Improved Approximation Algorithms for the Quality of Service Steiner Tree Problem

M. Karpinski Bonn UniversityI. Măndoiu UC San DiegoA. Olshevsky GaTechA. Zelikovsky Georgia State

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

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

14

14

6

3

2

7

2

8

2

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

14

14

6

3

2

7

2

8

2

cost = 68 +44= 64

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

14

14

6

3

2

7

2

8

2

cost = 69 +42= 62

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

Max node rate r1 Max node rate r2

e

r(e)=min{r1,r2}T1 T2

tree T

Formal QoSST Problem

Given: Undirected graph G=(V,E, l, r) with

rates r: VR+ on nodes (r = 0 means Steiner point)

lengths l: ER+ 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

Previous Work

• Introduced by Current[1986] in the context of network design and Maxemchuk[1997] 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 al[2001] 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-Schieber[2001].

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 [ (r1+r2) t2+ r1 t1 ]

(1-r) c1 + r2 c2 < f · opt, where r = r1/r2

min(c1, c2) < f opt /(1-r+r2) ( 4/3 f ) opt

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 ef - approximation

- rounding to integer powers of e with a random offset y, ey+i

- output union of Steiner trees for each rounded rate

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

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

(a) Splitting high-rate binary trees into paths(b) 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

k-Restricted Steiner TreesA 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 formingpath b/w pair of diametrical Terminals u and v

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,

i=2,...,n iopti with i i = 1Zelikovsky (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 tk+1 + rk tk+2 + … + rk tN)

Savings: the sum in parenthesis is not multiplied by

Case of Two Rates

New Algorithm: Output the cheapest out of two ST’s 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 = 1+ln 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(1+ln 3/2 )

1

maxf \al maxr

Case of Unbounded # of RatesAlgorithm: (randomized rounding - similar to Charikar et al (01))

- rounding to integer powers of e with a random offset y, ey+i

- 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 (1+ln 3/2 ) 4.059

Conclusions: Results

Algorithm LCA RNS BR MST

runtime polynomial polynomial O(rn3) O(rn log n + rm)

r = # rates 2 any 2 any 2 any 2 any

Previous ratio 2.066+ε

4.211+ε

2.22+ε 4.531+ε

2.444 4.934 2.667 5.44

Our ratio 1.96+ε 3.802+ε

2.059+ε

3.802+ε

2.237 4.059 2.414 4.311

r = number of rates, n = # vertices, m = # edges

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 GLOBECOM’03– Better up to 7% in simulations – No proof of better ratio – Needs advance in primal-dual analysis!

Thank You!