Power-aware Manhattan routing on chip multiprocessors Anne Benoit 1 - PowerPoint PPT Presentation

Framework Theoretical results Heuristics Simulations Manhattan vs XY; single source and destination Theorem Given that q = O ( p ) , an upper bound of P XY / P max is in O ( p ) . Pitt Benoit, Melhem, Renaud, Robert Power-aware Manhattan routing on CMPs 11 / 34

Framework Theoretical results Heuristics Simulations Manhattan vs XY; single source and destination Theorem Given that q = O ( p ) , an upper bound of P XY / P max is in O ( p ) . K : sum of all communications K (1) k : the sum of the γ i that cross D (1) k In this case, K (1) = K for each k k Pitt Benoit, Melhem, Renaud, Robert Power-aware Manhattan routing on CMPs 11 / 34

Framework Theoretical results Heuristics Simulations Manhattan vs XY; single source and destination Theorem Given that q = O ( p ) , an upper bound of P XY / P max is in O ( p ) . K : sum of all communications K (1) k : the sum of the γ i that cross D (1) k In this case, K (1) = K for each k k P XY = ( p + q ) × K α Pitt Benoit, Melhem, Renaud, Robert Power-aware Manhattan routing on CMPs 11 / 34

Framework Theoretical results Heuristics Simulations Manhattan vs XY; single source and destination Theorem Given that q = O ( p ) , an upper bound of P XY / P max is in O ( p ) . K : sum of all communications K (1) k : the sum of the γ i that cross D (1) k In this case, K (1) = K for each k k P XY = ( p + q ) × K α Lower bound on P max . Ideal sharing of one communication: Pitt Benoit, Melhem, Renaud, Robert Power-aware Manhattan routing on CMPs 11 / 34

Framework Theoretical results Heuristics Simulations Manhattan vs XY; single source and destination Theorem Given that q = O ( p ) , an upper bound of P XY / P max is in O ( p ) . p − 1 � K (1) � α q − 1 � K (1) � α � � k k P max ≥ 2 k + (2 p − 1) 2 k 2 p − 1 k =1 k = p q + p − 2 � � α K (1) � k + 2( q + p − k − 1) , 2( q + p − k − 1) k = q Pitt Benoit, Melhem, Renaud, Robert Power-aware Manhattan routing on CMPs 11 / 34

Framework Theoretical results Heuristics Simulations Manhattan vs XY; single source and destination Theorem Given that q = O ( p ) , an upper bound of P XY / P max is in O ( p ) . p − 1 � K (1) � α q − 1 � K (1) � α � � k k P max ≥ 2 k + (2 p − 1) 2 k 2 p − 1 k =1 k = p q + p − 2 � � α K (1) � k + 2( q + p − k − 1) , 2( q + p − k − 1) k = q � p k =1 k 1 − α ≥ K (1) = K and � p − 1 1 dx / x α − 1 k Pitt Benoit, Melhem, Renaud, Robert Power-aware Manhattan routing on CMPs 11 / 34

Framework Theoretical results Heuristics Simulations Manhattan vs XY; single source and destination Theorem Given that q = O ( p ) , an upper bound of P XY / P max is in O ( p ) . p − 1 � K (1) � α q − 1 � K (1) � α � � k k P max ≥ 2 k + (2 p − 1) 2 k 2 p − 1 k =1 k = p q + p − 2 � � α K (1) � k + 2( q + p − k − 1) , 2( q + p − k − 1) k = q � p k =1 k 1 − α ≥ K (1) = K and � p − 1 1 dx / x α − 1 , hence k � 1 1 q − p � � 1 − p 2 − α � P max ≥ K α 2 × + . 2 α − 1 2 − α (2 p − 1) α − 1 Pitt Benoit, Melhem, Renaud, Robert Power-aware Manhattan routing on CMPs 11 / 34

