Belief-Propagation Assisted Scheduling in Input-Queued Switches S. Atalla 1, D. Cuda 2, P. Giaccone 1, M. Pretti 2 1 Politecnico di Torino 2 Italian National Research Council Hot Interconnects 2010 August 2010

Outline  Background motivations  System model  Basic belief-propagation algorithm for MWM  Assisted scheduling  Belief-propagation for assisted scheduling  Performance evaluation  Hardware implementation  Conclusions 2Hot Interconnects 2010

Background motivations  Internet traffic is steadily increasing  Routers and switches require to process growing amount of data faster and faster  Input Queued (IQ) switches can be considered as a reference architecture  Memory speed = line rate  IQ switches require suitable scheduling algorithms that  Ensure good performance (throughput, delay,)  Run fast (few ns to take each scheduling decision)  Are implementable in hardware (HW) 3Hot Interconnects 2010

System model  NxN crossbar with Virtual Output Queuing  one FIFO queue for each input output pair  total of N 2 queues  Synchronous architecture:  time is slotted  fixed sized packets Hot Interconnects 20104

Scheduling algorithm  At each timeslot, the scheduler selects a set of head-of-line packets compatible with the crossbar constraint:  At the most one packet can be transferred to/from each output/input port  equivalent to choose a matching in a bipartite graph  Inputs: lengths of the VOQ  Outputs: matching described through binary variable: x ij =1 iff input i transfer packet to output j q ij Scheduler (MWM, iSLIP, iLQF, …) x 00 =1 x 33 =0 5Hot Interconnects

Scheduling algorithm dichotomy  Maximum Weight Matching (MWM) is  Optimal in terms of performance  Difficult to implement in HW  O(N 3 ) operations, difficult to be parallelized  Heuristic algorithms mimicking MWM  E.g., iSLIP, iLQF, WFA (and many others)  Efficient to be implemented in HW  e.g., iSLIP was implemented in CISCO serie  Possible traffic losses under critical traffic patterns Hot Interconnects 20106

Basic belief-propagation for MWM  Recently, Belief-Propagation (BP) algorithm has been proposed to solve MWM problem [1,2]  BP algorithms are message passing algorithms firstly conceived to study Graphical Models (GMs)  GMs combine graphic theory and probability theory  BP is exact for MWM over bipartite graph (see [1]), but  To ensure convergence, MWM must be unique  Small random noise can be added to queue length  It takes O(N 3 / ε ) to converge  ε : difference in weight between the first two heaviest matchings  not known a priori Hot Interconnects [1]M. Bayati, D. Shah, and M. Sharma, “Max-product for maximum weight matching: Convergence, correctness, and LP duality,” Information Theory, IEEE Transactions on, vol. 54, no. 3, pp. 1241–1251, Mar [2]M. Bayati, B. Prabhakar, D. Shah, and M. Sharma, “Iterative scheduling algorithms,” in INFOCOM 2007, IEEE, , pp. 445 –453.

Basic belief-propagation for MWM 0 0 8Hot Interconnects

Basic belief-propagation for MWM 0 0 9Hot Interconnects

Basic belief-propagation for MWM Hot Interconnects

Basic belief-propagation for MWM Hot Interconnects

Basic belief-propagation for MWM 0 0 After convergence, each output it is matched to the input associated with the largest message. 12Hot Interconnects 2010

Assisted scheduling  Our major contribution is the introduction of the concept of assisted scheduling:  Instead of the queue length, scheduling algorithms are modified to use messages computed by BP as weights  We show that BP assisted scheduling boosts performance of existing schedulers while keeping backward compatibility 13Hot Interconnects 2010

Assisted scheduling  We introduce the Belief-Propagation Message-Processing module between the VOQs and the Scheduler  BP-MP computes message values as a function of the queue length Q(t), based on a BP algorithm  The scheduler works in the usual way, but scheduling decisions are based on the messages F(t) computed by the BP-MP module instead that on Q(t)  F(t) can be see as a correction of the VOQ lengths Q(t) BP-MP few I Scheduler 14Hot Interconnects 2010

Assisted scheduling  BP propagation has been improved with:  Relaxation of the MWM uniqueness constraint  We do not need BP to converge anymore  No random noise  Finite (and small) number of iterations  Integer number representation  Memory  Self-Asynchronous update Hot Interconnects

Messages for assisted scheduling  It runs for a fixed (and small) number of iterations I Hot Interconnects Messages are bounded  Messages  represented through integer numbers  Same numerical range of the queue length (around log 2 Q max bits)

Memory for assisted scheduling  Queues exhibit a strong correlation that is reflected in the message dynamics  Queue length can change at the most by 1 at each timeslot Memory: messages are initialized to the last computed messages Memory speeds up convergence 17Hot Interconnects 2010

