Layout DA (Physical Design)
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1 Layout DA (Physical Design) n Floor Planning n Placement and Partitioning n Global Routing n Routing n Layout Compaction 1
2 Routing n Types of Local Routing Problems n Area Routing u Lee s Algorithm n Channel Routing u Channel Routing Models u Constraint Graph u Algorithm n Global Routing 2
3 Types of Routing Problems n global routing (loose routing) n detail routing (local routing) n area routing n channel routing 3
4 Types of Local Routing Problems characterizing parameters. n The number of wiring layers. n The orientation of wire segments. u Reserved-layer models. n Gridded or gridless. n The presence or absence of obstacles. n The position of terminals. n Terminal s with a fixed or floating position. n Permutability of terminals. n Electrically equivalent Terminals. 4
5 Area Routing n Terminals are allowed anywhere in the area available for routing. n Lee s Algorithm (1961). maze routing or path connection algorithm. n wave propagation, backtracing, and cleanup. 5
6 The basic version of Lee s Algorithm. (a) problem. (b) the wave propagated. (c) the path found. (d) cleanupped. 6
7 Lee Algorithm n O(n 2 ) n If a path exist, the algorithm always finds the shortest connection. n Multi( 3)-terminals net connection. u not the shortest connection. u the shortest connection is NP-Hard problem. n Ordering problem : nets have to be routed sequentially. 7
8 The basic version of Lee s Algorithm. 8
9 The basic version of Lee s Algorithm. (Wave propagation phase) 9
10 The basic version of Lee s Algorithm. (Backtracing phase) 10
11 Channel Routing n A natural problem occurs in standard-cell and building-block layout styles. n Routing nets across a rectangular channel. n The nets have fixed terminals at the top and bottom of channel and floating terminals at the open sides at the left and the right. n Height of channel is not fixed. n Similar but more difficult problem : u SWITCH BOX ROUTING 11
12 Channel Routing Models (classical model) n All wire run along orthogonal grid lines with uniform separation. n There are two wiring layer. n Horizontal segments are put on one layer and vertical segments on the other one. n For each net, the wiring is realized by a single horizontal segment, with vertical segments connecting it to terminals. n An exception : Cycles in the vertical constraint graph occurs. 12
13 An instance of the channel routing problem and a solution. 13
14 Variations n Gridless routing model u each wire has specific width. n Non-Orthogonal grids model u 45 turns allowed. n Algorithm can deal with more than 2-layers. n Non-reserved layer model. n The use of doglegs. 14
15 Non-reserved layer model Doglegging 15
16 The Vertical constraint graph (VCG) n Each column having two terminals in the sane layer gives rise to a vertical constraint. n Vertices : the endpoints of the terminal segments. n Directed edges : the relation shuold be located above. 16
17 (a) A simple channel routing problem. (b) VCG based on individual column. 17
18 (c) the VCG when one horizontal segment per net is used. (d) the unique solution within the classical model 18
19 (a) A simple problem. (b) A circular VCG. (c) The cycle is broken by splitting one of vertices. (d) Solution with dogleg. 19
20 Horizontal Constraint and the Left-edge algorithm n General channel routing problem is NP-Complete. n the left-edge algorithm (by Hashimoto-Stevens) optimally solves an instance of the channel routing problem without vertical constraint. n Worst-case time complexity of the left-edge algorithm is O(nlogn + dn) u n : number of intervals u d : the density (number of rows in solution) 20
21 Horizontal Constraint and the Left-edge algorithm n Horizontal Constraint : The segments put on the same row which belong to different nets cannot overlap. n Interval graph G(V,E) u for each interval i, vertex v V and edge (vk,vl) exist, if the corresponding intervals ik and il overlap. n The minimum number of rows for instance without vertical constraints equivalent to finding a vertex coloring of the interval graph with a minimal number of colors. 21
22 An instance of channel routing problem without vertical constraint. 22
23 Horizontal Constraint : the interval graph and its coloring. 23
24 The left-edge algorithm 24
25 Solution as found by the left-edge algorithm. 25
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