1. Network Design with (Dis)economies of Scale Lisa Zhang Joint with Matthew Andrews, Spyridon Antonakopoulos and Steve Fortune

2. All Rights Reserved © Alcatel-Lucent 2006, ##### 2 | Presentation Title | Month 2006 Power: New Resource in Networking  Network power efficiency is a new metric  Mismatch in growth, 2020 projection –Traffic x30-100 –CMOS power efficiency x3-6  Goal: Power proportional - power follows load  Current situation: –Power consumption largely independent of traffic –Under-utilized network elements (routers, switches) due to over- provisioned network for traffic fluctuation and redundancy  Opportunity for optimization  Routing: example of global optimization for a network of servers

3. All Rights Reserved © Alcatel-Lucent 2006, ##### 3 | Presentation Title | Month 2006 Routing  Input  Network topology and matrix of traffic demands  Cost function f that models power consumption  Question  Find routing paths that satisfy the demands and minimize total power consumption of network. –Min  e f ( x e ) where x e is traffic load on e  Well studied problem for a range of f( )  What is appropriate f( ) in the context of power? ABC… A053… B501… C310… ⋮⋮⋮⋮⋱ A B C D E F

4. All Rights Reserved © Alcatel-Lucent 2006, ##### 4 | Presentation Title | Month 2006 Some simple cost functions  Perfectly linear cost function  shortest-paths routing for each traffic demand  Solvable in polynomial time [Dijkstra]  Step-function  Steiner forest problem  2-approximation: can find a solution at most twice the optimal cost in polynomial time [Agrawal-Klein-Ravi, Goemans-Williamson] link load cost link load cost perfectly linearstep-function

5. All Rights Reserved © Alcatel-Lucent 2006, ##### 5 | Presentation Title | Month 2006 Buy-at-bulk cost functions  Sub-additive cost function  buy-at-bulk network design problem  Sub-additive: f(x 1 + x 2 ) ≤ f(x 1 ) + f(x 2 ); models economies of scale  Can find O(log n) approx for uniform case [Awerbuch-Azar]; polylog(n) approx for non- uniform case [Chekuri-Hajiaghayi-Kortsarz-Salavatipour]  Cannot approximated better than  (log ¼ n) [Andrews] link load cost sub-additive average

6. All Rights Reserved © Alcatel-Lucent 2006, ##### 6 | Presentation Title | Month 2006 Energy costs and (dis)economies of scale What if the cost function has the form f(x) =  + x  for x > 0, f(0) = 0?  Motivation: describes power consumption of CMOS circuits with dynamic frequency and voltage scaling  Reflects a combination of economies and diseconomies of scale  Similarly-shaped cost curves commonly encountered in many industries  potential for wide model applicability link load cost average

7. All Rights Reserved © Alcatel-Lucent 2006, ##### 7 | Presentation Title | Month 2006 Results  If the cost function has the form f(x) =  + x  for x > 0, f(0) = 0  If  = 0, constant approx  If  > 0, –Polylog(n) approx –For every , (log ¼ n) lower bound  What is the opportunity for power savings in real networks?  Experimental study that motivates power-aware optimization

8. All Rights Reserved © Alcatel-Lucent 2006, ##### 8 | Presentation Title | Month 2006 Polylog Approximation Joint with M. Andrews and S. Antonakopoulos link load cost

9. All Rights Reserved © Alcatel-Lucent 2006, ##### 9 | Presentation Title | Month 2006 Connection to Capacitated Network Design (CND)  CND: use capacitated links of least total cost to carry given traffic demands  Replace each link by a collection of parallel edges, with a fixed capacity equal to  1 / , and flat (step-function) costs: f(  1 /  ) – f(0) = 2 , f(2  1 /  ) - f(  1 /  ) = (2  – 1) , f(3  1 /  ) - f(2  1 /  ) = (3  – 2  ) , and so on.  Cheap edges will be used before expensive ones.  CND not equivalent to Steiner forest, because of edge capacities. f(x)f(x)  (3  – 2  )  (2  – 1)  22 … 3  1 /  2  1 /   1 /  1 st edge 2 nd edge 3 rd edge

