1. LEMAR: A Novel Length Matching Routing Algorithm for Analog and Mixed Signal Circuits H. Yao, Y. Cai and Q. Gao EDA Lab, Department of CS, Tsinghua University, Beijing, China ASPDAC 2012

2. Outline Introduction Problem Formulation and Overall Flow Tile-based Detouring Tile Path Searching and Detouring Experimental Results Conclusions

3. Introduction The design automation for analog and mixed signal circuit components in SOCs is attracting increasing interests. Mismatches in electrical characteristics (e.g., parasitic capacitance and resistance) between two matching nets will cause circuit malfunctions. It is very important to enforce the matching constraint on certain analog signals.

4. Introduction

5. Problem Formulation This paper focused on the single-layer length matching routing problem for two nets. Given:  A single-layer routing area R with a set of routing obstacles O and two nets n 1 and n 2. Find:  Routing paths p 1 and p 2 for nets n 1 and n 2 to minimize the total wirelength. Subject to:  The routing obstacles in O are avoided.  There are no design rule violations.  p 1 and p 2 are of the same length.

6. Overall Flow

7. Tile-Based Detouring Partitioning of the routing area  Expand obstacles by WS min /2 called obstacle tiles.  Represent the source and target pins as rectangular tiles called pin tiles.  The routing area is partitioned by the boundaries of all the pin tiles and obstacle tiles.

8. Tile-Based Detouring There is a special type of space tiles called tiny tiles, whose width or height is smaller than a pitch. pitch = WS min + WW min. During the partitioning of the routing area, merge the tiny tiles into the adjacent tiles.

9. Tile-Based Detouring Estimation of the detouring length  Nets can make detours in the free tiles.  The detouring length L d can be calculated as follows: S a : the area of the original tile  S: the area of the wasted region

10. Tile-Based Detouring Transformation of Different Tiles  According to the entrance and exit points within the tile, free tiles can be classified into two categories: Entrance and exit are on the diagonal corners. Entrance and exit are on the same edge.  The two different cases can be transformed into single type of tile by some operations.  All the free tiles are transformed into a single case, where entrance and exit are on the bottom left and top right corners of tile.

11. Tile-Based Detouring Transformation of Different Tiles

12. Tile-Based Detouring Detouring Pattern  To obtain maximum detouring length, adopt two detouring patterns according to the geometric shape of a free tile: Narrow pattern: width or height of a tile < 4 X pitch Standard pattern: width and height of a tile > 4 X pitch Narrow patternStandard pattern

13. Tile Path Searching and Detouring A*-Search Algorithm for tile path Terminate condition Backtrack strategy costF = costG + costH costG: cost from t s to t cur costH: Manhattan distance from t new to t t.

14. Tile Path Searching and Detouring The backtrack strategy is used in the tile searching process. If t cur is t t, but the max length from t s to t t of the current tile path is smaller then L obj, the algorithm would set t cur free. t cur would be equal to a neighbor tile of t t, and expand to t t again. This would force tile path to detour around the t t to increase the detouring length.

15. Tile Path Searching and Detouring Minimum Length and Maximum Length  If there is not a turn at t cur, the relative position of source point in t new is the same as t cur.

16. Tile Path Searching and Detouring Minimum Length and Maximum Length  If the first turn of tile path is at t cur, the width or height of t cur is pitch.

17. Tile Path Searching and Detouring Detouring after Searching  The routing path is detoured in the tiles one by one from t t to t s.  The desired wirelength in each tile can be calculated as follows:  The maximum detour length of each tile: L max t  The detouring length L d of each tile:

18. Tile Path Searching and Detouring Adjust the height of some detouring parts

19. Experimental Results

22. Conclusions This paper presents a novel detailed routing algorithm for length matching problem. Future work includes extending the algorithm for matching more than two nets simultaneously and evenly distributed detouring wires to avoid local congestion.