Unit 6: Routing. Course contents: Readings. Maze/A*-search routing Global routing Routing trees Channel routing Full-chip routing

Transcription

1 Course contents: Maze/A*-search routing Global routing Routing trees Channel routing Full-chip routing Readings W&C&C: Chapter 12 S&Y: Chapters 5, 6, and 7 Unit 6: Routing Filling Unit 6 1

2 Routing Unit 6 2

3 Routing Constraints 100% routing completion + area minimization, under a set of constraints: Placement constraint: usually based on fixed placement Number of routing layers Geometrical constraints: must satisfy design rules Timing constraints (performance-driven routing): must satisfy delay constraints Signal integrity constraint: Crosstalk? Manufacturability & reliability: CMP, MPL, EUV, antenna effect? Unit 6 3

4 Classification of Routers Global routing Graph search Steiner tree Hierarchical Progressive Routers Detailed routing General purpose Restricted regions Maze Line search A* search Channel Switchbox Over-the-cell Specialized routing Clock tree Power & ground Unit 6 4

5 Maze Router: Lee Algorithm Lee, An algorithm for path connection and its application, IRE Trans. Electronic Computer, EC-10, Discussion mainly on single-layer routing Strengths Guarantee to find connection between 2 terminals if it exists. Guarantee minimum path. Weaknesses Requires large memory for dense layout Slow Applications: global routing, detailed routing Unit 6 5

6 Lee Algorithm Find a path from S to T by wave propagation. Filling Time & space complexity for an M N grid: O(MN) (huge!) Unit 6 6

7 Reducing Memory Requirement Akers's Observations (1967) Adjacent labels for k are either k-1 or k+1. Want a labeling scheme such that each label has its preceding label different from its succeeding label. Way 1: coding sequence 1, 2, 3, 1, 2, 3, ; states: 1, 2, 3, empty, blocked (3 bits required) Way 2: coding sequence 1, 1, 2, 2, 1, 1, 2, 2, ; states: 1, 2, empty, blocked (need only 2 bits) Unit 6 7

8 Reducing Running Time Starting point selection: Choose the point farthest from the center of the grid as the starting point. Double fan-out: Propagate waves from both the source and the target cells. Framing: Search inside a rectangle area % larger than the bounding box containing the source and target. Need to enlarge the rectangle and redo if the search fails. Unit 6 8

9 Hadlock's Algorithm Hadlock, A shortest path algorithm for grid graphs, Networks, Uses detour number (instead of labeling wavefront in Lee's router) Detour number, d(p): # of grid cells directed away from its target on path P. MD(S, T): the Manhattan distance between S and T. Path length of P, l(p): l(p) = MD(S, T) + 2 d(p). MD(S, T) fixed! Minimize d(p) to find the shortest path. For any cell labeled i, label its adjacent unblocked cells away from T i+1; label i otherwise. Time and space complexities: O(MN), but substantially reduces the # of searched cells. Finds the shortest path between S and T. Unit 6 9

10 Hadlock's Algorithm (cont'd) d(p): # of grid cells directed away from its target on path P. MD(S, T): the Manhattan distance between S and T. Path length of P, l(p): l(p) = MD(S, T) + 2d(P). MD(S, T) fixed! Minimize d(p) to find the shortest path. For any cell labeled i, label its adjacent unblocked cells away from Ti+1; label i otherwise. Unit 6 10

11 Soukup's Algorithm Soukup, Fast maze router, DAC-78. Combined breadth-first and depth-first search. Depth-first (line) search is first directed toward target T until an obstacle or T is reached. Breadth-first (Lee-type) search is used to bubble around an obstacle if an obstacle is reached. Time and space complexities: O(MN), but times faster than Lee's algorithm. Find a path between S and T, but may not be the shortest! Unit 6 11

12 Features of Line-Search Algorithms Time and space complexities: O(L), where L is the # of line segments generated. Unit 6 12

13 Mikami-Tabuchi's Algorithm Mikami & Tabuchi, A computer program for optimal routing of printed circuit connectors, IFIP, H47, Every grid point is an escape point. Unit 6 13

14 Hightower's Algorithm Hightower, A solution to line-routing problem on the continuous plane, DAC-69. A single escape point on each line segment. If a line parallels to the blocked cells, the escape point is placed just past the endpoint of the segment. Unit 6 14

15 Comparison of Algorithms Soukup, Mikami, and Hightower all adopt some sort of line-search operations cannot guarantee shortest paths. Unit 6 15

16 A*-Search Routing Maze search is also called blind search because it searches the routing region in a blind way. A*-search is also called best-first search, which uses function f(x) = g(x) + h(x) to evaluate the cost of a path x g(x): the cost from the source to the current node of x h(x): the estimated cost from the current node of x to the target A*-search first searches the routes that is most likely to lead towards the target. BFS is a special case of A*-search where h(x) = 0 for all x Property: If h(x) is admissible, then A*-search is optimal Admissibility: the router never overestimates the actual cost from the current node to the target Unit 6 16

17 Example of A*-Search Routing (1/2) Cost function for a path x: f(x) = g(x) + h(x) g(x): the label from the source S to the current node of x (i.e., the lable used in maze routing) h(x): max(dist x (T, x), dist y (T, x)) T T T S 6 5 S 5 4 S wavefront explored # in exploration Unit 6 17

18 Example of A*-Search Routing (2/2) Cost function for a path x: f(x) = g(x) + h(x) g(x): the label from the source S to the current node of x (i.e., the lable used in maze routing) h(x): max(dist x (T, x), dist y (T, x)) S T S T 7 S wavefront explored # in exploration Unit 6 18

19 Multi-layer Routing 3-D grid: Two planner arrays: Neglect the weight for inter-layer connection through via. Pins are accessible from both layers. Unit 6 19

20 Net Ordering Net ordering greatly affects routing solutions. In the example, we should route net b before net a. Unit 6 20

21 Net Ordering (cont'd) Order the nets in the ascending order of the # of pins within their bounding boxes. Order the nets in the ascending (or descending??) order of their lengths. Order the nets based on their timing criticality. A mutually intervening case: Unit 6 21

22 Rip-Up and Re-routing Rip-up and re-routing is required if a global or detailed router fails in routing all nets. Approaches: the manual approach? the automatic procedure? Two steps in rip-up and re-routing 1. Identify bottleneck regions, rip off some already routed nets. 2. Route the blocked connections, and re-route the ripped-up connections. Repeat the above steps until all connections are routed or a time limit is exceeded. Unit 6 22

23 INR (Iterative Negotiation-based Ripping-up/rerouting) Proposed in PathFinder [McMurchie and Ebeling, FPGA 95] to reduce overflows Spreads congested wires iteratively At the i-th iteration, the cost of a global-routing edge e: B e : base cost of using e (say, delay and/or length of e) P e : # of nets passing e () i ( e ) H e (i) : historical cost on e, b h p e h () i e h ( i1) e ( i1) he if e has overflow otherwise Used by recent academic global routers, Archer [ICCAD 07], BoxRouter [ICCAD 07], FastRoute [ICCAD 06], FGR [ICCAD 07], NCTUgr [ASP-DAC 09], NTHU-Route [ICCAD 08], NTUgr [ICCAD 08], SGR [DAC 10], etc. INR may get stuck as the number of iterations increases [Ozdal, ICCAD 07] [Gao et al., ASP-DAC 08] e 1 Unit 6 23

24 Graph Modeling and Global Routing Graph Often model a chip as a graph such that the graph topology can represent the routing structure and resource in the chip Global routing graph Each cell is represented by a vertex. Two vertices are joined by an edge if the corresponding cells are adjacent to each other. Partitioned Layout Resource Modeling Global Routing Graph Unit 6 24

25 Graph Models for Global Routing: Grid Graph Each cell is represented by a vertex. Two vertices are joined by an edge if the corresponding cells are adjacent to each other. The occupied cells are represented as filled circles, whereas the others are as clear circles. Unit 6 25

26 Graph Model: Channel Intersection Graph Channels are represented as edges. Channel intersections are represented as vertices. Edge weight represents channel capacity. Extended channel intersection graph: terminals are also represented as vertices. Unit 6 26

27 Global-Routing Problem Given a netlist N = {N 1, N 2,, N n }, a routing graph G = (V, E), find a Steiner tree T i for each net N i, 1 i n, such that U(e j ) c(e j ), e j E and is minimized, where c(e j ): capacity of edge e j x ij =1 if e j is in T i ; x ij = 0 otherwise : # of wires that pass through the channel corresponding to edge e j L(T i ): total wirelength of Steiner tree T i. For high-performance, the maximum wirelength ( ) is minimized (or the longest path between two points in T i is minimized). Unit 6 27

28 Global Routing in Different Design Styles Full custom Flexible channel Most general problem Standard cell Global routing Flexible channel Fixed feedthrough Gate array Fixed channel FPGA Fixed routing track Switchbox constraint Unit 6 28

29 Objective Global Routing in Standard Cell Minimize total channel height. Assignment of feedthrough: Placement? Global routing? For high performance, Minimize the maximum wire length. Minimize the maximum path length. Unit 6 29

30 Objective Global Routing in Gate Array Guarantee 100% routability. For high performance, Minimize the maximum wire length. Minimize the maximum path length. Unit 6 30

31 Objective Global Routing in FPGA Guarantee 100% routability. Consider switch-module architectural constraints. For performance-driven routing, Minimize # of switches used. Minimize the maximum wire length. Minimize the maximum path length. Unit 6 31

