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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

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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

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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

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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

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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 2010 33 33 00 0 0

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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 12000 serie Possible traffic losses under critical traffic patterns Hot Interconnects 20106

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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 20107 [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. 2008. [2]M. Bayati, B. Prabhakar, D. Shah, and M. Sharma, “Iterative scheduling algorithms,” in INFOCOM 2007, IEEE, 6-12 2007, pp. 445 –453.

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Basic belief-propagation for MWM 0 0 8Hot Interconnects 2010 33

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Basic belief-propagation for MWM 0 0 9Hot Interconnects 2010 33

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Basic belief-propagation for MWM 0 0 10Hot Interconnects 2010 33

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Basic belief-propagation for MWM 0 0 11Hot Interconnects 2010 33

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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

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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

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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

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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 201015

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Messages for assisted scheduling It runs for a fixed (and small) number of iterations I Hot Interconnects 201016 Messages are bounded Messages represented through integer numbers Same numerical range of the queue length (around log 2 Q max bits)

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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

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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

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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 201019

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

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

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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

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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

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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 201024

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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

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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” 1 2 3 4 5 6 7 Take or not to take (2,1)? w 32 w 21 w 42 w 61 w 71 26Hot Interconnects 2010

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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 1 2 3 4 5 w 21 1 2 3 4 5 27Hot Interconnects 2010

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Example: MWM computation over a tree Node “1” decision: Node “1” adds edge (1,2) to the MWM if: or equivalently 1 2 3 4 5 6 7 Take or not to take (2,1)? w 32 w 21 w 42 w 61 w 71 28Hot Interconnects 2010

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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

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VOQBP-MPScheduler 30Hot Interconnects 2010

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

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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

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Basic belief-propagation for MWM 0 0 33Hot Interconnects 2010 33

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