Improved Approximation Algorithms for the Quality of Service Steiner Tree Problem

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Improved Approximation Algorithms for the Quality
of Service Steiner Tree Problem
M. Karpinski Bonn University I. Mandoiu
UC San Diego A. Olshevsky GaTech A.
Zelikovsky Georgia State
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Outline

• QoS for multimedia distribution
• Quality of Service Steiner Tree Problem
• Previous work
• - case of two rates
• - general case with multiple rates
• Reusing higher rate connections
• Main ideads for better ratios
• - k-restricted Steiner trees
• convex approximations of Steiner trees
• cases of 2 and multiple rates
• Conclusions

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QoS for Multimedia Distribution

• Given
• source in the network
• set of customers requesting high volume data at
  different rates (bandwidths)
• cost of link is proportional to
• link length (or data unit cost)
• rate (or bandwidth)
• Find
• Minimum cost tree connecting source to each
  customer
• S.t. each customer gets data with at least
  requested rate

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QoS for Multimedia Distribution

• Given
• source in the network
• set of customers requesting high volume data at
  different rates (bandwidths)
• cost of link is proportional to
• link length (or data unit cost)
• rate (or bandwidth)
• Find
• Minimum cost tree connecting source to each
  customer
• S.t. each customer gets data with at least
  requested rate

2
S
4
1
4
3
6
6
1
7
8
4
2
2
4
6
cost 6?8 4?4 64
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QoS for Multimedia Distribution

• Given
• source in the network
• set of customers requesting high volume data at
  different rates (bandwidths)
• cost of link is proportional to
• link length (or data unit cost)
• rate (or bandwidth)
• Find
• Minimum cost tree connecting source to each
  customer
• S.t. each customer gets data with at least
  requested rate

2
S
4
1
4
3
6
6
1
7
8
4
2
2
4
6
cost 6?9 4?2 62
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Removing Source and Directions

• We may get rid of source and directions by
• assigning to source the maximum rate and
• introducing rates on edges rate of edge e, r(e)
  is the lowest rate between the maximum node rate
  in two components T1 and T2 in the routing
  tree T-e

tree T
e
T1
T2
r(e)minr1,r2
Max node rate r1
Max node rate r2
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Formal QoSST Problem

• Given Undirected graph G(V,E, l, r) with
• rates r V?R on nodes (r 0
  means Steiner point)
• lengths l E?R on edges (l is a metric)
• Find Spanning tree T of minimum cost
• cost(T) ?e ?E r(e) l(e),
• where rate of edge e, r(e), is the smaller
  among maximum node rates in connected components
  of T-e

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Previous Work

• Introduced by Current1986 in the context of
  network design and Maxemchuk1997 in the context
  of network routing
• Known as Multi-Tier/QoS/Grade-of-Service STP
• Case of a single rate classical Steiner tree
  problem
• Case of few rates explored in a series of papers
  by Mirchandani-Balakrishnan-Magnanti 1994, 1996
• Mirchandani at al 1996 and Xue et al2001
  obtained better results for the case of two and
  three rates
• First constant-factor approximation algorithm for
  arbitrary number of rates given by
  Charikar-Naor-Schieber2001.

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Case of Two Rates

• Let S1 and S2 be sets of nodes of rate r1 and r2
  
• Algorithm Output the smaller cost tree out of
  two Steiner Trees
• ST(S1 ? S2 ) and ST(S1) ? ST( S2 )
  (1994-2001)
• Approximation ratio is at most 4/3 f,
• where f is the Steiner tree approximation ratio
• cost of optimum QoSST opt r2 t2 r1 t1
• where ti length of all edges of rate ri ,
  i 1, 2
• c1 cost ST(S1 ? S2) and c2 cost ST(S1) ?
  ST(S2)
• c1 f r2 t2 r2 t1 and c2 f
  (r1r2) t2 r1 t1
• (1-r) c1 r2 c2 lt f opt,
  where r r1/r2
• min(c1, c2) lt f opt /(1-rr2) ? (
  4/3 f ) opt

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General Case with Multiple Rates

• Case of 3 rates much more involved
• 4-5 pages of calculations nonlinear
  optimization (Mirchandani et al, 1996) and
  elementary derivation in (Xue et al, 2001)
• Unbounded number of rates (Charikar et al 2001)
• rounding rates to integer powers of 2 ? 4f -
  approximation
• randomized rounding ? e?f - approximation
• - rounding to integer powers of e
  with a random offset y, eyi
• - output union of Steiner trees for each
  rounded rate