32 Classification of Global-Routing Algorithm Sequential approach: Assigns priority to nets; routes one net at a time based on its priority (net ordering?). Concurrent approach: All nets are considered at the same time (complexity?) hierarchical progressive Integer linear programming Multi-commodity flow Unit 6 32

33 Global Routing by Integer Programming Suppose that for each net i, there are n i possible trees t 1i, t 2i,, t ni i to route the net. Constraint I: For each net i, only one tree t ji will be selected. Constraint II: The capacity of each cell boundary c i is not exceeded. Minimize the total tree cost. Question: Feasible for practical problem sizes? Keys: Hierarchical approach, progressive approach, multilevel framework Boundary 1 Unit 6 33

34 An Integer-Programming Example g i, j : cost of tree t ji g 1,1 =2, g 1,2 =3, g 1,3 =3, g 2,1 = 2, g 2,2 =3, g 2,3 =3, g 3,1 =2, g 3,2 =2. Unit 6 34

35 Hierarchical Global Routing Marek-Sadowska, Router planner for custom chip design, ICCAD-86. At each level of the hierarchy, an attempt is made to minimize the cost of nets crossing cut lines. At the lowest level of the hierarchy, the layout surface is divided into R R grid regions with boundary capacity equal to C tracks. Let R l be the # of grid regions of a given cut line l; a cut line can be divided into M = R l / C sections. Global routing can be formulated as a linear assignment problem: x i,j = 1 if net i is assigned to section j; x i,j = 0 otherwise. Each net crosses the cut line exactly once: Capacity constraint of each section: w ij : cost of assigning net i to section j. Minimize chip Unit 6 35

36 Global routing can be formulated as a linear assignment problem: x i,j = 1 if net i is assigned to section j; x i,j = 0 otherwise. Hierarchical Global Routing (cont d) Each net crosses the cut line exactly once: Reasonable assumption that simplifies the problem Capacity constraint of each section: w ij : cost of assigning net i to section j. Minimize a cut line a b b chip plane Unit 6 36

37 The Routing-Tree Problem Problem: Given a set of pins of a net, interconnect the pins by a routing tree. Minimum Rectilinear Steiner Tree (MRST) Problem: Given n points in the plane, find a minimum-length tree of rectilinear edges which connects the points. MRST(P) = MST(P S), where P and S are the sets of original points and Steiner points, respectively. Unit 6 37

38 Theoretic Results for the MRST Problem Hanan s Thm: There exists an MRST with all Steiner points (set S) chosen from the intersection points of horizontal and vertical lines drawn points of P. Hanan, On Steiner's problem with rectilinear distance, SIAM J. Applied Math., Hwang s Theorem: For any point set P, Hwang, On Steiner minimal tree with rectilinear distance, SIAM J. Applied Math., Better approximation algorithm with the performance bound 61/48 Foessmeier et al, Fast approximation algorithm for the rectilinear Steiner problem, Wilhelm Schickard-Institut für Informatik, TR WSI , 93. Unit 6 38

39 A Simple Performance Bound Easy to show that 2 Given any MRST T on point set P with Steiner point set S, construct a spanning tree T' on P as follows: 1. Select any point in T as a root. 2. Perform a depth-first traversal on the rooted tree T. 3. Construct T' based on the traversal. Unit 6 39

40 Coping with the MRST Problem Ho, Vijayan, Wong, New algorithms for the rectilinear Steiner problem, TCAD Construct an MRST from an MST. 2. Each edge is straight or L-shaped. 3. Maximize overlaps by dynamic programming. About 8% smaller than Cost(MST). Unit 6 40

41 Iterated 1-Steiner Heuristic for MRST Extended by Kahng & Robins, A new class of Steiner tree heuristics with good performance: the iterated 1-Steiner approach, ICCAD, Algorithm: Iterated_1-Steiner(P) P: set P of n points. 1 begin 2 S ; /* H(P S): set of Hanan points */ /* MST(A, B) = Cost(MST(A)) - Cost(MST(A B)) */ 3 while (Cand {x H(P S) MST(P S, {x}) > 0 } ) do 4 Find x Cand which maximizes MST(P S, {x}); 5 S S {x}; 6 Remove points in S which have degree 2 in MST(P S); 7 Output MST(P S); 8 end Remove degree-2 Steiner point Unit 6 41

42 Steiner Minimal Trees Based on Spanning Graphs Zhou, Efficient Steiner tree construction based on spanning graphs, ISPD-03 (TCAD-04) Select Borah et al. s edge-substitution approach as the basis (TCAD-94) Connect a point to an edge and delete the longest edge on the created cycle Enhance it with Zhou et al. s spanning graph (ISPD-03) a a p p b b c Edge substitution e d Unit 6 42 c e Spanning graph d

43 Edge-Substitution Approach Borah, Owens, and Irwin s algorithm for the rectilinear SMT (TCAD-94) Start with a minimal spanning tree Iteratively consider connecting a point to a nearby edge (e.g., point p to edge (a, b)) Delete the longest edge on the cycle thus formed (e.g., edge (b, c)) a a p p b b c e e d Unit 6 43 c d

44 Implementing the Edge-Substitution Approach How to construct the initial spanning tree? How to find the edge-point pair candidates for connection? How to find the longest edge on the formed cycle? Spanning graphs may answer the questions!! a b p e c Spanning graph d Unit 6 44

45 Spanning Graph An MST on the graph is an MST of the given points geometrical MST = spanning graph + graph MST Spanning graphs represent geometrical proximity information Also provide candidates for point-edge connection E.g., no edge (b, d) since d is blocked by edge (c, e) It is a sparse graph: O(n) edges a p b c e d Unit 6 45 b a c p e Spanning graph d

46 Octal Partition for Finding a Spanning Graph Find a spanning graph in O(n lg n) time Yao, SIAM J. Computing, 1982 For each point s, the plane is divided into 8 octal regions pq < max( ps, qs ) Only the closest point in each region needs to be connected to s R7 R8 R1 R2 p R1 q R6 s R3 R5 R4 s Unit 6 46

47 Finding Closest Points in R1 Sweep points by increasing x+y Keep points waiting for closest point in A (active set) Checking current point with A make connections delete connected points add current point in A Active set can be linearly ordered according to x Use a binary search tree Finding an insertion place: O(lg n) Deleting/inserting a point: O(lg n) Each point is inserted and deleted at most once: O(n lg n) x+y Unit 6 47

48 Finding the Neighbors a b R1 R2 R3 R4 p e For each point, use the octal partition to find its neighbors. c a b c d e p R1 a R2 p e R3 p e d d b a Spanning graph p e R4 c c d e c d Unit 6 48

49 Rectilinear Steiner Minimal Tree Algorithm Construct a rectilinear spanning graph Sort the edges in the graph Use Kruskal to build MST, at the same time Build the merging binary tree and Find possible point-edge candidates (by the spanning graph) Use least common ancestors on the merging binary tree to find the longest edges in cycles Iteratively update point-edge connections Unit 6 49

50 Kruskal's MST Algorithm Add a safe edge at a time to the growing forest by finding an edge of least weight (from those connecting two trees in the forest). Time complexity: O(E lge + V) (= O(E lgv + V), why?) Lines 2--3: O(V). MST-Kruskal(G,w) 1. A ; 2. for each vertex v V[G] 3. Make-Set(v); 4. Sort the edges of E[G] by nondecreasing weight w; 5. for each edge (u,v) E[G], in order by nondecreasing weight 6. if Find-Set(u) Find-Set(v) 7. A A {(u, v)}; 8. Union(u,v); 9. return A Line 4:O(E lge). Lines 5--8: O(E) operations on the disjoint-set forest, so O(E (E, V)). Unit 6 50

51 Example: Kruskal's MST Algorithm Unit 6 51

52 Finding the Longest Edge on a Cycle E 3 A 5 B 4 2 D F 4 5 A C 1 Generate the merging binary tree corresponding to the minimal spanning tree C 1 B 3 2 E The longest edge for this pair is the least common ancestor D F Unit 6 52

53 Example Spanning Graph Construction 1. Construct a rectilinear spanning graph 2. Use Kruskal s algorithm to build MST 3. Update point-edge connections iteratively a a a p p p b b b e e e c d c Spanning graph d c MST d Unit 6 53

54 Point-Edge Connection Updating 1. Construct a rectilinear spanning graph 2. Use Kruskal s algorithm to build MST 3. Update point-edge connections iteratively point edge delete edge gain a (b,c) (a,b) 3 c (a,b) (b,c) 3 p (a,b) (b,c) 3 p (a,b) (c,e) 2 c (d,e) (c,e) 1 e (b,c) (c,e) 1 p (a,b) (e,p) 1 p (b,c) (c,e) 1 p (c,e) (e,p) 1 Building the table according to the spanning graph b a c MST p e d Unit 6 54

55 Point-Edge Connection Updating (cont d) 1. Construct a rectilinear spanning graph 2. Use Kruskal s algorithm to build MST 3. Update point-edge connections iteratively Connect the point-edge pair with the max gain (e.g., point a to edge (b, c)) Delete related point-edge candidates (point c to edge (a, b), etc.) point edge delete edge gain a (b,c) (a,b) 3 c (a,b) (b,c) 3 p (a,b) (b,c) 3 p (a,b) (c,e) 2 c (d,e) (c,e) 1 e (b,c) (c,e) 1 p (a,b) (e,p) 1 p (b,c) (c,e) 1 p (c,e) (e,p) 1 b a c p e d Unit 6 55

