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Efficient QoS Partition and Routing of Unicast and Multicast

Dean H.Lorenz,Ariel Orda,Danny Raz,Yuval Shavitt

Proceeding of IWQoS 2000, Pittsburgh, PA, June 2000

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Outline

Background,model and problems Pseudo-polynomial algorithms Approximation techniques Lower & upper bounds Conclusion

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Background

Supporting QoS :– Routing :

• Finding a path or tree with minimum cost

– Resource allocation : • Mapping end-to-end QoS into local ones

Problem :– How to provide the required QoS with

minimum cost?

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Model

Each link offers several QoS guarantees– Associated with different cost

Integer cost functions– Delay and cost are integer

Focus on additive QoS requirements– Harder than bottleneck ones

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QoS Routing Problem

Given :– Network graph G(V,E)– End-to-end delay requirement D– link cost functions

Objective :– Find minimum cost route

• Cost of optimal resource allocation

Ell dc )}({

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QoS Partition Problem

Given :– A route (path or tree)– End-to-end delay requirement D– link cost functions

Objective :– Find delay requirement for each link

• With minimum cost• Satisfy end-to-end delay requirement

Ell dc )}({

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Cost

Cost– Ensuring a specific guarantee on a route

Various considerations– Link perspective

• Resources reserved or consumption

– Network perspective • performance

– User perspective• pricing scheme• Feasible cheapest route

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Cost Function

General integer cost functions– The (delay,cost) pairs are integer– Always decreasing

d

Cl(d)

c

d

(d,c)

(d,c)D

(d,c)

(d,c)

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Outline

Background,model and problems Pseudo-polynomial algorithms Approximation techniques Lower & upper bounds Conclusion

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Restricted Shortest Path (RSP) Problem

Given :– Network graph : G(V,E)– End-to-end QoS requirement : D– Single delay and cost for each link– Upper bound of optimal cost value : U

Find :– Minimum cost path that satisfies QoS

requirement• delay(p) D, p means a path

E l l lc d } , {

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Recursive Form of RSP Problem

D(v,i) : – minimum delay from source to v with cost

no more than i Recursive formula until :

– )}},({min),1,(min{),(|

llicu

ciuDdivDivDl

s v(dl,cl)

)(vNu l

i-cl

i N(v) : neighbors of v

i : end-to-end cost

DitD ),(

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Optimal QoS Partition & Routing (OPQR) Problem

Given :– Network graph : G(V,E)– End-to-end requirement : D– Delay/cost function for each link – Upper bound of optimal value : U

Find :– Minimum cost path p and partition that

satisfies QoS requirement D•

Elll dc )}({

pl llpll dcpcd )()(,}{

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OPQR Problem

Idea – View each link l as set of links– Corresponding all possible cost 1,…,U– Delay

• Minimum value achieve specified cost

},...,,{ 21 Ulll

l

l1

l2

lU

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Recursive Form of OPQR Problem

Recursive formula until :– for j=1,2,…,i

)}},()({min),1,(min{),(

})(|min{)(

jiuDjdivDivD

jdcdjd

lu

ll

DitD ),(

)(vNu

l

i

(1,dl(1))

(i,dl(i))i-j N(v) : neighbors of v

i : end-to-end costs v

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Multicast Optimal QoS Partition(M-OPQ) Problem

Given :– A tree :T– Delay/cost function for each link :– End-to-end delay requirement : D

Find :– Optimal partition satisfying end-to-end

delay requirement• DTdelayd Tll )(,}{

Tll dc )}({

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M-OPQ Problem

Idea :– Use same technique in OPQR problem

Notation :– X,Y,Z are tables holding best delay for each cost– Size is U

s

t1

t2

xy

z

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Merge Procedure

Find best allocation between• Two branching sub-trees• A sub-tree and its root link

Merge two branching sub-trees– For c=1,…,U

Merge sub-tree and its root link– For c=1,…,U

)}(),(max{min)( 1 xcZxYcD cx

)}(),({min)( 1 xcWxXcD cx

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Outline

Background,model and problems Pseudo-polynomial algorithms Approximation techniques Lower & upper bounds Conclusion

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Logarithmic Sampling and Linear Scaling

Idea – Log Sampling

• Improve methods of OPQR• Check delays only corresponding to specific

costs• Specific costs are 1, , ,…,U

– Scaling• Applied to all costs• Smaller costs for OPQR problem

• Scale factor :

2)1( )1(

1n

L

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Outline

Background,model and problems Pseudo-polynomial algorithms Approximation techniques Lower & upper bounds Conclusion

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Finding Upper & Lower Bounds

Test procedure Test( ):– Check whether is a valid upper bound

General idea – Call Test( ) for = {1,2,4,8,…}– For some

• Test( ) return fail and Test(2 ) succeed •

– f-Approximated test procedure :• Bound C by C f(2)

2C

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Test Procedure

TEST() For each link e

– Set de() min{ d | ce(d) }– Put on each link

Find Shortest-Path p w.r.t. {de()} If Delay(p) D

C n = f() Else

< C

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Outline

Background,model and problems Pseudo-polynomial algorithms Approximation techniques Lower & upper bounds Conclusion

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Conclusion

Characteristics :– Establish fully polynomial approximation schemes

for problem OPQR– First FPAS for problem M-OPQ

Future Works :– Cost model– Multicast routing problem under this framework