On the Mapping between Logical and
Physical Topologies
Yun Hou*, Murtaza Zafer†, Kang-Won Lee†,
Dinesh Verma†, Kin K. Leung*
* Imperial College
† IBM
T. J. Watson Research
© 2009 IBM Corporation
Graph Mapping Problem
b
2
1
15 Mbps
5 Mbps
a
10 Mbps
10 Mbps
10 Mbps
d
3
5 Mbps
c
Demand
Constraints
(Logical network)
(Physical network)
Question: How to map logical network to physical network such that
the demand is satisfied?
COMSNETS 2009, Bangalore
Jan 08, 2009
© 2009 IBM Corporation
Graph Mapping Problem
10 Mbps
2
1
15 Mbps
1
a
b
10 Mbps
10 Mbps
10 Mbps
5 Mbps
5 Mbps
10 Mbps
5 Mbps
3
2
d
5 Mbps
3
c
5 Mbps
Demand
Constraints
(Logical network)
(Physical network)
Question: How to map logical network to physical network such that
the demand is satisfied?
COMSNETS 2009, Bangalore
Jan 08, 2009
© 2009 IBM Corporation
Motivations
ƒ Cloud computing
ƒ Social networking
ƒ Smarter networking
ƒ Related work
– Task-processor assignment
– Wavelength assignment in optical networks (WDM)
– Multi-commodity flow optimization
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© 2009 IBM Corporation
Problem Statement
ƒ Given
– A logical graph GL = (VL, X)
– A physical graph GP = (VP, C)
ƒ Determine
– Node assignment: place 1 logical node at 1 physical node.
– Flow assignment: find a (multi-path and multi-hop) routing that
satisfies the logical requirements and physical capacities.
COMSNETS 2009, Bangalore
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© 2009 IBM Corporation
The two-step approach
ƒ To solve the problem, we take a 2-step approach
Node assignment
(NA)
Flow assignment
(FA)
NA : Find a feasible node assignment
Feasible NA: under the NA, one can find at least
one feasible routing
FA : Find a feasible routing
Feasible routing: satisfies the physical capacities
while well meets the logical requirements
When NA is given, FA is the well known Multi-Commodity Flow problem
Our contribution: conditions of NA feasibility and how to find out the feasible NA
COMSNETS 2009, Bangalore
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© 2009 IBM Corporation
Pop quiz
2
5
Logical
Network
5
10
1
4
5
5
3
?
b
7
Physical
Network
8
5
a
8
COMSNETS 2009, Bangalore
d
8
c
Jan 08, 2009
© 2009 IBM Corporation
No, because the demand cannot be met…
Logical
Network
Physical
Network
All 1-cut checks are passed
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The blue 2-cut check is failed
Jan 08, 2009
© 2009 IBM Corporation
Feasibility testing - the m-cut checks
ƒ Definitions
– Cut = a set of edges that partitions a graph into two sides
– m-Cut: a cut that partitions the graph into a set of m nodes and the other
set of N-m nodes
– Cut capacity: sum of the weight of edges in the cut
m
nodes
N-m
nodes
ƒ The m-cut feasibility check
– A given NA and a random m-cut
– if the PHY m-cut capacity ≥ the LOG m-cut capacity, the check is
passed
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Good news and bad news
ƒ Feasibility condition for NA
Consider m-cut feasibility checks for m = 1, 2, …, N/2. If
all the checks are satisfied, the particular node
assignment is feasible; vice-versa, if a particular node
assignment is feasible, it must pass all the m-cut checks.
ƒ This condition is shown to be necessary and sufficient
using max-flow theory
ƒ There are
N
C Nm =  
m 
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m-cuts!
Jan 08, 2009
© 2009 IBM Corporation
NA algorithm 1
- Random Assignment Plus Check (RAPC)
ƒ The all-cut RAPC algorithm
1) Pick up a random node assignment
2) Check all m = 1,2,…,N/2 cut checks
ƒ Complexity of RAPC
–
If two network can be mapped by some NA
• The RAPC goes over k-1 unfeasible NA’s
• Reach a feasible NA at k-th trial
• Complexity : O (2 N )
–
If two networks cannot be mapped via any NA
• The RAPC goes over all possible N! NA’s
• Complexity : O ( N !)
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Jan 08, 2009
© 2009 IBM Corporation
Catch Probability
Catch probability Pm
1.01
1
0.99
0.98
0.97
0.96
0.95
0.94
0.94
0.93
0.92
0.91
0.9
1-cut
0.992
0.999
1.