56 Point-Edge Connection Updating (cont d) 1. Construct a rectilinear spanning graph 2. Use Kruskal s algorithm to build MST 3. Update point-edge connections iteratively Find the remaining point-edge pair with the max gain (point c to edge (d, e)) point edge delete edge gain a (b,c) (a,b) 3 c (a,b) (b,c) 3 p (a,b) (b,c) 3 P (a,b) (c,e) 2 c (d,e) (c,e) 1 e (b,c) (c,e) 1 p (a,b) (e,p) 1 p (b,c) (c,e) 1 p (c,e) (e,p) 1 b a c p e d Unit 6 56

57 Summary on the RSMT Algorithms Optimal algorithms: Hwang, Richards, Winter [ADM-92] Warme, Winter, Zachariasen [AST-00] GeoSteiner package Near-optimal algorithms: Griffith et al. [TCAD-94] Batched 1-Steiner heuristic (BI1S) Mandoiu, Vazirani, Ganley [ICCAD-99] Low-complexity algorithms: Borah, Owens, Irwin [TCAD-94] Edge-based heuristic, O(n log n) Zhou [ISPD-03] Spanning graph based, O(n log n) Algorithms targeting low-degree nets (VLSI applications): Soukup [Proc. IEEE-81] Single Trunk Steiner Tree (STST) Chen et al. [SLIP-02] Refined Single Trunk Tree (RST-T) Chu [ICCAD-04, TCAD-08]: very efficient & effective O(n log n)-time algorithm (see Appendix C) Obstacle-avoiding RSMT construction Lin, et al. [ISPD-07, TCAD-08] (see Appendix D), Ajwani, et al., [TCAD-11]; many more Multilayer (3D) obstacle-avoiding RSMT construction Lin, et al. [ICCAD-07, TCAD-08], Liu, et al. [TCAD-14]; many more 57

58 Detailed Routing: Channel/Switchbox Routing Unit 7 58

59 Order of Routing Regions and L-Channels a) No conflicts if routing in the order of 1, 2, and 3. b) No ordering is possible to avoid conflicts. c) The situation of (b) can be resolved by using L-channels. d) An L-channel can be decomposed into two channels and a switchbox. Unit 7 59

60 Routing Considerations Number of terminals (two-terminal vs. multi-terminal nets) Net widths (power and ground vs. signal nets) Via restrictions (stacked vs. conventional vias) Boundary types (regular vs. irregular) Number of layers (two vs. three, more layers?) Net types (critical vs. non-critical nets) Unit 7 60

61 Grid-based model: Routing Models A grid is super-imposed on the routing region. Wires follow paths along the grid lines. Pitch: distance between two gridded lines Gridless model: Any model that does not follow this gridded approach. Unit 7 61

62 Models for Multi-Layer Routing Unreserved layer model: Any net segment is allowed to be placed in any layer. Less popular, but wrong-way jogs? Reserved layer model: Certain type of segments are restricted to particular layer(s). More popular model Two layers: HV (horizontal-vertical), VH Three layers: HVH, VHV Modern design can afford 10+ layers Unit 7 62

63 Terminology for Channel Routing Problems Local density at column i, d(i): total # of nets that crosses column i. Channel density: maximum local density # of horizontal tracks required channel density. Unit 7 63

64 Channel Routing Problem Assignments of horizontal segments of nets to tracks. Assignments of vertical segments to connect. horizontal segments of the same net in different tracks, and the terminals of the net to horizontal segments of the net. Horizontal and vertical constraints must not be violated. Horizontal constraints between two nets: The horizontal span of two nets overlaps each other. Vertical constraints between two nets: There exists a column such that the terminal on top of the column belongs to one net and the terminal on bottom of the column belongs to the other net. Objective: Channel height is minimized (i.e., channel area is minimized). Unit 7 64

65 Horizontal Constraint Graph (HCG) HCG G = (V, E) is undirected graph where V = { v i v i represents a net n i } E = {(v i, v j ) a horizontal constraint exists between n i and n j }. For graph G: vertices nets; edge (i, j) net i overlaps net j. Unit 7 65

66 Vertical Constraint Graph (VCG) VCG G = (V, E) is directed graph where V = { v i v i represents a net n i } E = {(v i, v j ) a vertical constraint exists between n i and n j }. For graph G: vertices nets; edge i j net i must be above net j. Unit 7 66

67 2-Layer Channel Routing: Basic Left-Edge Algorithm Hashimoto & Stevens, Wire routing by optimizing channel assignment within large apertures, DAC-71. No vertical constraint. HV-layer model is used. Doglegs are not allowed. Treat each net as an interval. Intervals are sorted according to their left-end x-coordinates. Intervals (nets) are routed one-by-one according to the order. For a net, tracks are scanned from top to bottom, and the first track that can accommodate the net is assigned to the net. Optimality: produces a routing solution with the minimum # of tracks (if no vertical constraint). Unit 7 67

68 Basic Left-Edge Algorithm Algorithm: Basic_Left-Edge(U, track[j]) U: set of unassigned intervals (nets) I 1,, I n ; I j =[s j, e j ]: interval j with left-end x-coordinate s j and right-end e j ; track[j]: track to which net j is assigned. 1 begin 2 U {I 1, I 2,, I n }; 3 t 0; 4 while (U ) do 5 t t + 1; 6 watermark 0; 7 while (there is an I j U s.t. s j > watermark) do 8 Pick the interval I j U with s j > watermark, nearest watermark; 9 track[j] t; 10 watermark e j ; 11 U U - {I j }; 12 end Unit 7 68

69 Basic Left-Edge Example U = {I 1, I 2,, I 6 }; I 1 = [1, 3], I 2 = [2, 6], I 3 = [4, 8], I 4 = [5, 10], I 5 = [7, 11], I 6 = [9, 12]. t =1: Route I 1 : watermark = 3; Route I 3 : watermark = 8; Route I 6 : watermark = 12; t = 2: Route I 2 : watermark = 6; Route I 5 : watermark = 11; t = 3: Route I 4 Unit 7 69

70 Basic Left-Edge Algorithm If there is no vertical constraint, the basic left-edge algorithm is optimal. If there is any vertical constraint, the algorithm no longer guarantees optimal solution. Unit 7 70

71 Constrained Left-Edge Algorithm Algorithm: Constrained_Left-Edge(U, track[j]) U: set of unassigned intervals (nets) I 1,, I n ; I j =[s j, e j ]: interval j with left-end x-coordinate s j and right-end e j ; track[j]: track to which net j is assigned. 1 begin 2 U { I 1, I 2,, I n }; 3 t 0; 4 while (U ) do 5 t t + 1; 6 watermark 0; 7 while (there is an unconstrained I j U s.t. s j > watermark) do 8 Pick the interval I j U that is unconstrained, with s j > watermark, nearest watermark; 9 track[j] t; 10 watermark e j ; 11 U U -{I j }; 12 end Unit 7 71

72 Constrained Left-Edge Example I 1 = [1, 3], I 2 = [1, 5], I 3 = [6, 8], I 4 = [10, 11], I 5 = [2, 6], I 6 = [7, 9]. Track 1: Route I 1 (cannot route I 3 ); Route I 6 ; Route I 4. Track 2: Route I 2 ; cannot route I 3. Track 3: Route I 5. Track 4: Route I 3. Unit 7 72

73 Dogleg Channel Router Deutsch, A dogleg channel router, 13rd DAC, Drawback of Left-Edge: cannot handle the cases with constraint cycles. Doglegs are used to resolve constraint cycle. Drawback of Left-Edge: the entire net is on a single track. Doglegs are used to place parts of a net on different tracks to minimize channel height. Might incur penalty for additional vias. Unit 7 73

74 Dogleg Channel Router Each multi-terminal net is broken into a set of 2-terminal nets. Two parameters are used to control routing: Range: Determine the # of consecutive 2-terminal subnets of the same net that can be placed on the same track. Routing sequence: Specifies the starting position and the direction of routing along the channel. Modified Left-Edge Algorithm is applied to each subnet. Unit 7 74

75 Framework Evolution Billions of transistors may be fabricated in a single chip for nanometer technology. Need frameworks for very large-scale designs. Framework evolution for EDA tools: Flat Hierarchical Multilevel Multilevel Hierarchical Flat Source: Intel (ISSCC-03) Unit 7 75

76 Sequential approaches Maze searching Flat Routing Framework Line searching Concurrent approaches Network-flow based algorithms Linear assignment formulation Drawback: hard to handle larger problems Sequential Concurrent Unit 7 76

77 Hierarchical Routing Framework The hierarchical approach recursively divides a routing region into a set of subregions and solve those subproblems independently. Drawbacks: lack the global information for the interaction among subregions. Unit 7 77

78 Hierarchical Routing Revisited Global routing can be formulated as a linear assignment problem: x i,j = 1 if net i is assigned to section j; x i,j = 0 otherwise. Each net crosses the cut line exactly once: Capacity constraint of each section: w ij : cost of assigning net i to section j. Minimize a cut line a b b chip plane Unit 7 78

