Improved Approximation Algorithms for the Quality of Service Steiner Tree Problem M. Karpinski Bonn University I. Măndoiu UC San Diego A. Olshevsky GaTech A. 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

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 cost = 6  8 +4  4= 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 cost = 6  9 +4  2= 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 T 1 and T 2 in the routing tree T-e Max node rate r 1 Max node rate r 2 e r(e)=min{r 1,r 2 } T1T1 T2T2 tree T

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

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 S 1 and S 2 be sets of nodes of rate r 1 and r 2 Algorithm: Output the smaller cost tree out of two Steiner Trees ST(S 1  S 2 ) and ST(S 1 )  ST( S 2 ) ( ) Approximation ratio is at most 4 / 3 f, where f is the Steiner tree approximation ratio cost of optimum QoSST: opt = r 2 t 2 + r 1 t 1 where t i = length of all edges of rate r i, i= 1, 2 c 1 = cost ST(S 1  S 2 ) and c 2 = cost ST(S 1 )  ST(S 2 ) c 1 ≤ f [ r 2 t 2 + r 2 t 1 ] and c 2 ≤ f [ ( r 1 +r 2 ) t 2 + r 1 t 1 ] (1-r) c 1 + r 2 c 2 < f · opt, where r = r 1 / r 2 min(c 1, c 2 ) < f opt /(1-r+r 2 )  ( 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  e  f - approximation - rounding to integer powers of e with a random offset y, e y+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 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 opt k ≤  k opt A full k-restricted tree with thick extreme edges forming path 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 i opt i with  i i = 1 Zelikovsky (91)/Berman-Ramaiyer (92)/Promel-Steger(00) are convex, loss –algorithms e.g. Robins-Zelikovsky (00) is not convex t k = the length of edges of rate r k in the optimal tree, i.e. opt =  r k t k T k = Steiner tree computed for s and all nodes of rate r k by a convex  - approximation Steiner tree algorithm after collapsing all nodes of rate strictly higher than r k into the source s and treating all nodes of rate lower than r k as Steiner points. cost(T k ) ≤  r k t k + (r k t k+1 + r k t k+2 + … + r k t N ) Savings: the sum in parenthesis is not multiplied by 

Case of Two Rates New Algorithm: Output the cheapest out of two ST’s T 1 = ST(S 1  S 2 ) and T 2 = ST(S 2 )  ST( S 1  ST(S 2 ) ), ST(S 2 ) = contracted ST(S 2 ), where for T 1 use f 1 - approximation and for T 2 use convex f 2 - approximation cost of optimum QoSST: opt = r 2 t 2 + r 1 t 1 c 1 = cost(T 1 ) ≤ f 1 [ r 2 t 2 + r 2 t 1 ] and c 2 = cost(T 2 ) ≤ f 2 r 2 t 2 + f 2 r 1 t 1 + r 1 t 2 From these we obtain min(c 1, c 2 ) ≤, where r = r 1 / r 2 The best known values f 1 = 1+ln 3/2  1.55 (Robins-Zelikovsky, 00) and f 2 = 5/3  1.66 (Promel-Steger, 00) give ratio vs previous = 4/3(1+ln 3/2 ) 1

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, e y+i - sort rounded rates in descending order - repeat for each rounded rate r: - find Steiner tree T r with convex f-approximation algorithm - contract the tree T r - output union of Steiner trees T r 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 vs e (1+ln 3/2 )  4.059

Conclusions: Results AlgorithmLCARNSBRMST runtimepolynomial O(rn 3 )O(rn log n + rm) r = # rates2any2 2 2 Previous ratio ε ε 2.22+ε ε Our ratio1.96+ε ε ε ε 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!

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