Maze Routing مرتضي صاحب الزماني

Improvement to Lee’s Algorithm Improvement on memory: Aker’s Coding Scheme Improvement on time: Starting point selection Double fan-out Framing Hadlock’s Algorithm Soukup’s Algorithm مرتضي صاحب الزماني

Aker’s Coding Scheme to Reduce Memory Usage “A Modification of Lee’s Path Connection Algorithm”, S.B. Akers, IEEE Transactions on Electronic Computers, pages 97-98, Feb. 1967. مرتضي صاحب الزماني

Aker’s Coding Scheme For the Lee’s algorithm, labels are needed during the retrace phase. But there are only 2 possible labels for neighbors of each vertex labeled i, which are, i-1 and i+1. So, is there any method to reduce the memory usage? مرتضي صاحب الزماني

Alternative Paths A B 2 3 4 5 6 7 8 9 10 11 12 1 مرتضي صاحب الزماني

Aker’s Coding Scheme One bit (independent of grid size) is enough to distinguish the two labels. S Sequence: ??? ...… (what sequence?) (Note: In the sequence, the labels before and after each label are always different.) T مرتضي صاحب الزماني

Improving Lee’s Algorithm The choice of the starting pin can affect the speed of the algorithm, one guideline is to start on the pin farthest from the center of the chip Use double fan out - begin at both pins and continue until a point of contact is made Use framing, an artificial rectangular boundary 10 to 20% larger than the boundary formed by the pins مرتضي صاحب الزماني

Schemes to Reduce Run Time 1. Starting Point Selection: 2. Double Fan-Out: 3. Framing: T S S T S S T T مرتضي صاحب الزماني

Multi-Terminal Nets For a k-terminal net, connect the k terminals using a rectilinear Steiner tree with the least wire length on the maze. This problem is NP-Complete! So, just want to find some good heuristic. This problem can be solved by extending the Lee’s algorithm. Any idea ….. مرتضي صاحب الزماني

Connecting Multipoint Nets One point is selected as the source and all the other points are the target propagate from the source until one target is reached find the path from the source to that target all the cells on the path are labeled as source cells and the remaining unconnected pins are targets repeat the steps مرتضي صاحب الزماني

Example 4 4 5 3 3 2 2 1 1 S T Start at the source and run the maze router until you hit a target Every cell on the path is a source – run the maze router مرتضي صاحب الزماني

Soukup’s Algorithm to Reduce Run Time “Fast Maze Router”, J. Soukup, DAC 1978, p.100-102. مرتضي صاحب الزماني

Soukup Router مرتضي صاحب الزماني

Basic Idea 2 2 2 1 1 S 1 1 1 1 T 2 1 1 2 2 مرتضي صاحب الزماني

Soukup Router Algorithm مرتضي صاحب الزماني

Hadlock’s Algorithm to Reduce Run Time “A Shortest Path Algorithm for Grid Graphs”, F.O. Hadlock, Networks, 7:323-334, 1977. مرتضي صاحب الزماني

Hadlock’s Algorithm Detour Number: For a path P from S to T, d(P) = # of grids directed away from T, then L(P) = MD(S,T) + 2d(P) So minimizing L(P) and d(P) are the same. Length Manhattan Distance D D D: Detour d(P) = 3 MD(S,T) = 6 L(P) = 6+2x3 = 12 D S T مرتضي صاحب الزماني

Hadlock’s Algorithm Label vertices with detour numbers instead of distance. Vertices with smaller detour number are expanded first. Therefore, favor paths without detour. 3 2 2 3 2 1 1 1 S 1 1 2 T 2 1 1 2 2 3 2 2 2 2 2 مرتضي صاحب الزماني

Hadlock’s Algorithm مرتضي صاحب الزماني

Multilayer Routing Give a system with multiple wire layers Parallel grids vertically stacked, one for each layer Use vias to access other layers Label cells as to whether a via is permitted at its location How do we find wire paths in such a structure? مرتضي صاحب الزماني

Multilayer Routing مرتضي صاحب الزماني

Aside: VIAs Vias: Issue: size “Vertical” electrical connection On chips, vias are usually a lot bigger than wire widths so you have to be careful where you put them You can’t put vias as close to each other as you can put wires مرتضي صاحب الزماني

Example Given two metal layers with vias allowed in some cells (labeled as ‘v’) 2 4 5 3 1 Expansion may go up and down as well as to adjacent cells v S T v v Layer 1 مرتضي صاحب الزماني Layer 2

References David Pan, VLSI Physical Design Automation, Lecture Slides, University of Texas, 2009. مرتضي صاحب الزماني