79 Multilevel Full-Chip Routing Framework Lin and Chang, A novel framework for multilevel routing considering routability and performance, ICCAD-2002 (TCAD, 2003). Multilevel framework: coarsening followed by uncoarsening. Coarsening (bottom-up) stage: Constructs the net topology based on the minimum spanning tree. Processes routing tiles one by one at each level, and only local nets (connections) are routed. Applies two-stage routing of global routing followed by detailed routing. Uses the L-shaped & Z-shaped pattern routing. Performs resource estimation after detailed routing to guide the routing at the next level. Uncoarsening (top-down) stage Completes the failed nets (connections) from the coarsening stage. Uses a global and a detailed maze routers to refine the solution. Unit 7 79

80 A Multilevel Full-Chip Routing Framework Unit 7 80

81 Coarsening Stage Build MSTs for all nets and decompose them into twopin connections. Route local nets (connections) from level 0. Two-stage routing (global + detailed routing) for a local net. an MST edge global route detailed route level k level 0 level k Unit 7 81

82 Global Routing Apply pattern routing for global routing Use L-shaped and Z-shaped connections to route nets. Has lower time complexity than maze routing. Lower L-Shaped connection Upper L-Shaped connection Z-Shaped connection Unit 7 82

83 Via minimization Detailed Routing Modify the maze router to minimize the number of bends. Local refinement Apply general maze routing to improve the detailed routing results. Resource estimation Update the edge weights of the routing graph after detailed routing. Unit 7 83

84 Via Minimization Back Wave Trace Propagation b (u) d (u) S Unrouteable Obstacle T Unit 7 84

85 Local Refinement Local refinement improves detailed routing results by merging two connections which are decomposed from the same net. an MST edge global route detailed route level k level 0 level k Unit 7 85

86 Resource Estimation Global routing cost is the summation of congestions of all routed edges. Define the congestion, C e, of an edge e by Ce 2 where p e and d e are the capacity and density, respectively. Update the congestion of routed edges to guide the subsequent global routing. 1 ( p e de ), Unit 7 86

87 Uncoarsening Global Routing Use maze routing. Iterative refinement of a failed net stops when a route is found or several tries have been made. Coarsening stage Uncoarsening stage Unit 7 87

88 Routing Solution for Prim2 0.18um technology, pitch = 1 um, 8109 nets. Two layers, 100% routing completion. Unit 7 88

89 Summary: Detailed Routing Channel routing is considered a well solved problem. Hierarchical and multilevel are keys to handle largescale routing problems. Routing frameworks: go parallel?? Λ-shaped routing: ICCAD-02 (TCAD-03); ASP-DAC-05 (TCAD-07) V-shaped routing: ASP-DAC-06 Two-pass bottom-up routing: DAC-06 (TCAD-08) Routing considerations for nanometer technology Noise (crosstalk) & electro-migration constraints Buffer insertion for timing optimization Additional design rules: antenna effect, redundant via, OPC (optical proximity correction), CMP (chemical mechanical polishing), double/multiple patterning, e-beam, EUV (extreme ultraviolet lithography), directed self-assembly (DSA), etc. Initial detailed routing: ISPD-18 & -19 contests Unit 7 89

90 Appendix A: FLUTE: Fast Lookup Table Based Wirelength Estimation Technique Chris Chu ICCAD-04, TCAD-08 (Slides provided by the authors) Unit 6 90

91 FLUTE Features Fast and accurate rectilinear Steiner minimum tree (RSMT) construction technique Finds optimal RSMTs for up to nine pins Near minimal (~1%) RSTs for larger nets Based on a precomputed lookup table for low degree nets Uses a net-breaking technique for high degree nets O(n log n) runtime 91

92 Net Representation Degree-n nets are partitioned into n! groups Potentially optimal wirelength vectors (POWVs) represent linear combination of distances between adjacent pins Few POWVs for each group are precomputed and stored in a table A potentially optimal Steiner tree (POST) associated with each POWV is also stored To find the optimal RSMT of a net wirelengths corresponding to POWVs for the net group are computed the POST of the POWV with minimum wirelength is returned 92

93 POWV and POST Pins for a sample net POWV: #segments in h i /v i All POSTs for the sample net (1,2,1,1,1,1,2) cannot be shorter than (1,2,1,1,1,1,1) Wirelength needs to be calculated for (1,2,1,1,1,1,1) vs. (1,1,1,1,1,2,1) 93

94 FLUTE-Based RSMT Computation RSMT is easy to find if all the POWVs and POSTs are precomputed in a lookup table Infinite number of different nets Nets that can share the same POWVs are grouped together Steiner trees are topologically equivalent if they can be transformed into each other by changing the edge lengths Same position sequence To obtain RSMT for a net, look up the vectors for the corresponding group from the table Wirelength is computed based on vector entries and edge lengths Minimum wirelength vector s POST gives RSMT 94

95 Lookup Table Generation Boundary size reduction to reduce the number of Steiner trees generated Grid size is reduced by compacting one of the four boundaries (pins on boundary shifted to the adjacent grid line) Original Steiner trees are generated by expanding A non-compactible grid 95

96 Grid Compactibility A grid is compactible if it has boundary with only one pin A grid is compactible if it has a corner with one pin and both boundaries adjacent have exactly one other pin A grid is compactible if it has up to six pins on all four boundaries 96

97 POWV Generation For a grid with seven pins, boundary compaction together with near-ring structures can generate all POWVs Near-ring structure: bounding box that surrounds the grid with edges connecting one of the 7 pairs of the adjacent pins removed. For eight or more pins Connect-adj-pins() used to generate extra trees It connects two or more adjacent pins on the same boundary and introduces a branch along that boundary Then pins are replaced by a pseudo-pin Gen-Lut() function is called to generate POSTs of reduced grid Original POSTs are generated by connecting the branch with the POSTs of reduced grid Boundary compaction together with Connect-adj-pins() is sufficient to generate all POWVs for nets with up to 10 pins 97

98 Algorithm: Lookup Table Generation Gen-LUT(G) Algorithm Input: A grid G with some pins at grid nodes Output: One POST for each POWV of the group associated with G begin If G is simple enough, generate and return the set of POSTs for G else if any boundary b contains only one pin, return Expand-b(Gen-LUT(Compact-b(G)) else if there is a corner with one pin such that both its adjacent boundaries b1 and b2 have one other pin, return Prune(Expand-b1(Gen-LUT(Compact-b1(G))) U (Expand-b2(GEN-LUT(Compact-b2(G))) else if there are 7 pins with all 7 pins on boundaries, S = {Trees with near-ring structure connecting all pins} else if there are more than 8 pins with more than seven pins on boundaries S = Connect-adj-pins(G,d) where d = # of pins -3 return Prune(S U Expand-left(Gen-LUT(Compact-left(G))) U Expand-right(Gen-LUT(Compact-right(G))) U Expand-top(Gen-LUT(Compact-top(G))) U Expand-bot(Gen-LUT(Compact-bot(G)))) end 98

99 Lookup Table Reduction In order to reduce storage requirements for POWVs and associated POSTs in the lookup table, similar POWVs can be grouped together and differences can be stored Numbers of POWVs or the POSTs do not decrease Alternatively, equivalence of groups can be exploited to generate less POWVs and POSTs Storage requirements and table generation time decreases Boundary reduction causes two different nets to have same set of POWVs POSTs can also be shared Equivalency of groups can reduce the number of groups generated and stored by 25.8x 99

100 Wirelength Computation Speedup Instead of adding edge lengths (to reduce number of addition operations), an MST-based approach can be used to reduce minimum wirelength computation time q nodes corresponding to q POWVs in the set and one more node to represent wirelength vector (1,,1) Weight of each edge is the number of addition/subtraction required to convert from wirelength of one vector to other Total edge weight of MST gives the number of additions/subtractions required to compute POWVs Based on the fact that most POWVs are similar to each other i.e., wirelength computations differ by a few additions/subtractions Can significantly speed up evaluation of high-degree nets 100

101 Net Breaking Net Breaking for High-degree Nets Table lookup approach is only practical for low-degree nets due to storage and computation time requirements A lookup table is constructed up to degree D = 9 Higher-degree (>9) nets are broken into subnets with degrees 2 to 9 Net Breaking Heuristics If a net is broken at r, then two subnets: pin 1 to r, and pin r to n A score is computed to evaluate the most desirable way of breaking (based on if the edge is likely to be counted in both partitions or not) Score S(r) = S 1 (r) αs 2 (r) βs 3 (r) γs 4 (r) (α=0.3, β=7.4/(n+10), γ=4.8/(n-1)) S 1 (r) = y r+1 -y r-1 (it is better to break at pin r if large) S 2 (r) = 2(x 3 x 2 ) if s r = 1 or 2 = x sr+1 x sr-1 if 3 s r n-2 = 2(x n-1 x n-2 ) if s r = n - 1 or n S 3 (r) = s (n+1)/2 x h + r (n+1)/2 x v, h =(x n-1 x 2 )/(n-3) and v =(y n-1 y 2 )/(n-3) S 4 (r) = total HPWL of the two subnets (direct way to predict resulting wirelength) 101

102 RSMT Construction After subtrees of subnets are generated, they are combined to form ST Redundant edges detected in constant time Removed via extra Steiner node Local refinement technique using FLUTE to reduce wirelength by reconstructing subtree connecting the neighborhood of breaking pin 102

103 Appendix B: Obstacle-Avoiding Rectilinear Steiner Tree Construction Lin, Chen, Li, Chang, and Yang ISPD-07, TCAD-08 Unit 6 103

