1 efficient qos partition and routing of unicast and multicast dean h.lorenz,ariel orda,danny...
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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
3
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
10
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