Interconnect Topology Optimization. Interconnect Topology Design

Transcription

1 Interconnect opology Optimiation Interconnect opology Design otal wire length minimiation Minimum spanning tree Steiner tree Path length minimiation Bounded radius bounded cost BRBC tree A-tree Delay minimiation WBA-tree P-tree Rats-tree 1

2 Prim Algorithm for Minimum Spanning rees Grow a connected subtree from the source, one node at a time. At each step, choose the closest un-connected node and add it to the subtree. X Y s Steiner ree Steiner points and Steiner tree problem Heuristic algorithms for Steiner tree problem Edge-merging based algorithm 1-Steiner algorithm Edge-insertion based algorithm 2

3 Merging-based Steiner ree 1-Steiner ree 3

4 Edge-Insertion based Steiner ree Interconnect opology Optimiation Under Linear Delay Model [Cong-Kahng-Robin, -CAD 92] Conventional Routing Algorithms Are Not Good Enough Minimum spanning tree may have very long source-sin path. Shortest path tree may have very large routing cost. 4

5 Example: A Routing Problem Source Source Critical sin Critical sin Definitions Given Net N with source s and connected by tree. Radius of net N: distance from the source to the furthest sin. Radius of a routing tree r: length of the longest path from the root to a leaf. Cost of an edge: distance between two endpoints other weights are o. Cost of a routing tree cost: sum of the edge costs in. minpath G u,v: shortest path from u to v in G dist G u,v: cost of minpath G u,v. r s source radius of the net routing tree 5

6 Problem Formulation Basic Idea: Restrict the tree radius while minimiing the routing cost Bounded radius minimum spanning tree problem BRMS: Given A net N with radius R Find A minimum cost tree with radius r 1+ εr source ε= radius=1,cost=4.95 source ε=1 radius=1.77,cost=4.26 source ε= radius=4.3,cost=4.3 trade-off between radius and the cost of routing trees Parameter ε controls the trade-off between radius and cost ε = minimum spanning tree; ε = shortest path tree BPRIIM Algorithm Bounded-Radius Minimum Spanning rees Given net N with source s and radius R, and parameter ε. Grow a connected subtree from the source, one node at a time At each step, choose the closest pair x and y N- If dist s, x + costx,y 1+εR, add x,y Else bactrac along minpath s,x to find x such that dist s, x + costx, y R, then add x, y X X Y Y S dist s,x+costx,y S X 1+ εr dist s,x +costx,y R Slac εr is introduced at each bactrace so we do not have to bactrace too often. 6

7 An Improved Algorithm -- BRBC [Cong-Kahng-Robin, -CAD 92] Construct MS and SP, Q = MS; Construct L -- a depth-first tour of MS; raverse L while eeping a running total S of traversed edge costs, when reaching L i If S ε dist SP S, L i then add minpath SP s, L i to Q and reset S =, Else continue traverse L; Construct the shortest path tree of Q. s s s L depth-first tour L MS L i L i Graph Q if S=dist L L i,l i ε dist SP S, L i then add minpath SP s, L i to Q and reset S = BRBC rees Have Bounded Radius heorem 1: r 1+ ε R Proof: For any vertex x, let y be the last vertex before x in L that we add minpath SP s, L. By the choice of y, we have dist L y,x ε dist SP s,x ε R herefore, dist Q s,x dist Q s,y +dist Q y,x dist SP s,y +dist L y,x R+εR =1+ εr x y dist SP s,y R s Graph Q tour L y x dist L y,x ε dist SP S, x εr s 7

8 BRBC rees Have Bounded Cost heorem 2: cost 1+2/ ε costms. Proof: Let v 1 v 2 v be the vertices that we add minpath SP s,v i Note that is a subgraph of Q cost Q = cost MS + cost MS + i = 1 i = 1 1 cost MS + cost L 2 cost MS + cost MS dist 1 SP dist s, L v cost MS dist L v i-1,v i ε dist SP S, v i Related wor by [Awarbuch - Barat - Peleg, PODC-9] v i i 1, v i v i-1 s s v i Graph Q tour L Interconnect opology Design Formulation Under Distributed RC Delay Model t = = all nodes t 1 t 1 t 2 t t 3 4 R C + = = = = t 2 R d R R R = all nodes + t 3 C C all sins all nodes C all sins d + t 4 pl length C R d + pl R pl C MS SP + QMS Constant C Objective: Minimie α length + β all sins pl + γ all nodes pl 8

9 Impact of Resistance Ratio Definition: R d /R Driver resistance versus unit wire resistance Determined by the echnology: Reduce device dimension R R d R d /R Impact on Interconnect Optimiation: t length 1 = R d C... t = 2 R C pl all sins... t = 3 R C pl... all nodes MS SP QMS Comparison of hree ypes of rees OS SP QMS OS cost 9optimal 11 1 SP cost optimal 31 QMS cost optimal 9