104 Introduction to OARSMT Problem Given a set of pins and a set of obstacles, an obstacleavoiding rectilinear Steiner minimal tree (OARSMT) Connect those pins, possibly through some Steiner points Use only vertical and horizontal edges Avoid running through any obstacle Have a minimal total wirelength It becomes more important than ever for modern nanometer IC designs. The design needs to consider numerous routing obstacles incurred from Prerouted nets Large-scale power networks IP blocks, etc IP block IP block Prerouted nets Unit 6 104

105 Construction-by-Correction Approach Construct an initial tree without considering obstacles Correct edges overlapping obstacles Construction Correction Lack a global view of obstacles Have a limited solution quality Feng et al., ISPD-06 Construct OARSMT in the λ-geometry plane Unit 6 105

106 Connection-Graph Based Approach Construct a connection graph in which there is a desired OARSMT Apply searching techniques to find the desired OARSMT Spanning graph Spanning tree OARSMT Prune many redundant edges to reduce problem size Obtain much better solution quality Shen et al., ICCD-05 Achieve good results but can be further improved Unit 6 106

107 Problem Formulation Given m pins {p 1, p 2,, p m } and k obstacles {o 1, o 2,, o k }, construct an OARSMT such that the total wirelength of the tree is minimized. Obstacles overlap each other Obstacles are point-touched or line touched A pin locates inside an obstacle A pin is at the corner or on the boundary of an obstacle An edge intersects an obstacle An edge is point-touched or line-touched with an obstacle Unit 6 107

108 An OARSMT Example Smaller wirelength pin-vertex obstacle Larger wirelength Unit 6 108

109 Algorithm Flow Obstacle-Avoiding Spanning Graph Construction Obstacle-Avoiding Spanning Tree Construction Obstacle-Avoiding Rectilinear Spanning Tree Construction OARSMT Construction Unit 6 109

110 Algorithm Flow Spanning Graph Spanning Tree pin-vertex obstacle corner-vertex turning-vertex Steiner-vertex Rectilinear Spanning Tree OARSMT Unit 6 110

111 Spanning Graph Construction The plane is divided into four regions for each vertex. pin-vertex obstacle corner-vertex R R 2 R 2 of c i,2 3 R 1 R 4 R 1 of c i,2 c i,1 R 3 of c i,2 c i,2 c i,3 o i c i,4 v 1 and v 2 are connected if no other vertex or obstacle is inside or on the boundary of the bounding box of v 1 and v 2. p 1 c p 1,3 2 o 3 p 1 c p 1,3 2 o 3 o 1 c 1,4 c 3,1 o 1 c 1,4 c 3,1 c 1,1 c 1,1 o 2 c 2,3 c 4,2 o 2 c 2,3 c 4,2 o 4 o 4 Spanning graph for R 2 of c 4,2 Unit 6 111

112 Spanning Graph Construction Example Apply a sweeping line algorithm for each region Use blocking information to check a vertex is blocked or not p 1 c p 1,3 2 o 3 o 1 c c 3,1 1,4 c 1,1 o 2 c 2,1 c 2,3 c 2,4 c 4,2 o 4 Candidate set c 1,1 c 2,3 c 1,4 p 2 c 3,1 Blocking information o 1 o 2 o 3 Spanning graph for R 2 of c 4,2 Unit 6 112

113 Properties of Our Spanning Graph (1/2) Our spanning graph guarantees a shortest path of any two vertices. p 1 p 2 p 1 p 2 OARSMT pin-vertex obstacle corner-vertex p 1 p 2 p 1 p 2 OARSMT waste waste Spanning graph of Shen et al. (ICCD-05) does not guarantee it. Unit 6 113

114 Properties of Our Spanning Graph (2/2) p 1 p 2 p 3 p r p r+1 4 p 1 p 2 p 3 p r p r Our spanning graph 4r OARSMT total wirelength = 4r p 1 p 2 p 3 p r p r p 1 p 2 p 3 p r p r Spanning graph of Shen et al. 4r OARSMT total wirelength = 6r+2 pin-vertex obstacle corner-vertex Unit 6 114

115 Algorithm Flow Spanning Graph Spanning Tree pin-vertex obstacle corner-vertex turning-vertex Steiner-vertex Rectilinear Spanning Tree OARSMT Unit 6 115

116 Spanning Tree Construction Spanning graph Pin-vertices shortest path computation G Minimum spanning tree (MST) construction T 1 pin-vertex obstacle corner-vertex G 2 T 2 T 3 Edges mapping Edges insertion MST construction Unit 6 116

117 Algorithm Flow Spanning Graph Spanning Tree pin-vertex obstacle corner-vertex turning-vertex Steiner-vertex Rectilinear Spanning Tree OARSMT Unit 6 117

118 Rectilinear Spanning Tree Construction Transform slant edges into vertical and horizontal edges Longer edges are transformed first. Three cases for a slant edge e and its neighboring edge e : 1. They are in opposite regions. e e is transformed randomly e 2. They are in neighboring regions. e e and e are transformed with edge overlap e 3. They are in the same region. e e v b v a e and e are transformed with edge overlap (v a,v b ) is transformed randomly Unit 6 118

119 Rectilinear Spanning Tree Construction Example p 1 c 1 c 2 p 1 c 1 p 1 c 1 c 2 c 1 c 2 t 5 t t p 2 3 t 4 3 p3 p 3 p3 p 2 p 5 p 2 t 1 p 5 p 2 p4 p 4 1. They are in opposite regions. e e e e 2. They are in neighboring regions. pin-vertex corner-vertex turning-vertex e is transformed randomly e and e are transformed with edge overlap 3. They are in the same region. e e v b v a e and e are transformed with edge overlap (v a,v b ) is transformed randomly Unit 6 119

120 Algorithm Flow Spanning Graph Spanning Tree pin-vertex obstacle corner-vertex turning-vertex Steiner-vertex Rectilinear Spanning Tree OARSMT Unit 6 120

121 OARSMT Construction (1/2) Overlapping edge removal Redundant vertex removal A redundant-vertex is a non-pin-vertex with the degree of 2, and the two edges connecting to it are parallel. c 1 c 1 c 2 t 4 Overlapping edge removal Redundant vertex removal Unit 6 121

122 OARSMT Construction (2/2) U-shaped pattern refinement A vertex satisfies the U-shaped pattern refinement rules if it is not a pin-vertex, and its degree is 2. Two cases for the U-shaped pattern refinement: v 1 e 1 e 1 e 2 e 2 One of the vertices v 1 and v 2 must satisfy the refinement rule. v 2 e 3 v 2 e 3 v 1 e 1 e 1 e 2 e 3 e 5 e 5 e 2 e 3 Both vertices v 1 and v 2 must satisfy the refinement rules. v 2 e 4 e 4 Unit 6 122

123 Properties of Our OARSMT Our OARSMT is an optimal solution for: Any 2-pin net Any 3-pin net without obstacles Any net whose topology of an optimal solution contains only simple paths pin-vertex Steiner-vertex Two simple paths Three paths with Steiner-vertex They are not guaranteed by any previous work. They give the sufficient but not necessary conditions for an optimal solution. More optimal solutions may still be generated in other cases. Unit 6 123

124 Complexity Spanning Graph Construction Spanning Tree Construction Worst Case Rectilinear Spanning Tree Construction OARSMT Construction Overall time complexity O(n 2 lgn) O(n 3 ) O(n lgn) O(n 2 ) Practical Applications (# of edges in spanning graph is O(n).) O(n lgn) O(n 2 lgn) O(n lgn) O(n 2 ) O(n 3 ) in the worst case O(n 2 lg n) for practical applications n: # of pin-vertices and corner-vertices Unit 6 124

125 Experimental Results Routing Result of rt3 Unit 6 125

126 Appendix C: Modern Global Routing & NCTU-GR Wen-Hao Liu, Yih-Lang Li, et al. DAC 10, TVLSI-12 (NCTU-GR), TCAD-13 (NCTU-GR 2.0) (Slides provided by Dr. Wen-Hao Liu, with revisions) Maze Routing Solution Space Escaping Monotonic L Pattern Unit 6 126

127 Global Routing Problem Given a grid graph and a set of nets (each net consists of a set of pins). The goal of global routing is to identify the global routing paths to connect each net. The optimization order is to minimize overflows, total wirelength (wire edges and vias), and finally runtime. 127

128 Global Routing Strategies Two strategies are generally adopted to resolve the global routing problem. Directly perform global routing on a 3D graph. 2D global routing with layer assignment. Due to the runtime concern, most global routers adopt the second strategy. 2 3D graph 2D graph 2D routing result 3D routing result P3 P3 P3 P3 P1 0 3 P2 P1 Graph Compaction 5 P2 P1 Global Routing P2 P1 Layer Assignment P2 12 8

129 Mainstream Global Routing Flow Start Graph Compaction Net Decomposition Initial Routing Rip-up and Rerouting stage Maze Routing Overflow No Yes Layer Assignment Finish 12 9

130 Global Routing Studies The studies of existing global routing researches Net decomposition [6,7,8,14] Routing nets ordering [2,7,10,13] Rip-up and rerouting scheme [2,8,11,13,16] Routing cost formulation [1,2,3,6,7,8,10,13,14,15] Routing algorithms [4,6,9,10,13,15,19] Layer assignment approaches [6,7,12,13] Multi-threaded routing [14,16,17,18] NTHU-Route [1,2] FastRoute [3-6] FGR [7,8] MGR [9] NTUgr [10] Box-Router [11,12] NCTU-GR [13,14] Archer [15] GRIP [16,17] HybridGR [18] Maize-Router [19] 13 0

