Fast Matching Algorithms for Repetitive Optimization Sanjay Shakkottai, UT Austin Joint work with...
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Transcript of Fast Matching Algorithms for Repetitive Optimization Sanjay Shakkottai, UT Austin Joint work with...
Fast Matching Algorithms for Repetitive Optimization
Sanjay Shakkottai, UT Austin
Joint work with Supratim Deb (Bell Labs) and Devavrat Shah (MIT)
Outline
Refresher: MWMBackground: Switch SchedulingAlgorithm and Main ResultUnsolved Problems, Extensions, Other ApplicationsConclusions
Maximum Weight Matching in a Bipartite Graph
Weight for each edge
Weight of matching =Sum of weights of matched edges
MWM maximizes the weight of the matching
Popular algorithms for obtaining MWM are O(N3)
Scheduling in Input Buffered Switches
Slotted System Slot Duration=Packet transfer time
At each slot, an input port can deliver packet to at most one outputAn output port can receive packet from one input portThe schedule corresponds to a matching
Popular Scheduling Schemes
iSLIP (used in Cisco Routers) Low complexity and High Delay
Batch Scheduling Apply MWM once every L slots Does not provide good tradeoff between delay and complexity
MWM based on queue-lengths High complexity and low delay
Why MWM?Excellent Delay Properties
Comparable to output-buffered switches
Total queue-length grows linearly with switch size
Provides 100% throughput
Goal
Can we improve the complexity of MWM? Use matching from the previous slot Queue-lengths do not change by much in successive slots
Model and Notations
An arrival happens at an input port i and destined to output port k in a slot with probability ik Stability if and only if
qik(t) is the number of packets at input port i, destined for output port k at time t
Primal and the Dual Problem
Primal:Max
Subject to
Dual:Min
Subject to
Facts:
1. Can ignore the integrality constraint
2. There exists integral dual solutions
Key Idea
(x,r,p) optimal if xij=1 ) dij=ri+pj-qij=0 (complementary slackness CS) (x,r,p) feasible (F)
As the qij ‘s change by +1 or –1, adjust the r’s and p’s by adding +1 and –1 so that CS and F are maintained
Basic Algorithm
Suppose q11 increases by +1
If d11>0, CS and F not violated
Basic Algorithm
Suppose q11 increases by +1
If d11>0, CS and F not violated
If d11=0, add +1 to r’s and subtract –1 from p’s till CS and F are satisfied
Basic Algorithm
Suppose q11 increases by +1
If d11>0, CS and F not violated
If d11=0, add +1 to r’s and subtract –1 from p’s till CS and F are satisfied
Basic Algorithm
Suppose q11 increases by +1
If d11>0, CS and F not violated
If d11=0, add +1 to r’s and subtract –1 from p’s till CS and F are satisfied
Basic Algorithm
Suppose q11 increases by +1
If d11>0, CS and F not violated
If d11=0, add +1 to r’s and subtract –1 from p’s till CS and F are satisfied
Basic Algorithm
Suppose q11 increases by +1
If d11>0, CS and F not violated
If d11=0, add +1 to r’s and subtract –1 from p’s till CS and F are satisfied
Basic Algorithm
Suppose q11 increases by +1
If d11>0, CS and F not violated
If d11=0, add +1 to r’s and subtract –1 from p’s till CS and F are satisfied
Complexity
Run the basic algorithm for each qij that changes
Complexity is O(N2 + NE) where E=no of non-empty queues Need to take special care of nodes having zero queues
Theorem
If < 0.5, given an MWM from the previous slot, a new MWM can be computed in expected O(N2) operations
Conjectured to be true for <1 Require total queue-length to be O(N) under MWM (simulations suggest so)
Conjecture: The expected complexity is O(Nlog(N))
Extensions and Applications
Improve the complexity bound
Devise good incremental MWM algorithm for a more general graph