Framework Theoretical results Heuristics Simulations Manhattan vs XY; single source and destination Theorem Given that q = O ( p ) , an upper bound of P XY / P max is in O ( p ) . p − 1 � K (1) � α q − 1 � K (1) � α � � k k P max ≥ 2 k + (2 p − 1) 2 k 2 p − 1 k =1 k = p q + p − 2 � � α K (1) � k + 2( q + p − k − 1) , 2( q + p − k − 1) k = q � p k =1 k 1 − α ≥ K (1) = K and � p − 1 1 dx / x α − 1 , hence k � 1 1 q − p � � 1 − p 2 − α � P max ≥ K α 2 × + . 2 α − 1 2 − α (2 p − 1) α − 1 Altogether, P max = O ( K α ) and P XY = O ( p × K α ), hence the result. Pitt Benoit, Melhem, Renaud, Robert Power-aware Manhattan routing on CMPs 11 / 34

Framework Theoretical results Heuristics Simulations Manhattan vs XY; single source and destination Theorem The upper bound of P XY / P max in O ( p ) is tight. Pitt Benoit, Melhem, Renaud, Robert Power-aware Manhattan routing on CMPs 12 / 34

Framework Theoretical results Heuristics Simulations Manhattan vs XY; multiple sources and destinations Theorem Given that q = O ( p ) , an upper bound of P XY / P max is in O ( p α − 1 ) . Theorem The upper bound of P XY / P max in O ( p α − 1 ) can be achieved with a 1 - MP routing on a square CMP. Pitt Benoit, Melhem, Renaud, Robert Power-aware Manhattan routing on CMPs 13 / 34

Framework Theoretical results Heuristics Simulations NP-completeness of Manhattan routing Theorem Finding a s- MP routing that minimizes the total power consumption while ensuring that link bandwidths are not exceeded is a NP-complete problem. Pitt Benoit, Melhem, Renaud, Robert Power-aware Manhattan routing on CMPs 14 / 34

Framework Theoretical results Heuristics Simulations Outline of the talk Framework 1 Theoretical results 2 Heuristics 3 Simulations 4 Pitt Benoit, Melhem, Renaud, Robert Power-aware Manhattan routing on CMPs 15 / 34

Framework Theoretical results Heuristics Simulations Summary of the heuristics Simple greedy ( SG ): greedily assigns communications, hop by hop, on the least loaded link. Pitt Benoit, Melhem, Renaud, Robert Power-aware Manhattan routing on CMPs 16 / 34

Framework Theoretical results Heuristics Simulations Summary of the heuristics Simple greedy ( SG ): greedily assigns communications, hop by hop, on the least loaded link. Improved greedy ( IG ): virtually pre-assigns communications onto links, then almost like SG . Pitt Benoit, Melhem, Renaud, Robert Power-aware Manhattan routing on CMPs 16 / 34

Framework Theoretical results Heuristics Simulations Summary of the heuristics Simple greedy ( SG ): greedily assigns communications, hop by hop, on the least loaded link. Improved greedy ( IG ): virtually pre-assigns communications onto links, then almost like SG . Two-bend ( TB ): for each communication, chooses the best path with two bends. Pitt Benoit, Melhem, Renaud, Robert Power-aware Manhattan routing on CMPs 16 / 34

Framework Theoretical results Heuristics Simulations Summary of the heuristics Simple greedy ( SG ): greedily assigns communications, hop by hop, on the least loaded link. Improved greedy ( IG ): virtually pre-assigns communications onto links, then almost like SG . Two-bend ( TB ): for each communication, chooses the best path with two bends. XY improver ( XYI ): starts from XY assignment, and moves communications from the highest loaded link. Pitt Benoit, Melhem, Renaud, Robert Power-aware Manhattan routing on CMPs 16 / 34

Framework Theoretical results Heuristics Simulations Summary of the heuristics Simple greedy ( SG ): greedily assigns communications, hop by hop, on the least loaded link. Improved greedy ( IG ): virtually pre-assigns communications onto links, then almost like SG . Two-bend ( TB ): for each communication, chooses the best path with two bends. XY improver ( XYI ): starts from XY assignment, and moves communications from the highest loaded link. Path remover ( PR ): virtually pre-assigns communications onto links, and iteratively prevents communications from using highly loaded links. Pitt Benoit, Melhem, Renaud, Robert Power-aware Manhattan routing on CMPs 16 / 34