131 Routability-Driven Placement & Routing Add a congestion estimator during placement for routability consideration A Fast Global-routing-based Routing Congestion Estimator Placement Stage Design Placer Routing Stage Global Router NCTU-GR Grace Congestion Estimator No Routable? Yes Routability Optimizer Post 3D Global Router Detailed Router Routing result 13 1

132 Routing Effectiveness vs. Runtime Effectiveness and runtime of each routing stage for an industry design Grid size : 444 x 518 Net number : 1.8M Maze Routing Solution Space Escaping Monotonic L Pattern L-shaped Pattern Routing Route 100 % nets 11 seconds Monotonic Routing Reroute 11% nets 14 seconds Escaping Routing Reroute 3% nets 23 seconds Maze Routing Reroute 1% nets 120 seconds 13 2

133 Unilateral Monotonic Routing Unilateral monotonic routing can seek a detoured path and running in the time complexity linear to its searching region Fixed Monotonic Routing Horizontally Unilateral Monotonic Routing T T S S Fixed Fixed 133

134 Unilateral monotonic routing is based on dynamic programming. Unilateral Monotonic Routing Given a bounding box B, unilateral monotonic routing identifies a parent for each node within B in a bottom-up manner. Vertically Monotonic Routing T Fixed The time complexity of unilateral monotonic routing is O( B ). S 134

135 Hybrid Unilateral Monotonic (HUM) Routing HUM path consists of two paths P s,u and P t,u P s,u is either horizontally or vertically t monotonic path from s to internal node u. u P t,u is either the horizontally or vertically monotonic path from t to internal node u. s t t t t u u u u s s s s Vertically monotonic path from s to u Horizontally monotonic path from s to u Vertically monotonic path from t to u Horizontally monotonic path from t to u 135

136 Perform four shots of unilateral monotonic routing to try all values of u in the bounding box, and then pick the best one to be internal node. Perform horizontally monotonic routings from S to the left and right boundaries. HUM Routing Horizontally Vertically Monotonic Routing T Perform vertically S monotonic routings from S to the top and bottom boundaries. 136

137 HUM Routing Perform horizontally monotonic routings from T to the left and right boundaries. Horizontally Vertically Monotonic Routing Perform vertically monotonic routings from T to the top and bottom boundaries. T Select the best node u S to minimize the routing cost of P s,u + P t,u u 137

138 Comparison The time complexities of 3-bend routing, escape routing and HUM routing are same, but HUM can get a better routing result. 3-bend routing Escape routing HUM routing u t t t u s s s 138

139 Congestion-aware Bounding Box Expansion If a net passes through a congested region, the bounding box of the net expands and then the net is rerouted. Over expanding the bounding box may increase the runtime, so use a congestion-aware bounding box expansion scheme to avoid over expanding. Traditional box expansion scheme Congestion-aware box expansion scheme Improve 45% runtime!! 139

140 Grace: Fast Congestion Estimator 140

141 Grace: Fast Congestion Estimator Horizontally Monotonic Routing Vertically Monotonic Routing HUM Routing t t t s s s 141

142 NCTU-GR 2.0 Design Flow 3D grid graph, a set of nets Compact 3D graph into 2D graph Building Skeleton Edge Set & RMST Decomposition Initial Routing Guide Negotiation-based Rip-up & Rerouting Sort two-pin nets Bounded Length Relaxed Create RSMT-aware Routing Scheme Guide Heuristic-BLMR Guide Post Refinement No Overflow Yes Update history cost Layer Assignment 3D Global Routing Result 14 2

143 How to Get Good Routing Results Carefully utilizing routing resource can get a routing result with better congestion, wirelength and runtime Which grid edge is a critical routing resource Who needs a critical routing resource How to avoid wasting routing resources Wirelength Runtime Congestion Initial Routing Bad Routing Result Good Routing Result A B 14 3

144 RSMT-aware Routing Scheme Rectilinear Steiner minimum tree (RSMT) has shorter wire length, but has less routing flexibility and more complicated data structure Rectilinear minimum spanning tree (RMST) offers better flexibility, but may have longer routing wirelength RSMT Decomposition RMST Decomposition n 1 n 2 n 1 n 2 n 3 n 4 n

145 RSMT-aware Routing Scheme Decompose each multi-pin net to two-pin subnets by RMST, and then guide routing by an ideal RSMT Strike a good balance between flexibility and wirelength Restore the routing solution to the idea RSMT as the congestions are eliminated Every net knows which grid edges are critical to itself Encourage the routing paths passing through the skeleton edges of RSMT n 1 n 2 n 1 n 2 n 3 n

146 Bounded-Length Maze Routing (BLMR) Heuristic-BLMR can identify a minimal-cost path from s to t such that wirelengh of the found path cannot exceed BL Discard path P s,v if wl(p s,v )+Manh(v,t) > BL Preserve a better path if more than one path from s to v BL=15 t t t s v v v P 1 s P 2 P 3 s wl(p 1 )=12 wl(p 2 )=10 wl(p 3 )=4 146

147 Bounded-Length Relaxation The BL of net n at iteration k is formulated as follows: k n n n BL Manh( s, t ) (arctan( k ) ) arctan(k 9)+2.5 Iteration k Initial Routing BL = 13 BL = 18 T T T Reroute Reroute S S S 14 7

148 Advantages of Heuristic-BLMR Specifically control the routing wirelength Restrict the searching region for accelerating routing Has good routing resource utilization by avoiding redundant detours. BL = 9 Maze routing within Bounding Box Heuristic-BLMR s t S T S T 148

149 Dynamically Adjusted History Cost A grid edge with high history cost implies that its routing resource is critical The proposed history cost can more precisely capture the criticality of grid edges h k 1 e h h k e k e h inc if e is overflowed otherwise h k 1 e h k e of k e conventional NCTU-GR A B C A B C A B C k k k of e 1 of e 2 of e

150 Appendix D: Box Router Cho and Pan DAC 2006 Unit 6 150

151 BoxRouter: Progressive Global Routing Cho & Pan, BoxRouter: A new global router based on box expansion and progressive ILP, DAC Incremental box expansion From the most congested region ( 擒賊先擒王 ) 2. Progressive ILP With adaptive maze routing 3. Post routing Rerouting w/o rip-up Global routing cell (G-cell) Try to minimize the total routing overflow Unit 6 151

152 Motivation for Box Expansion c c a d b b a d Different resource management for wires inside/outside of a box Box expansion pushes/diffuses congestion outwards progressively Unit 6 152

153 Overall Flow of BoxRouter Min Steiner tree construction Net decomposition Prerouting & initial box selection ILP routing BoxRoute Maze routing Box expansion all wires routed? N Iterative box routing Initial boxes Y Post routing Unit 6 153

154 Steiner Tree & Net Decomposition & Prerouting b a net a-b-c-d d c Flat wire (no via) wire a-e Minimum Steiner Tree: FLUTE [ICCAD-04], Spanning graph [ISPD-03], Geosteiner 3.1 Decompose into 2-pin wires Each wire becomes an atomic routing object Preroute as many flat wires as possible: 60% of nets Overall congestion captured, runtime improved Start with the box of the most congested region Unit b a e f d wire f-c c

155 目前無法顯示此圖像 BoxRouting in Action va vc b xb2 f xb1 Maze router xf1 h xh2 f b [DAC03] xh1 vb h vd x b x x 1 f 1 x h1 1 solution 1 b 2 max : Unit s.t : 0 x x x b1 b2 f 1 f 2 h1 h2 b1, xb2, x f 1, x f 2, xh 1, xh2 {0,1} b1 x f 1 xh 1 xb 1 xh 1 x x x b 2 xh 2 C C C AB BD AC b 2 xh 2 CCD 1 b , f h 2 1 x b x x f x x h x x x x x routing cap. constraint solution constraint

156 Progressive ILP with Box Expansion ILP typically incurs very high time complexity How to reduce the time complexity? Key: Progressive ILP Incorporate the solution from inner boxes Solve it between two consecutive boxes Unit 6 156

157 Comparison between ILP Formulations max : s.t : { xi 1 xi2} x i x i 1, 2 1 i 2 {0,1} x i x 1 x 0 i2 x C e x ij ij e i W box i W box i W box W flat i W box min : D s.t : x i, x i 2 x i x 1 1 i 2 x i2 x e x ij {0,1} 1 0 D ij i W box i W box i W box i W box W flat i W box ILP formulation with constant bounds Maximize # routed nets (min. total overflow) Work together with maze routing ILP formulation with bounds as objective Minimize max overflow Much slower!! (Try not to use formulation of this kind!!) Unit 6 157

158 Post Routing Reroute w/o ripping up other nets In the same order as box expansion Enhance routing path Tradeoff between wirelength and congestion Controlled by a parameter K va b xb2 vc x f h f b vb h vd x K => overflow, wirelength y K K>6? 20+K y 26 Unit 6 158