Self-asynchronous update for assisted scheduling  Studies in BP showed that messages updated in a random sequential order are beneficial for the convergence (asynchronous update)  Not easy to implement in HW Self-asynchronous update: exploits randomness of the arrival process updates only messages associated with queues which have changed from the previous timeslot mimics asynchronous update 18Hot Interconnects 2010

Scheduling algorithms  iLQF vs. BP assisted iLQF (BP-iLQF)  Distributed greedy algorithm  Each input (each output) is equipped with an arbiter which selects output (input) associated with the longest queue  Greedy MWM (GMWM) vs. BP assisted GMWM (BP-GMWM)  centralized scheduling, iterating N times  at each iteration it selects the unmatched input/output couple associated with the longest queue  iSLIP  as iLQF, but sending only a binary information (queue empty/not-empty) Hot Interconnects

Performance evaluation settings  Simulation settings:  Traffic patterns:  Critical traffic pattern 20Hot Interconnects 2010

Performance evaluation results BP assisted scheduling improves performance (I=3) Memory No Memory 21Hot Interconnects 2010 Self-asynchronous Synchronous Asynchronous

Hardware design: General overview 2N modules running in parallel BP-MP Backward messages Forward messages When n=I, IM sends F(t) to the scheduler IM and OM perform the same operations VOQ Scheduler 22Hot Interconnects 2010

Hardware design: IM details Self-asynchronous: if w ij (t)≠ w ij (t-1) e ij =1 else e ij =0 Flags associated with VOQ at input i Memory: registers storing messages computed during the previous timeslot Max operation Tournament implementation  log 2 (N-1)  stages and (N-2) comparisons c used to select between 0 and the result of the subtraction operation Subtraction operation When n=I messages are sent to the scheduler 23Hot Interconnects 2010 N registers of size log 2 Q max

Conclusion  We proposed BP assisted scheduling to boost performance of existing scheduling algorithms keeping backward compatibility  BP runs for few iterations  We simplified and improved basic BP algorithm:  Relaxation of MWM uniqueness constraint  Integer messages (backward compatibility)  Message memory  Self-asynchronous update  We provided a high-level description of a possible HW implementation of the BP-MP:  BP-MP can be efficiently implemented in HW and it is compatible with existing implementations Hot Interconnects

Belief-Propagation Assisted Scheduling in Input-Queued Switches S. Atalla 1, D. Cuda 2, P. Giaccone 1, M. Pretti 2 1 Politecnico di Torino 2 Italian National Research Council Hot Interconnects 2010 August 2010 Any questions? Thank you for your attention! 25Hot Interconnects 2010

Example: MWM computation over a tree  Node “1” must decide to add or not edge (1,2) to the matching  Node “1” takes its decision based on the information provided only by nodes belonging to its neighborhood  E.g., Node “2” sends to “1” two messages:  : MWM of the sub-tree rooted at “2” comprising (2,1) given that (2,1) is part of the MWM rooted at “1”  : MWM of the sub-tree rooted at 2 comprehending (2,1) given that (2,1) is part of the MWM rooted at “1” Take or not to take (2,1)? w 32 w 21 w 42 w 61 w 71 26Hot Interconnects 2010

Example: MWM computation over a tree Message definitions:  If (2,1) is part of the MWM, then (3,2), (4,2), (5,2) can not be in the MWM  if (2,1) is not the MWM, then at the most one (or none) among (3,2), (4,2), (5,2) can part of the MWM  It is possible to reduce the number of exchanged messages combining into a single message w Hot Interconnects 2010

Example: MWM computation over a tree Node “1” decision:  Node “1” adds edge (1,2) to the MWM if:  or equivalently  Take or not to take (2,1)? w 32 w 21 w 42 w 61 w 71 28Hot Interconnects 2010

Graphical models  BP algorithms are message passing algorithms conceived firstly to study Graphical Models (GMs)  GMs are a “marriage” between probability theory and graph theory lo direi solo a voce, non significa niente qui  GMs are becoming a powerful tool in several fields of science (AI, speech recognition, coding/decoding, bioinformatics) to compute marginal probabilities and maximum a posteriori probability (max-product algorithm)  “BP” and “max-product “ are usually simply referred as “BP” since computing the maximum a posteriori probability requires first to compute the marginal distributions io questa frase non l’ho capita e mi pare rischiosissima!!! 29Hot Interconnects 2010

VOQBP-MPScheduler 30Hot Interconnects 2010

Scheduler: iLQF  If the MWM is unique, BP assisted iLQF, running with weights computes exactly the MWM 31Hot Interconnects 2010

Performance evaluation: results BP assisted scheduling improves performance (I=3) Average delays : delays BP-iLQF/GWM are at the most 1.37 times delays of iLQF/GWM. Memory No Memory 32Hot Interconnects 2010 Self-asynchronous Synchronous Asynchronous

Basic belief-propagation for MWM Hot Interconnects