1. A Polynomial-time Tree Decomposition for Minimizing Congestion
Chris Harrelson
Kirsten (Kris) Hildrum
Satish Rao

2. Multicommodity flow T2 T1 S1 T3 Congestion = 2 S3 S2
Cloud is a graph. (Undirected graph)
Sources, Sinks. Route flow from source to sink.
Show paths.
Notice two paths use same edge. Define congestion.
NP-hard to compute minimum congestion paths, but can get a log n approximation (Leighton Rao).
Transition:
We want something harder – oblivious routing.

3. Oblivious Routing T2 T1 S1 T3 S3 S2 Show paths for flows.
Measure of success is comparison to optimal. S3 S2

4. Problem Description Given A weighted graph
Find oblivious routing scheme
That is, a set of paths for each Si Ti pair
Competitive Analysis 
For any set of demands, bound ratio of our congestion to congestion of optimal flow

5. Some results Leiserson ’85 – Fat trees implies O(log n) on grids
Räcke ’02 – Ratio of O(log3 n), but construction not polytime
Bienkowski, Korzeniowski, and Räcke ‘03 – O(log4 n) construction and simpler O(log3 n) existence
Harrelson,H,Rao ‘03 O(log2 n log log n) construction
Azar, Cohen, Fiat, Kaplan and Räcke ‘03 – Linear program to find optimal oblivious scheme, not hierarchical
Azar – No new bound, but give optimal scheme, and appeal to Raecke to for bound on ratio.
Remember to say that at time of submission, ours was essentially the same as theirs.

6. Algorithm Outline
Edge weights are size of cut

7. Tree provides lower bound
Each node of the tree is a group of nodes in graph
Congestion on tree is lower bound for congestion on graph
Need to mimic tree on graph with low congestion
Best tree has smallest cuts

8. Example decomposition and treeV S1 S2 S3 S4 3 2 1 4 5 6 7 S3 6 7 1 S4 2 4 5 S2 3 S1
Next slide: Using the tree

9. Ti
If I had a title, this slide would be “using the tree” Si

10. Algorithm
Decompose graph into clusters
A step moves flow from border of one cluster to border of parent cluster
First to border of sibling clusters
Then to border of parent
The rest of this talk: creating clusters that allow for this flow with low-congestion
Result for O(log3 n)
Better accounting for O(log2n log log n)

11. Need Uniform border flow
Let w(u) be the border weight of u.
Then the flow from u to v is:
Border nodes can all communicate.
If k border nodes, each sends flow of 1/k to every border node.
Getting rid of border nodes makes the problem easier.
Decompose, and check if we can route the flow, if not fix decomposition.

12. Main Tool: Flow or Cut Given a set of demands we can find either
A way of routing the demands with congestion O(log2 n), or
A cut such that
(Follows from eg. Leighton-Rao)
No flow exists with congestion,
Red = demand, black = graph

13. Merging Need Uniform flow between border nodes
Start with nodes in sets of their own
Is there a uniform border flow with congestion O(log2 n) ?
Yes, done
No, find cut
Merge smaller side, repeat
Number of border nodes decreases
Start with a set and we are going to refine it.
One level partition. Very optimistic. Everyone is a border node.
Previous talk enforced an additional condition on
After merge, this is a good cut; only want to sent one unit of flow out, and have one edge to do so.

14. But What If…
Border nodes from higher levels still border nodes after merge
Make no progress!
Called: Bad Child Event
Child said “No Way” to parent.

15. Bandwidth sets Child can’t do anything
Give up and send offending cut back to parent
Solves child’s problem, but creates problem for higher level decomposition
Could happen repeatedly!
Cut, flow gap. Gave our children freedom, and they disobeyed.
Need other colors. Green node will be border node will be upper level one, now extra blue nodes need to talk.
The red nodes could talk, but new blue node,
We fail, so make cut at upper level, cloud of clouds, with new blue nodes.
My advisor calls this a bad child event, but maybe “bad parent event” would be better, since the parent has given the child an impossible task. P

16. Border Nodes from Bad Children
Bad Child Event creates new border nodes
By construction, red nodes can talk to red nodes in other pieces
Use max flow on smaller side to connect light blue to red
Number of new border nodes is factor of log n smaller than number of old
If max flow doesn’t work, we can find a cut where it does. Move border in picture?
Mention pileup.
Number of new nodes is *much smaller*

17. Bounding Pileup from Bad Child Events
extra load for every bad child event
Only log n bad child events
Total pileup:
Lots more red nodes than black nodes because of the way we set our congestion bounds.

18. Total congestion Max flow: O(log n)
2 each time edge in max flow
In max flow at most log n times
Each child is at most ½ of parent, so O(log n) depth
One multicommodity flow per level, each with congestion O(log2 n)
Total: O(log3 n) + O(log n) = O(log3 n)
Can get O(log2 n log log n) by charging based on piece size

19. Observations Bottom partition has done a lot more work!
allow more congestion in this case
Better accounting
Worst case no better
So sum congestion over all levels

20. Getting down to (log2n log log n)
Need Uniform flow between border nodes
Start with nodes in sets of their own
Is there a uniform border flow with congestion O(log2 n) ?
Yes, done
No, find cut
Merge smaller side, repeat
Number of border nodes decreases
δ(S)log n
Start with a set and we are going to refine it.
One level partition. Very optimistic. Everyone is a border node.
Previous talk enforced an additional condition on
After merge, this is a good cut; only want to sent one unit of flow out, and have one edge to do so. S

21. New edge congestion
If P is parent, and S is current set, allow congestion
If an edge is in sets S0 to Sm, then total congestion is

22. Overload from bad child events
O(log3 n)
O(log2 log log n)
On a bad child event, new border nodes route via original border nodes
Number of new nodes was 1/log n, so to bound overload:
Same scheme, but number of new border nodes is
With each division, |P|/|S| increases
Total of log n possible divisions
 Overload still bounded

23. Conclusions/Future Work
Applications to preconditioners
Maggs, Miller, Parekh, Ravi and Wu
Notes
Decomposition that makes internal connections approximately equal to external connections
Use for finding communities in graphs?
“That’s nice, but the answer should be log n.” (Edith Cohen)
Mention R.
We sent this paper to Cohen, and this is what she said…

24. The End