Network Coding in a Multicast Switch

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Network Coding in a Multicast Switch

• Jay Kumar Sundararajan
• joint work with

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The Multicast Switch Model
Line rate 1 packet per slot
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The Multicast Switch Model
Crossbar constraints

• INPUT
• Can transmit only one packet at a time
• Can send it to many outputs

• OUTPUT
• Can receive from at most one input at a time

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The Multicast Switch Model
I
O
Example i 3 J 2,3,4
Flow
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Network Coding for Higher Throughput
Conclusion If inputs are allowed to transmit
linear combinations of packets of a flow, i.e.,
linear intra-flow network coding is allowed, then
we can get a strictly higher throughput.
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Questions

• Given the rates, find a schedule using rate
  decomposition
• Unicasts Birkhoff-von Neumann decomposition
  (Chang et al. 00)
• Understand the rate region
• Without network coding Convex hull of modified
  departure vectors (Marsan et al. 03)
• What kind of traffic patterns are difficult to
  handle?

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Enhanced Conflict Graph
TRAFFIC PATTERN
ENHANCED CONFLICT GRAPH
ENHANCED CONFLICT GRAPH
V One vertex for every subflow G (
V, E ) E Connect two vertices if the subflows
conflict Weights Rates of the subflows
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Enhanced Conflict Graph
TRAFFIC PATTERN
ENHANCED CONFLICT GRAPH
ENHANCED CONFLICT GRAPH
Valid service configuration Stable set (A
set of vertices no two of which are connected)
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Enhanced Conflict Graph
TRAFFIC PATTERN
ENHANCED CONFLICT GRAPH
ENHANCED CONFLICT GRAPH
Time-sharing of valid service configurations
(Schedule) Convex combination of stable sets
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Enhanced Conflict Graph
TRAFFIC PATTERN
ENHANCED CONFLICT GRAPH
ENHANCED CONFLICT GRAPH
Rate region Convex hull of stable sets
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Rate Region

• Virtual queue for a subflow (i, J, k) (similar
  to Ho-Viswanathans backpressure algorithm)
• Maintains the number of linearly independent
  degrees of freedom yet to be delivered from input
  i to output k for flow (i, J), for each
• Arrival Whenever there is an arrival for flow
  (i, J)
• Departure When an innovative packet from flow
  (i,J) is conveyed to output k

Result Let r be the vector of arrival rates to
the virtual queues. r is achievable iff r is in
the stable set polytope of the enhanced conflict
graph.
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Proof outline Achievability
Schedule over a frame
M slots
(n,k) MDS code
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Understanding the traffic pattern

• Computing offline schedule Fractional weighted
  coloring of ECG
• If ECG is perfect, then
• Polynomial algorithm for finding schedule
• Entire admissible region is achievable
• Minimum speedup needed to sustain given rates
  Fractional weighted chromatic number of ECG

Enhanced conflict graph (ECG) summarizes the
nature of the traffic pattern
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How much benefit?
2 x 3 switch all possible flows
RATE REGION WITHOUT CODING
without coding 1 with coding (i.e. no
speedup)
Need speedup of
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Online max-weight algorithm

• ECG vertex weight the size of the virtual queue
  corresponding to a subflow (i.e., number of
  degrees of freedom yet to be delivered)
• Maximum weighted stable set algorithm In each
  slot
• Compute the maximum weighted stable set in the
  ECG
• Activate this set of queues
• For queues of the same flow, compute a linear
  combination that is simultaneously innovative to
  all recipients (Requires large enough field size)
• We can use similar techniques as in TE92 to
  prove stability of the virtual queues
• Using a batching process, we can stabilize the
  physical buffers as well

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Simulation results
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Conclusions

• Network coding in multicast switches gives
• Larger rate region than fanout splitting
• Insightful graph theoretic characterization of
  the rate region
• Scope for use of results/algorithms from graph
  theory literature
• Scheduling algorithm, throughput, speedup

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Two Service Strategies