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