Yashar Ganjali High Performance Networking Group Stanford University September 17, 2003 Minimum-delay Routing

February 13, 2003 Minimum-delay Routing2 Outline 1.Network and flow model 2.Delay model 3.Problem statement 4.Previous work 5.New algorithm 6.A simple example 7.Outline of the optimality proof

February 13, 2003 Minimum-delay Routing3 Network & Flow Model Network G=(V,E) –N nodes –M links K commodities –Source s i –Destination t i –Demand d i s1s1 t1t1 s2s2 s3s3 t2t2 t3t3 d1d1

February 13, 2003 Minimum-delay Routing4 Flow Constraints Conservation of flow constraint –For any node v and commodity i d i (v)+  uv f i (uv)-  vu f i (vu) = 0 Capacity constraint –For any link uv and commodity i  i f i (uv) <= C uv uv C uv f 1 +f 2 +f 3 f 1 (uv) f 1 (vx) f 1 (wv) f 1 (vy) f 1 (vz) v

February 13, 2003 Minimum-delay Routing5 Delay (cost) Model Delay at each link –D uv = f uv /(C uv -f uv ) –Increasing –Convex u V f uv D uv

February 13, 2003 Minimum-delay Routing6 Problem Statement Goal: Minimizing the total delay in the network. Total delay =  uv D uv (f uv ) Problem: How to divide flows at each node of the network, i.e. finding routing tables.

February 13, 2003 Minimum-delay Routing7 Previous Results [Cantor 74] Linear Programming –Centralized [Gallager 77] Distributed algorithm –Network dependent [Bertsekas et al. 97] Distributed & Fast Approximation –Single commodity [Plotkin et al. 95] Distributed Multicommodity Flow Algorithm –Linear cost function Our method –Distributed –Fast convergence –Multicommodity

February 13, 2003 Minimum-delay Routing8 Relaxing Conservation of Flow Constraint We relax the conservation of flow constraint: d i (v)+  uv f i (uv)-  vu f i (vu) = g i (f,v) We call g i (f,v) the excess of commodity i at node v and flow f is called a pre-flow. We will use this quantity to find points of high pressure in the network.

February 13, 2003 Minimum-delay Routing9 Minimum-delay Routing Potential Function  We define  1 =  uv,i exp(  g i (f,uv)/d i ) and  2 =  uv D uv (f uv )/B  1 measures how close the current pre- flow f is to a flow.  2 measures how close the cost of the current flow is to the budget B. We let  =  1 +   1 x  2

February 13, 2003 Minimum-delay Routing10 Minimum-delay algorithm: Starting from zero flows our goals is to minimize   O(  -1 log(m  -1 )) phases  O(  -1 ) iterations –Increase capacities by a factor of  –Increase demands by a factor of  –Update the amount of excess at each node –BALANCE EXCESSES Rescale capacities and demands Update  if needed Our Algorithm

February 13, 2003 Minimum-delay Routing11 Balancing Excesses Each node divides its excess evenly among adjacent links. Each link locally minimize . uv C uv uv

February 13, 2003 Minimum-delay Routing12 Example 12 34

February 13, 2003 Minimum-delay Routing13 Example 12 34

February 13, 2003 Minimum-delay Routing14 Example 12 34

February 13, 2003 Minimum-delay Routing15 Example 12 34

February 13, 2003 Minimum-delay Routing16 Outline of the Proof 1.If  is small enough we are close to the optimal solution. 2.In each ITERATION the increase in  is small. 3.At the end of each PHASE the amount of  is divided by two.

February 13, 2003 Minimum-delay Routing17 Small   Close to Optimal We know  =  1 + …x  2  is small means both  1 and  2 are small.  1 is small means conservation of flows is almost satisfied.  2 is small means cost is close to optimal. We can show that if  < (1+  )  at least (1-  ) of each demand is satisfied and our cost is at most (1+  ) times the optimal cost

February 13, 2003 Minimum-delay Routing18 Increase in  is small Consider an optimal flow f* We have f* <= C and  f* <=  C Therefore, in each iteration we can have  f =  f* We can show the increase in  is small in this case (cost function is convex and has a bounded derivative). Therefore, if we minimize  the amount of increase is small.

February 13, 2003 Minimum-delay Routing19 In each phase  decreases by a factor < 1 At the end of each phase we divide all demands and flows by two. Therefore excesses and delays are reduced. We can show this reduces  by a factor which is less than 1.

February 13, 2003 Minimum-delay Routing20 Complexity of the Algorithm Original  is bounded. We know  is multiplied by a factor less than one in each phase. We can conclude that the algorithm has O(  -1 log(m  -1 )) phases. Each phase consists of O(  -1 ) iterations. Running Time: O*(  -2  -2 KM 2 N) Improved running time: O*(  -3  -3 KMN 2 )

February 13, 2003 Minimum-delay Routing21 Future Work Sensitivity and stability analysis –How sensitive the algorithm is to perturbations? Cost function –Realistic cost function Implementations issues –Where? –How to incorporate with existing routing protocols?

February 13, 2003 Minimum-delay Routing22 Thank you!