June 10, 2003STOC 2003 Optimal Oblivious Routing in Polynomial Time Harald Räcke Paderborn Edith...
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Transcript of June 10, 2003STOC 2003 Optimal Oblivious Routing in Polynomial Time Harald Räcke Paderborn Edith...
June 10, 2003 STOC 2003
Optimal Oblivious Routing in Polynomial Time
Harald Räcke Paderborn
Edith CohenAT&T Labs-Research
Yossi Azar Amos Fiat Haim KaplanTel-Aviv University
June 10, 2003 STOC 2003
Routing, Demands, Flow, Congestion
• Routing: a unit s-t flow for each origin-destination pair:
fab(i,j) 0 routing for OD pair a,b on edge (i,j)
• Demands: Dab >= 0 for each OD pair a,b• Flow on edge e=(i,j) when routing D with f: flow(e,f,D)=ab fab(i,j) Dab
Congestion on edge e=(i,j) when routing D with f: cong(e,f,D)=flow(e,f,d)/capacity(e)
June 10, 2003 STOC 2003
Congestion, Oblivious Routing
• Congestion of demands D with routing f: cong(f,D)= maxe cong(e,f,D)
• Optimal routing for D: min possible congestion: opt(D) = minf cong(f,D)
• Oblivious ratio of f: obliv(f)= maxD cong(f,D)/opt(D)• Optimal Oblivious Ratio of G: obliv-opt(G)=minf obliv(f)
June 10, 2003 STOC 2003
Example
1
4
3
2 1 2 3 4
1 1 1 1 1
2 1 1 1 1
3 1 1 1 1
4 1 1 1 1
Routing f: Route each OD pair on direct edgeDemands D: unit demand for all pairscong(e,f,D)=2 for all edgesThus, cong(f,D)=2 (f is optimal for D)
June 10, 2003 STOC 2003
Example
1
4
3
2 1 2 3 4
1 0 0 1 0
2 0 0 0 0
3 0 0 0 0
4 0 0 0 0
Routing f: Route each OD pair on direct edgeDemands D: unit demand for ONE paircong(e,f,D)=1 for used edge, 0 otherwise.Thus, cong(f,D)=1 (f is NOT optimal for D)
June 10, 2003 STOC 2003
Example
1
4
3
2 1 2 3 4
1 0 0 1 0
2 0 0 0 0
3 0 0 0 0
4 0 0 0 0
Routing f: Route each OD pair on the 3 1,2 hop pathsDemands D: unit demand for one paircong(e,f,D)=1/3 for used edgescong(f,D)=1/3 “direct” routing has oblivious ratio >= 3
June 10, 2003 STOC 2003
Example
1
4
3
2 1 2 3 4
1 1 1 1 1
2 1 1 1 1
3 1 1 1 1
4 1 1 1 1
Routing f: Route each OD pair on the 3 1,2 hop pathsDemands D: unit demand for all pairscong(e,f,D)=10/3 for all edges (10 pairs use each edge)cong(f,D)=10/3 (f is NOT optimal for D) 2-hop routing has oblivious ratio >= 5/3
June 10, 2003 STOC 2003
Optimal oblivious routing
• Balances performance across all demand matrices.
• Why is it interesting?– Demands are dynamic– Changes to routing are hard– Sometimes we don’t know the
demands
June 10, 2003 STOC 2003
History• Specific networks, VC routing
– Raghavan/Thompson 87…Aspnes et al 93– Valiant/Brebner 81: Hypercubes
• Räcke 02: Any undirected network has an oblivious
routing with ratio O(log^3 n)!!
• Questions: – Poly time algorithm. – Get an optimal routing. – Directed networks?
June 10, 2003 STOC 2003
LP for Optimal Oblivious Ratio
• Minimize r s.t. fab(i,j) is a routing (1-flow for every a,b)
For all demands Dab >= 0 which can be routed with congestion 1:
For all edges e=(i,j) : (cong(e,f,D) <= r)
ab fab(i,j) Dab/capacity(e) <= r
But… Infinite number of constraints use Ellipsoid
June 10, 2003 STOC 2003
Separation Oracle• Given a routing fab(i,j), find its oblivious ratio
and a demand matrix D which maximizes the ratio (the “worst” demands for f).
For each edge e=(i,j) solve the LP (and then take the maximum over these LPs):
• Maximize ab fab(i,j) Dab/capacity(e)
• gab(i,j) is a flow of demand Dab >= 0
• For all edges h, gab(h) <= capacity(h)
** Need to insure that the numbers don’t grow too much
June 10, 2003 STOC 2003
Directed Networks (Asymmetric link
capacities)• Our algorithm computes optimal oblivious
routing for undirected and directed networks.• Räcke’s O(log^3 n) bound applies only to
undirected networks. • We show that some directed networks have
optimal oblivious ratio of (sqrt(n)).
June 10, 2003 STOC 2003
t
{i,j}
( )k2
i j kk/2
Any flow from {i,j} to t is split on the two possible paths.Thus, a routing is determined by the split ratio for each {i,j}.For any routing f, there is at least one mid-layer node i thatroutes >= half the flow for >= k/2 pairs.
“Bad” demands for f: 1 on pairs {i,*} to t, 0 otherwise.congestion is >= k/4 with f. But optimal is 1 (via alternate paths)
June 10, 2003 STOC 2003
Extensions
• Subset of OD pair demands• Ranges of demands• Node congestion• Limiting dilation
June 10, 2003 STOC 2003
Follow up/subsequent work
• Polytime construction of a Räcke-like decomposition
(two SPAA 03 papers: Harrelson/Hildrum/Rao Bienkowski/Korzeniowski/Räcke)
• More efficient polynomial time algorithm (Applegate/Cohen SIGCOMM 03)
• Oblivious routing on ISP topologies (Applegate/Cohen SIGCOMM 03)
• Online oblivious routing (Bansal/Blum/Chawla/Meyerson SPAA 03)
June 10, 2003 STOC 2003
Open Problems
• Tighten Räcke’s bound O(log^3 n) (log n) (Currently, O(log^2 n log log n) by Harrelson/Hildrum/Rao 03)• Single source demands: Is there a constant optimal
oblivious ratio ?