1.00
Random generated PHY
and LOG of size N and M
in the range of 2 – 10
Uniformly distributed edge
weights
2-cut
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3-cut
4-cut
5-cut
Jan 08, 2009
© 2009 IBM Corporation
NA algorithm 2
- Simplified RAPC (1-cut RAPC)
ƒ The 1-cut RAPC algorithm
1) Pick up a random node assignment
2) Check the all 1-cut check
3) If all the1-cut checks are passed, the NA is asserted to be
feasible
ƒ Complexity of 1-cut RAPC
–
If two network can be mapped by some NA
• Complexity : O(N)
– If two networks cannot be mapped via any NA
• The RAPC goes over all possible N! NA’s
• Complexity : O ( N !)
COMSNETS 2009, Bangalore
Jan 08, 2009
© 2009 IBM Corporation
NA algorithm 3
- Greedy 1-cut mapping (1-cut MA)
ƒ Instead of random assignment, we assign nodes in a
greedy manner
ƒ Greedy 1-cut algorithm
1) Sort the logical nodes in descending order of 1-cut logical
capacity β1 (1) ≥ β1 (2) ≥ L ≥ β1 (m) ≥ L ≥ β1 ( M )
2) Sort the physical nodes in descending order of 1-cut
capacity λ (1) ≥ λ (2) ≥ L ≥ λ (n) ≥ L ≥ λ ( N )
1
1
1
1
3) Starting from k = 1, map the k-th PHY node to the k-th LOG
node, if λ1 (k ) ≥ β1 (k ) ; if λ1 (k ) < β1 (k ), stop the algorithm
ƒ If a NA is formed by 1-cut MA, it must be feasible
ƒ Complexity of 1-cut MA = O(N)
COMSNETS 2009, Bangalore
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Simulation results - Complexity of the algorithms
ƒ A specific case for illustration
ƒ When M=N=4
(N, M) = (4, 4)
Complexity
Error Pr
566
523
4
0
5.3%
5.3%
All-cut RAPC
1-cut RAPC
1-cut MA
1-cut MA algorithm is dramatically faster than RAPC algorithms
The high complexity of RAPC is due to the NP-hard nature of the problem
Error-tolerant scenarios Æ choose 1-cut MA
Error-sensitive scenarios Æ choose all-cut RAPC
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Performance evaluation - Error probability
ƒ How accurate are the algorithms?
ƒ Erroneous decision: an NA is unfeasible, but the algorithm
identifies it as feasible
ƒ Error prob. = prob. of making erroneous decisions
All-cut RAPC: Pe = 0
Pe = (1-αε)(1-
1-cut MA:
P1)
Pe = (1-αε)(1-
1−α
1-cut RAPC:
P1)
1-P1 = the prob that the 1-cut
cannot catch an unfeasible NA
α
P1 = prob {the 1-cut check is
failed | the NA is unfeasible}
ε
1− ε
How effective is the 1-cut? = How big is P1?
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Simulation results - Probability of feasible mapping
Probability of feasbile mapping
ƒ For two randomly generated network, how often can they
be mapped (there is at least one NA feasible)?
High redundancy
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
3 logical nodes
5 logical nodes
7 logical nodes
1 extra node provides
large increase
Low redundancy
3
4
5
6
7
8
Number of Physical Nodes
9
10
1 extra node provides
small increase
The more redundancy → the higher prob of feasible mapping
Gain by redundancy is inversely proportional to the size of logical network
COMSNETS 2009, Bangalore
Jan 08, 2009
© 2009 IBM Corporation
Summary
ƒ This paper
– Presented a novel set of feasibility checks for node assignments
based on graph cuts
– Showed the conditions to be necessary and sufficient.
– Proposed a simple and fast algorithm for node assignment based
on 1-cut
ƒ Possible next steps
– Dynamic assignment problem
– Node assignment with capacity consideration
– Multiple LOG mapping
– Other interesting issues
COMSNETS 2009, Bangalore
Jan 08, 2009
© 2009 IBM Corporation
Acknowledgment
ƒ This research was supported through participation in the
International Technology Alliance (ITA) sponsored by
the U.S. Army Research Laboratory and the U.K. Ministry
of Defense under Agreement Number W911NF-06-30001.
COMSNETS 2009, Bangalore
Jan 08, 2009
© 2009 IBM Corporation