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Reuse of Higher Rate Edges

• The lower rate nodes can be connected to higher
  rate nodes not only to the source
• (Maxemchuk, 97) suggested a simple algorithm
• sort all rates and connect first the highest rate
  nodes,
• then repeatedly connect to the existing tree the
  nodes of the next highest rate
• In the worst case the error may be logarithmic
• The known before approximation bounds did not
  take in account saving from high rate edges reuse
• This paper Improved approximation bounds based
  on estimation of savings delivered by reuse of
  higher rate edges

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Estimation of Reuse Savings

• (a) General QoSST with two rates
• high rate nodes are thick and are connected via
  binary tree
• lower rate nodes and connections are hidden in
  triangles
• Splitting high-rate binary trees into paths
• High-rate path-spine with attached lower-rate
  binary trees (triangles)
• Conclusion the Steiner tree for lower rate nodes
  is shorter than the Steiner for the union of
  higher and lower rate nodes by the length of
  spine

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k-Restricted Steiner Trees

• A Steiner tree is called k-restricted if it can
  be decomposed into components of at
• most k terminals where every terminal is a leaf.
  For optimal k-rest ST optk ?kopt

A full k-restricted tree with thick extreme
edges forming path b/w pair of diametrical
Terminals u and v
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Convex Steiner Tree Approximation Algorithms

• Steiner tree approximation algorithm is convex
  if output tree length upper bounded by convex
  combination of the optimal k-restricted ST,
• ?i2,...,n ?iopti with ?i ?i
  1
• Zelikovsky (91)/Berman-Ramaiyer
  (92)/Promel-Steger(00) are convex, loss
  algorithms e.g. Robins-Zelikovsky (00) is not
  convex
• tk the length of edges of rate rk in the
  optimal tree, i.e. opt ? rk tk
• Tk Steiner tree computed for s and all nodes
  of rate rk by a convex ?-approximation Steiner
  tree algorithm after collapsing all nodes of rate
  strictly higher than rk into the source s and
  treating all nodes of rate lower than rk as
  Steiner points.
• cost(Tk) ? rk tk (rk tk1 rk tk2 rk
  tN)
• Savings the sum in parenthesis is not multiplied
  by ?

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Case of Two Rates
1

• New Algorithm Output the cheapest out of two
  STs T1 ST(S1 ? S2) and
• T2 ST(S2) ? ST( S1 ? ST(S2) ), ST(S2)
  contracted ST(S2), where
• for T1 use f1- approximation and for T2 use
  convex f2- approximation
• cost of optimum QoSST opt r2 t2 r1 t1
• c1 cost(T1) f1 r2 t2 r2 t1 and
• c2 cost(T2) f2 r2 t2 f2 r1 t1 r1 t2
• From these we obtain
• min(c1, c2) , where r r1/r2
• The best known values f1 1ln 3/2 ? 1.55
  (Robins-Zelikovsky, 00) and
• f2 5/3 ? 1.66 (Promel-Steger, 00) give ratio
  1.960
• vs previous 2.066 4/3(1ln 3/2 )

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Case of Unbounded of Rates

• Algorithm (randomized rounding - similar to
  Charikar et al (01))
• - rounding to integer powers of e
  with a random offset y, eyi
• - sort rounded rates in descending order
• - repeat for each rounded rate r
• - find Steiner tree Tr with
  convex f-approximation algorithm
• - contract the tree Tr
• - output union of Steiner trees Tr for
  each rounded rate r
• Approximation ratio is at most ,
  where f is the
• approximation ratio of convex Steiner
  approximation algorithm.
• For f 5/3 ? 1.66 (Promel-Steger, 00), the
  ratio is 3.802 vs e (1ln 3/2 ) ? 4.059

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Conclusions Results
Algorithm LCA LCA RNS RNS BR BR MST MST
runtime polynomial polynomial polynomial polynomial O(rn3) O(rn3) O(rn log n rm) O(rn log n rm)
r rates 2 any 2 any 2 any 2 any
Previous ratio 2.066e 4.211e 2.22e 4.531e 2.444 4.934 2.667 5.44
Our ratio 1.96e 3.802e 2.059e 3.802e 2.237 4.059 2.414 4.311
r number of rates, n vertices, m edges
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Conclusions Future Work

• Discussed algorithms are coarse all nodes of the
  same rate are up-rated together
• How to design better algorithm incorporating
  certain nodes of lower rate while connecting
  nodes of higher rate, i.e., up-rate specific
  nodes ?
• Primal-dual algorithm is in GLOBECOM03
• Better up to 7 in simulations ?
• No proof of better ratio ?
• Needs advance in primal-dual analysis!

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• Thank You!