10. All Rights Reserved © Alcatel-Lucent 2006, ##### 10 | Presentation Title | Month 2006 Algorithm outline  Goal: (polylog(n), polylog(n))-approx to Capacitated Network Design (CND)  Link capacity violated by polylog(n) factor  Total cost within polylog(n) times OPT LP, optimal fractional solution of CND  Implies polylog(n)-approx to integral routing under with diseconomy of scale  Simplify assumption: unit demands, unit link capacity  Algorithm outline for approximating CND:  While there exist unrouted demands: 1.Solve LP relaxation and decompose fractional solution into well-cut-linked flows 2.Construct an expander graph as virtual network topology 3.Route some fraction of demands via edge-disjoint paths on the virtual topology  Output the union of all partial routings We need to ensure that each partial routing (i) serves at least a polylog(n) fraction of the demands, and (ii) has cost at most polylog(n) times OPT LP.

11. All Rights Reserved © Alcatel-Lucent 2006, ##### 11 | Presentation Title | Month 2006 LP relaxation  Solve an LP relaxation of CND Min  e C e  i X i,e s.t. {flow conservation on X i,e }  i X i,e ≤ 1 0 ≤ X i,e ≤ 1  In the fractional solution OPT LP, each demand may be routed along more than one paths, each carrying only part of the demand’s bandwidth.

12. All Rights Reserved © Alcatel-Lucent 2006, ##### 12 | Presentation Title | Month 2006 Well-cut-linked decomposition  Well-cut-linked decompostion [Chekuri-Khanna-Shepherd]  A partition of original graph into node-disjoint subgraphs, and a function  (·) on the terminals.   (·) represents surviving demand flows within each subgraph –At least a polylog fraction of total demand survives. –Each surviving demand has at least polylog fraction of surviving flow  Property of subgraph under decomposition: –Any equal partition of any subgraph has large cut: fractional flow in OPT LP of edges in A-B cut ≥ min{(A), (B)} Large cut A B

13. All Rights Reserved © Alcatel-Lucent 2006, ##### 13 | Presentation Title | Month 2006 Min-cost flow  Each subgraph is well-cut-linked as a result of decomposition  Every equal partition has large cut using fractional flow in OPT LP  Cost of fractional links along fractional paths for the cut is bounded by OPT LP  Integrality theorem converts fractional flows into integral flows  Round up fractional links into integral links  Cost of integral links along integral flow paths is bounded by polylog * OPT LP Large cut

14. All Rights Reserved © Alcatel-Lucent 2006, ##### 14 | Presentation Title | Month 2006 Constructing an expander topology  When is a graph G′ = (V′, E′) an expander?  Expansion: for any S  V′ with |S|  |V′| / 2, the number of edges in the cut (S, V′ - S) is at least c|S|, for a constant c > 0.  Similar to the property of well-cut-linked terminals we saw earlier.  How to construct an expander?  Suppose we are given a routine that for any balanced partition (A, B) of a node set V* produces a perfect matching. Then, we can construct an expander by calling this routine O(log 2 |V*|) times. [Khandekar-Rao-Vazirani]

15. All Rights Reserved © Alcatel-Lucent 2006, ##### 15 | Presentation Title | Month 2006 Constructing an expander topology  How to build an expander on the terminals of each well-cut-linked subgraph of the decomposition?  Perfect matchings consist of entire paths joining terminals, not just edges; cost of these path matchings can be bounded polylog OPT LP  Well-cut-linkedness ensures the existence of such a path-matching.  Result: a virtual expander topology that uses each edge of the real topology at most a polylog number of times.