159 Appendix E: Bounded-Radius Minimum Spanning Tree Cong, Kahng, Robins, Sarrafzadeh, Wong ICCD-91, TCAD-92 Unit 6 159

160 Bounded-Radius Minimum Spanning Tree Problem: Given a parameter 0 and a signal net with radius R (diameter D), find a minimum-cost spanning tree T with radius r(t) (1 + )R (d(t) (1 + )D). Awerbuch, et al., Cost-sensitive analysis of communication protocols, ACM Symp. Principles of Distributed Computing, Cong, Kahng, Robins, Sarrafzadeh, Wong, Performance-driven global routing for cell based IC's, ICCD-91 (& TCAD, June 1992). R (D): the maximum length among all shortest paths from the source (among every pair of nodes). MST (minimum spanning tree) minimum cost; SPT (shortest path tree) minimum radius. Question: How to find a spanning tree with a good trade-off between cost and radius? Spanning tree T Unit 6 160

161 MST vs. SPT Trade-off Cost(SPT) may be ( n ) times larger than Cost(MST). Unit 6 161

162 Bounded Radius Bounded Cost Spanning Tree Algorithm: Bounded-Radius_Bounded-Cost_Spanning_Tree(G) 1 begin 2 Compute MST G and SPT G ; 3 Q MST G ; 4 L depth-first tour of MST G ; 5 S 0; 6 for i 1 to L S S + cost(l i, L i+1 ); 8 if S dist G (s, L i+1 ) then 9 Q Q minpath G (s, L i+1 ); 10 S 0; 11 T = shortest path tree of Q; 12 end L 2 L 1 Q minpath G (s, L 4 ) Q minpath G (s, L 8 ) L 6 L L 3 L 5 4 L L 7 8 S dist G (s, L 4 ) S dist G (s, L 8 ) Unit 6 162

163 Bounded-Radius Bounded-Cost Spanning Tree For any weighted graph G and parameter, the routing tree T constructed by the algorithm has radius r(t) (1+ )R. v i-1 : the last node before v on L for which we added minpath G (s, v i-1 ) to Q. Unit 6 163

164 Bounded-Radius Bounded-Cost Spanning Tree For any weighted graph G and parameter, the routing tree T constructed by the algorithm has cost cost(t) (1+2/)cost(MST G ). Let v 1, v 2,, v m be the set of nodes to which the algorithm added shortest paths from source s. Unit 6 164

165 Appendix F: Robust Channel Router Yoeli, TCAD Unit 7 165

166 Robust Channel Router Yoeli, A robust channel router, IEEE TCAD, Alternates between top and bottom tracks until the center is reached. The working side is called the current side. Net weights are used to guide the assignment of segments in a track, which favor nets that contribute to the channel density; favor nets with terminals at the current side; penalize nets whose routing at the current side would cause vertical constraint violations. Allows unrestricted doglegs by rip-up and re-route. Unit 7 166

167 Robust Channel Router Select the set of nets for the current side by solving the maximum weighted independent set problem for interval graphs. NP-complete for general graphs, but can be solved efficiently for interval graphs using dynamic programming. Main ideas: The interval for net i is denoted by [x imin, x imax ]; its weight is w i. Process channel from left to right column; the optimal cost for position c is denoted by total[c]; A net n with a rightmost terminal at position c is taken into the solution if total[c 1] < w n + total[x nmin 1]. Can apply maze routers to fix local congestion or to postprocess the results. (Why not apply maze routers to channel routing directly??) Unit 7 167

168 Interval Graphs There is a vertex for each interval. Vertices corresponding to overlapping intervals are connected by an edge. Solving the track assignment problem is equivalent to finding a minimal vertex coloring of the graph. Unit 7 168

169 Weight Computation d(1) = 1 d(2) = 2 d(3) = 2 d(4) = 3 (nets 2, 3, 4) d(5) = 2 Computation of the weight w i for net i: 1. favor nets that contribute to the channel density: add a large B to w i. 2. favor nets with current side terminals at column x: add d(x) to w i. 3. penalize nets whose routing at the current side would cause vertical constraint violations: subtract Kd(x) from w i, K = 5 ~ 10. Assume B = 1000 and K = 5 in the 1 st iteration (top side): w 1 = (0) + (1) + (-5 * 2) = -9 Net 1 does not contribute to the channel density One net 1 terminal on the top Routing net 1 causes a vertical constraint from net 2 at column 2 whose density is 2 Unit 7 169

170 Weight Computation (cont d) Computation of the weight w i for net i: 1. favor nets that contribute to the channel density: add a large B to w i. 2. favor nets with current side terminals at column x: add d(x) to w i. 3. penalize nets whose routing at the current side would cause d(1) = 1 d(2) = 2 d(3) = 2 d(4) = 3 (nets 2, 3, 4) d(5) = 2 vertical constraint violations: subtract Kd(x) from w i, K = 5 ~ 10. Assume B = 1000 and K = 5 in the 1 st iteration (top side): w 1 = (0) + (1) + (-5 * 2) = -9 w 2 = (1000) + (2) + (-5 * 3) = 987 w 3 = (1000) + (2+2) + (0) = 1004 w 4 = (1000) + (3) + (-5 * 2) = 993 Unit 7 170

171 Top-Row Net Selection w 1 = -9, w 2 = 987, w 3 = 1004, w 4 = 993. A net n with a rightmost terminal at position c is taken into the solution if: total[c 1] < w n + total[x nmin 1]. total[1] = 0 selected_net[1] = 0 total[2] = max(0, 0-9) = 0 selected_net[2] = 0 total[3] = 0 selected_net[3] = 0 total[4] = max(0, w 2 +total[1]) = 987 selected_net[4] = 2 total[5] = max(987, , 0+993) = 1004 selected_net[5] = 3 Select nets backwards from right to left and with no horizontal constraints: Only net 3 is selected for the top row. (Net 2 is not selected since it overlaps with net 3.) Unit 7 171

172 Bottom-Row Net Selection nd iteration: bottom-row selection w 1 = (1000) + (2) + (0) = 1002 w 2 = (1000) + (2) + (-5 * 2) = 992 w 4 = (1000) + (1) + (-5 * 2) = total[1] = 0 selected_net[1] = 0 total[2] = max(0, ) = 1002 selected_net[2] = 1 total[3] = 1002 selected_net[3] = 0 total[4] = max(1002, 0+992) = 1002 selected_net[4] = 0 total[5] = max(1002, ) = 1993 selected_net[5] = 4 Nets 4 and 1are selected for the bottom row. Unit 7 172

173 Maze Routing + Rip-up & Re-route rd iteration Routing net 2 in the middle row leads to an infeasible solution. Apply maze routing and rip-up and re-route nets 2 and 4 to fix the solution. Unit 7 173

174 Robust Channel Router /* for */ /* if */ /* for */ Unit 7 174

175 Appendix G: Gridless Routing Unit 7 175

176 Gridless Full-Chip Routing Chen and Chang, Multilevel full-chip gridless routing considering optical proximity correction, ASPDAC-05 (TCAD-08). Is based on the multilevel routing framework. Applies the implicit connection graph to transform the gridless structure into a grid -like structure. Adopts the interval tree to do range query for identifying obstacles and available spaces. Congestion metric is based on available routing space in a region. Gridless routing is needed for handling some nanometer effects like OPC, metal slotting, etc. Unit 7 176

177 Implicit Connection Graph Given a set of obstacles and a source s and sink t G S is an orthogonal grid graph T S Unit 7 177

178 Implicit Representation of G S Store x, y coordinates into two sorted arrays O(n) space & O(n lg n)-time pre-construction (n: # of rectangles) y 1 y 2 T y 3 y 4 y 5 y 6 y 7 S y 8 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 Unit 7 178

179 Partition the chip plane into rows and keep a query tree for each row. The query tree stores cut blocks in the internal nodes. Uncut blocks are stored in the leaf nodes. Question: Is q in free space? Need average O(lg n) time to answer. a Interval Tree c b q c d d b Unit 7 179

180 Block Insertion To insert a block (say, block e), we traverse the query tree from the root down to an internal node until it is cut, or a leaf node if it is uncut. Average time complexity: O(lg n). a c q e b d c b e d Unit 7 180

181 Block Deletion To delete a block (say, block c), we traverse the query tree from the root down to a node that stores the block name and delete the block. Average time complexity: O(lg n). a c q e b d c b e d Unit 7 181

182 Interval (query) tree Complexity Summary Average Complexity Memory Usage O(n 2 ) Block Insertion O(lg n) Block Deletion O(lg n) Point Finding O(lg n) Neighbor Finding O(1) n: number of blocks Unit 7 182

183 Routing Solution for s5378 The routing solution of s5378 with nonuniform (above) and uniform (below) nets Courtesy of Tai-Chen Chen Unit 7 183

184 Appendix H: Flip-Chip/InFO Routing Unit 7 184

185 Flip-Chip Technology The flip-chip package is a technique for connecting a die to an external circuitry such as package carrier Advantages More areas for I/Os, higher performance with shorter interconnections, better signal integrity and robustness Use redistribution layer (RDL) to connect I/O pads to bump pads Die (flippe d over) Cells and I/O buffers Metal 1 Top metal layer Redistribution layer Package carrier I/O pad Isolator Bump pad Bump bal

186 Integrated Fan-Out Wafer-Level Chip-Scale Package A chip is surrounded by a molding compound which spreads the package footprint outside the chip Allow I/Os to be both fanned in over the chip and fanned out across the molding compound An integrated fan-out (InFO) wafer-level chip-scale package (WLCSP) can integrate multiple chips Molding compound Chip1 Chip2 RDLs Fan-out region Micro-bump

187 Flip-Chip (RDL) Routing Flip-chip routing is to route I/O to bump pads on RDL Two types of the flip-chip package: (a) peripheral-i/o and (b) area-i/o I/O pad (a) Bump pads I/O pad (b) (c) Routing instance

