ao"nneAad %-lireOa >agheri C 201E *lsevier /td

old!erg -@% arAan R* - new a''roach to the$a5i$"$D;ow 'ro!le$ -Q

-h"Aa R% agnanti /% rlin [> Network ;ows:theor% algorith$s% and a''lications *ngland:earson *d"cation /i$itedK 2014

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Seminar ppt of a maxflow problem

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