10 Results on Interconnect opology Optimiation A-REE: Generalied Rectilinear Minimum Steiner Arborescence Steiner ree Advantage of A-tree: A-ree Always a SP t 2 is minimum Minimiing t 1 Minimiing t 3 for most A-trees Objective: Minimie the total wirelength of A-tree: Simultaneous minimiation of t 1,t 2 and t 3. Rectilinear Steiner Arborescence Algorithm [Rao-Sadayappan-Huang 92] RSA: Rectilinear Steiner Arborescence shortest path Steiner tree, A-tree in short RSA algorithm Start with a forest of n single-node A-trees, repeatedly combining two A-trees into a new one 1

11 Performance of RSA Algorithm ime Complexity On log n. Wirelength of the tree by RSA algorithm 2 lengthrsma RSMA: Rectilinear Steiner Minimum Arborescence A-tree Algorithm [Cong-Leung-Zhou, DAC 93] Start with a forest of n single-node A-trees Apply a sequence of moves D Grow an existing A-tree, or D Combine two A-trees into a new one erminate when only one A-tree is left 11

12 A-tree AlgorithmCont d p dominates q if p x >= q x and p y >= q y. DOMp, F : the set of nodes in F dominated by p. Given a node p, define NWp: {x, y x < p x, y > p y }. SEp, SWp, NEp, Np, Sp, Wp, Ep similarly. Given node p, define: MFp, F -- nodes in DOMp, F with minimum rectilinar distance from p, dfp, F -- the distance from p to any node in MFp, F. mf west p, F -- the one with smallest x-coordinate in MFp, F. mf south p, F defined similarly. A-tree AlgorithmCont d Given p as a root in F : mxp, F -- the node in NWp ROOF, not bloced from p and have the minimum horiontal distance from p. dxp, F -- the horiontal distance between p and mxp, F or if mxp, F doesn t exist myp, F, dyf similarly defined 12

13 ype-1 Safe Move Combine two A-trees into a new one mx p dxp, F >= dfp, F dyp, F >= dfp, F mf my Move: p to mf west p, F ype-2 Safe Move mx p p my mf south dxp, F >= dfp, F dyp, F < dfp, F mx p p mf south my Move: southward, p to p with length min{dist y mf south p, F, p, dyp, F } 13

14 ype-3 Safe Move mx p p mf west my dxp, F < dfp, F dyp, F >= dfp, F mx p p mf west Move: westward, p to p with length min{dist x mf west p, F, p, dxp, F } my Heuristic Moves ype-1 and ype-2 Heuristic Moves: Combines two A-trees into a new one H1: select node p such that p =mf west p, F is farthest away from the source and introduce a path fromp to p H2: select two nodes p 1, p 2 such that p =min{p 1 x, p 2 x }, min{p 1 y, p 2 y } is farthest away from the source and connect p 1 and p 2 to p 14

15 Definitions: Optimality of a Move F is the forest constructed after the -th move by the A-tree algorithm F is the minimum-cost rectilinear Steiner A-tree containing F as a subgraph F is the optimal A-tree F m is the A-tree constructed by our algorithm Optimality of a Move: If costf =costf +1 then the +1-th is called an optimal move heorem All three types of safe moves are optimal moves Lower Bound Computation of Optimal A-ree Lower Bound Computation: After the -th move, we can compute an upper bound ERROR of costf +1 - costf Note that i ERROR = if the -th move is a safe move ii ERROR > if the -th move is a heuristic move cost cost* + Σ ERROR : constructed by the A-tree algorithm *: optimal A-tree Results obtained from the Lower Bound Computation: 94% of the moves are safe moves 45% of the A-trees constructed by the algorithm are optimal the A-trees constructed by the algorithm are at most 4% from optimal 15

16 Steiner Elmore Routing reeser Heuristic [Boese-Kahng-Robins, DAC 93] Use Elmore Delay Model directly in construction of routing tree Add nodes to one-by-one lie Prim s MS algorithm. At each step, choose v and u s.t. i he linear weighted Elmore-delay to the critical sins has minimum increase SER-C algorithm or ii he maximum Elmore-delay to any sin has minimum increase SER algorithm. Examples of SER-C Construction source a Node 2 or 4 critical source d Node 6 critical source source 2 b Node 3 or 7 critical c Node 5 critical also 1-Steiner tree source source 2 e Node 8 critical f Node 9 critical also Steiner ER 16

17 Some heoretical Results on Optimal Routing ree Under Elmore Delay Model [Boese-Kahng-McCoy-Robins, DAC 94] When minimiing linear weighted Elmore delay to the critical sins, there exists optimal tree on the Hanan-grid. When minimiing maximum Elmore delay in routing tree, optimal tree may not lie on the Hanan-grid. 1,3c=,R=1.75 2,3c= 1.63,2c=.37 2,c= unit R=1, unit c=1 source 1.5, Performance Driven Multisource Routing Problem [Cong-Madden, ISCAS 95] Previous wor: Assume a single source driving one or more sin nodes Contribution: First wor to address nets where each pin may act as a source, sin, or both. Signal busses are examples of multisource nets. 17

18 An Example Multisource Routing Problem Minimie ΣWi,j delayp i,p j weighted delay L total tree length 18