Framework Theoretical results Heuristics Simulations Simple greedy ( SG ) Simple greedy ( SG ): greedily assigns communications, hop by hop, on the least loaded link. Improved greedy ( IG ): virtually pre-assigns communications onto links, then almost like SG . Two-bend ( TB ): for each communication, chooses the best path with two bends. XY improver ( XYI ): starts from XY assignment, and moves communications from the highest loaded link. Path remover ( PR ): virtually pre-assigns communications onto links, and iteratively prevents communications from using highly loaded links. Pitt Benoit, Melhem, Renaud, Robert Power-aware Manhattan routing on CMPs 18 / 34

Framework Theoretical results Heuristics Simulations Simple greedy ( SG ) Pitt Benoit, Melhem, Renaud, Robert Power-aware Manhattan routing on CMPs 19 / 34

Framework Theoretical results Heuristics Simulations Improved greedy ( IG ) Simple greedy ( SG ): greedily assigns communications, hop by hop, on the least loaded link. Improved greedy ( IG ): virtually pre-assigns communications onto links, then almost like SG . Two-bend ( TB ): for each communication, chooses the best path with two bends. XY improver ( XYI ): starts from XY assignment, and moves communications from the highest loaded link. Path remover ( PR ): virtually pre-assigns communications onto links, and iteratively prevents communications from using highly loaded links. Pitt Benoit, Melhem, Renaud, Robert Power-aware Manhattan routing on CMPs 20 / 34

Framework Theoretical results Heuristics Simulations Improved greedy ( IG ) Pitt Benoit, Melhem, Renaud, Robert Power-aware Manhattan routing on CMPs 21 / 34

Framework Theoretical results Heuristics Simulations Two-bend ( TB ) Simple greedy ( SG ): greedily assigns communications, hop by hop, on the least loaded link. Improved greedy ( IG ): virtually pre-assigns communications onto links, then almost like SG . Two-bend ( TB ): for each communication, chooses the best path with two bends. XY improver ( XYI ): starts from XY assignment, and moves communications from the highest loaded link. Path remover ( PR ): virtually pre-assigns communications onto links, and iteratively prevents communications from using highly loaded links. Pitt Benoit, Melhem, Renaud, Robert Power-aware Manhattan routing on CMPs 22 / 34

Framework Theoretical results Heuristics Simulations Two-bend ( TB ) Pitt Benoit, Melhem, Renaud, Robert Power-aware Manhattan routing on CMPs 23 / 34

Framework Theoretical results Heuristics Simulations XY improver ( XYI ) Simple greedy ( SG ): greedily assigns communications, hop by hop, on the least loaded link. Improved greedy ( IG ): virtually pre-assigns communications onto links, then almost like SG . Two-bend ( TB ): for each communication, chooses the best path with two bends. XY improver ( XYI ): starts from XY assignment, and moves communications from the highest loaded link. Path remover ( PR ): virtually pre-assigns communications onto links, and iteratively prevents communications from using highly loaded links. Pitt Benoit, Melhem, Renaud, Robert Power-aware Manhattan routing on CMPs 24 / 34

Framework Theoretical results Heuristics Simulations XY improver ( XYI ) Pitt Benoit, Melhem, Renaud, Robert Power-aware Manhattan routing on CMPs 25 / 34

Framework Theoretical results Heuristics Simulations Path remover ( PR ) Simple greedy ( SG ): greedily assigns communications, hop by hop, on the least loaded link. Improved greedy ( IG ): virtually pre-assigns communications onto links, then almost like SG . Two-bend ( TB ): for each communication, chooses the best path with two bends. XY improver ( XYI ): starts from XY assignment, and moves communications from the highest loaded link. Path remover ( PR ): virtually pre-assigns communications onto links, and iteratively prevents communications from using highly loaded links. Pitt Benoit, Melhem, Renaud, Robert Power-aware Manhattan routing on CMPs 26 / 34

Framework Theoretical results Heuristics Simulations Path remover ( PR ) Pitt Benoit, Melhem, Renaud, Robert Power-aware Manhattan routing on CMPs 27 / 34

Power-aware Manhattan routing on chip multiprocessors Anne Benoit 1 - PowerPoint PPT Presentation

Framework Theoretical results Heuristics Simulations Power-aware Manhattan routing on chip multiprocessors Anne Benoit 1 , Rami Melhem 2 , Paul Renaud-Goud 1 and Yves Robert 1 , 3 1 . Ecole Normale Sup erieure de Lyon, France, {

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