188 Free-assignment vs. Pre-assignment Routing Free-assignment routing Each bump pad can be connected to any I/O pad Router can assign each I/O pad to a bump pad Pre-assignment routing Connections between I/O and bump pads are predefined Router must connect a bump pad to its pre-assigned I/O pad Free-assignment Pre-assignment

189 Free- Assign ment Existing Works on RDL Routing Peripheral I/O, free-assignment Fang et al., ICCAD 05 (TCAD 07): Flow Liu et al., DAC 10: VD+Flow Area I/O, free-assignment Fang & Chang, ICCAD 08 (TCAD 09): Flow Yan & Chen, ASPDAC 09: VD Peripheral I/O Peripheral-I/O, unified FA/PA Lin et al., ICCAD 16: DP+Flow Peripheral I/O, pre-assignment Fang et al., DAC 07 (TCAD 09): ILP Lee et al., ICCAD 09 (TCAD 12): DP Yan & Chen, GLSVLSI 09: Greedy Area-I/O, unified FA/PA Area I/O, pre-assignment Fang et al., DAC 09 (TCAD 10): VD+Flow Pre- Assign ment VD: Voronoi Diagram Area I/O DP: Dynamic Programm

190 Network Flow for Free-assignment Routing Most current methods for free-assignment routing are based on Minimum-Cost Maximum-Flow (MCMF) I/O pad source node Bump pad sink node Wire unit flow Max-flow highest routability Min-cost shortest wirelength Routing result S T I/O pad Bump pad

191 Peripheral-I/O Free-assignment Routing First addressed with a network-flow based approach in [Fang et al., ICCAD 05] Divide chip into four regions and handle them separately Bottom side

192 Capacity Node Construction Tile node represents routing capacity of a rectangular tile formed by four pads Intermediate node represents routing capacity between two adjacent pads #nets routed through these nodes are constrained by their capacities Bump pad Bottom side I/O pad Intermediate node Tile node

193 Flow Network Construction Construct tile and intermediate nodes Construct flow edges from outside to inside Connect pads to super-source and super-sink Bottom side t s

194 Detailed Routing Detailed routing assigns exact location for each net After network-flow processing, split edges to get independent nets without any wire crossing Modify the net ordering determination [Hsu,DAC-83] Apply maze routing We shall focus on global routing and omit detailed routing since global routing is much more critical Trac k Intermediate node Bump Pad I/O Pad

195 Peripheral I/O, Free-assignment Routing Result Circuit: fs900

196 Area-I/O Free-assignment Routing First addressed in [Fang and Chang, ICCAD 08, TCAD 09] Unlike peripheral-i/o structure, the I/O pads is placed among tiles formed by bump pads Bump pad I/O pad

197 Modified Flow Network Flow edges go in every direction, instead of outside-toinside only Every pad has edges connecting to adjacent intermediate and tile nodes The capacity of a tile node is reduced if there are I/O pads inside Bump pad I/O pad Intermediate node Tile node Edge with unit capacity Edge with capacity C t or C d

198 Triangular-tile Based Method Fang et al. [DAC 09, TCAD 10] & Liu et al. [DAC 10] proposed a network-flow method based on Delaunay triangulation and Voronoi diagram Apply Delaunay triangulation on bump pad vertices, I/O pad centers, and auxiliary points on chip boundary Bump pad I/O pad Chip boundary

199 Flow-Network Construction with VD Construct Voronoi diagram with triangulation result Treat VD nodes and edges as flow network and add additional nodes and edges Bump pad I/O pad Chip boundary

200 Flow-Network Construction with VD VD network resembles the channels between pads and boundaries The additional edges simply connect sinks and sources to these channels Bump pad I/O pad Chip boundary

201 Capacity Tiles in Flow Methods In network-flow methods, it is crucial to model the routing resource with appropriate tile model There are usually two types of tile models Rectangular-tile model Triangular-tile model Bump pad Rectangular-tile model Triangular-tile model

202 Pre-assignment Routing Connections between I/O and bump pads are pre-defined Network-flow-based methods are no longer applicable Under the pre-assignment constraint, RDL routing becomes an NP-complete multi-commodity flow problem I/O pad Bump pad Pre-assignment routing

203 Peripheral-I/O Pre-Assignment Routing Fang, Hsu & Chang used integer linear programming (ILP) for pre-assignment routing [DAC 07, TCAD 08] Consider every possible path for a net, from outer ring into inner ring 1, 2 3 Bump pad b i b j I/O pad

204 ILP Formulation The ILP formulation is to minimize total wirelength under following constraints: Wire crossing avoidance Congestion control Flow conservation Reduction techniques are applied to reduce the problem sizes while preserving the optimality

205 Non-ILP Methods ILP methods are generally very time-consuming Some other methods utilize the regularity of flip-chip structure and achieve better efficiency

206 Sequence Exchange Lee et al. proposed a dynamic-programming, sequence-exchange based method to solve this problem efficiently [ICCAD 09, TCAD 12] If two net orders on bump side and I/O pad side are reversed: (a) The two nets would cross each other We can avoid net crossing by either: (b) Detour net 2 in I/O pad side; net sequence becomes [2,1] (c) Detour net 2 in bump pad side; net sequence becomes [1,2] n 1 n 2 n n 2 n 1 n (a) (b) (c)

207 Pre-assignment Routing Objective: Connect I/O pads and bump pads according to a predefined assignment between the I/O and bump pads such that the total wirelength is minimized Key: Route in the ring-by-ring manner to get linear order ( dynamic programming) I/O pad bump pad ring net assignment (1) (2) 207

208 Min. #Detours by Longest Common Subsequence Cut the rings into lines/segments (form linear orders) d 1 d 3 d 1 d 3 n 1 n 3 n 1 n 2 n 1 S b d 2 d 4 n 2 n 4 b 1 b 2 b 3 b 4 n 3 b 1 n 2 b 2 n 1 b 3 n 4 b 4 (1) (2) S d d 1 d 2 d 1 d 3 d 3 n 3 n 4 n 3 net sequence d 1 b 1 b 2 b 3 b 4 d 4 d 3 d 1 d 2 d 1 d 3 d 4 d 3 (3) (4) n 3 n 2 n 1 n 4 S d =[1,2,1,3,4,3] S b =[3,2,1,4] Need detour for only n 3 208

209 209 #Detours Minimization by Dynamic Programming Longest common subsequence (LCS) computation d 1 d 2 d 3 b 3 b 1 b 2 d 1 d 2 d 3 b 3 b 1 b 2 seq.1=<1,2,3> seq.2=<3,1,2> Common subseq = <3> d 1 d 2 d 3 b 3 b 1 b 2 (c) LCS= <1,2> (a) (b) d 1 d 2 d 3 b 3 b 1 b 2 d 1 d 2 d 3 b 3 b 1 b 2 d 1 d 2 d 3 b 3 b 1 b 2 (d) (e) b 4 b 5 b 6 b 4 b 5 b 6 b 4 b 5 b 6 chord set: {3,4,5} subset:{3} (f) MPSC = {4,5} Maximum planar subset of chords (MPSC) computation

210 Supowit's Algorithm for Finding MPSC Supowit, Finding a maximum planar subset of a set of nets in a channel, IEEE TCAD, Problem: Given a set of chords, find a maximum planar subset of chords. Label the vertices on the circle 0 to 2n-1. Compute MIS(i, j): size of maximum independent set between vertices i and j, i < j. Answer = MIS(0, 2n-1). Vertices on the circle

211 Supowit's Dynamic Programming Algorithm Apply dynamic programming to compute MIS(i, j ). 211

212 Ring-by-Ring Routing Example Decompose chip into rings of pads and route from inner rings to outer rings Keep applying LCS & MPSC between two adjacent rings (red ring and blue ring) current ring preceding ring Achieve over 100X speedups over ILP

213 Global Routing Results for fc2624 A routing path for each net is guided by a set of segments Segment 213

214 Detailed Routing: Phase 1 Segment-by-segment routing in counter-clockwise order As compacted as possible 214

215 Detailed Routing: Phase 2 Net-by-net re-routing in clockwise order Wirelength and #bends minimizations 215

216 Unified-Assignment Problem A routing instance could contain both free-assignment (FA) and pre-assignment (PA) nets Some critical nets are pre-assigned, while other nets are left free-assigned Considering PA and FA nets simultaneously would lead to better result than considering them separately

217 Area-I/O, Unified-assignment Algorithm Fang, Wong & Chang proposed an algorithm to handle unified-assignment by congestion estimation and routing FA and PA nets iteratively [DAC 09, TCAD 10] The routing network is constructed using Delaunay triangulation and Voronoi diagram

218 1. Congestion Estimation Iterative Improvement Apply min-cost max-flow (MCMF ) algorithm on the freeassignment (FA) flow network and identify the edges with potential congestion; the nets are not routed yet 2. PA Net Routing Apply sequential maze routing on PA nets; previously identified congested nets are avoided by high cost 3. FA Net Routing Apply MCMF to actually route FA nets; edges used in PA routing cannot be crossed 4. Iterative Improvement Iteratively rip-up and re-route to finish remaining unrouted wires

219 Unified-assignment Routing Result Circuit fcpb3: 346 I/O pads and 441 bump pads Red free-assignment wires, black pre-assignment wires, green I/O pads, and blue bump pads