16. All Rights Reserved © Alcatel-Lucent 2006, ##### 16 | Presentation Title | Month 2006 Edge-disjoint routing in the virtual expander topology  Given an expander and a set of node pairs, we can route at least a polylog fraction of those pairs via edge-disjoint paths.  In the real topology we can route at least a polylog fraction of the demands, while the load on every edge is at most polylog larger compared to the load under OPT LP.  We apply the same process on the remaining demands. No more than a polylog number of iterations required.  Theorem. (polylog(n), polylog(n))-approx to uniform Capacitated Network Design, where link cap is violated by polylog(n) factor and total cost within polylog(n) times OPT LP.  Theorem. Uniform network design with (dis)economies of scale is polylog(n)-approximable.

17. All Rights Reserved © Alcatel-Lucent 2006, ##### 17 | Presentation Title | Month 2006 Open questions  What happens to non-uniform CND, where links have different capacities?  What happens to non-uniform routing with (dis)economies of scale, where each link e has a distinct cost function f e ( )?  What happens to protection version of the problem?

18. All Rights Reserved © Alcatel-Lucent 2006, ##### 18 | Presentation Title | Month 2006 Experimental motivation Joint with S. Antonakopoulos and S. Fortune

19. All Rights Reserved © Alcatel-Lucent 2006, ##### 19 | Presentation Title | Month 2006 Goal: Quantification of opportunity for saving power as realistically as possible Opportunity for power saving  Optimizing traffic routes (global optimization)  Rate-adaptive network elements (local optimization) –Adaptivity characterized by Energy Proportional Index (EPI): ratio of base power to full traffic power –Full power at full link rate C –Base power to keep network element operational –Linearization  Both approaches have engineering challenges  Opportunity for each method alone and in combination perfect rate adaptivity c c 100% EPI50% EPI25% EPI0% EPI traffic power traffic power c traffic power c 75% EPI traffic power c traffic power always on c traffic power

20. All Rights Reserved © Alcatel-Lucent 2006, ##### 20 | Presentation Title | Month 2006 Data sets Focus on service-provider networks: Rocketfuel datasets by U. Washington packet-tracing technology to discover router-level graphs routers labeled with cities, multiple routers/city

21. All Rights Reserved © Alcatel-Lucent 2006, ##### 21 | Presentation Title | Month 2006 Routing Heuristic “Iterative greedy least-power routing” motivated by [Charikar-Karagiazova] Randomly permute demands Route each demand along route that incurs least marginal cost First use incurs base power Subsequent use incurs increment cost Iterate until no improvement is observed

22. All Rights Reserved © Alcatel-Lucent 2006, ##### 22 | Presentation Title | Month 2006 Plots for x% EPI in rate-power curve 0% EPI (perfect rate adaptivity) Min-hop routing optimal always on (current state) power traffic multiplier  0% 100% c traffic power c traffic power Overprovision in capacity

23. All Rights Reserved © Alcatel-Lucent 2006, ##### 23 | Presentation Title | Month 2006 Plots for x% EPI in rate-power curve 0% EPI (perfect rate adaptivity) always on (current state) saving due to rate adaptation traffic multiplier  0% 100% Min-hop routing Power-aware routing c traffic power saving due to routing

24. All Rights Reserved © Alcatel-Lucent 2006, ##### 24 | Presentation Title | Month 2006 AS 1775 c 75% EPI traffic power

25. All Rights Reserved © Alcatel-Lucent 2006, ##### 25 | Presentation Title | Month 2006 AS 1755  For lightly loaded networks, power saving opportunities are there.  For small rate adaptivity, opportunity for routing optimization is there.

26. All Rights Reserved © Alcatel-Lucent 2006, ##### 26 | Presentation Title | Month 2006 Open Problems  What happens to non-uniform CND, where links have different capacities?  What happens to non-uniform routing with (dis)economies of scale, where each link e has a distinct cost function f e ( )?  What happens to protection version of the problem?  What is analogue of Charikar—Karagiazova for routing with (dis)economies of scale?

27. All Rights Reserved © Alcatel-Lucent 2006, ##### 27 | Presentation Title | Month 2006 Backup