Routing 2

1. Outline Maze Routing Line Probe Routing Channel Routing Goal Understand maze routing Understand line probe routing Routing 2

2. Place wires and components on grid grid size = wire width + spacing works best if all wire widths and spacings are equal grid squares occupied Distance metric each square has 4 neighbors unit cost away Search graph node for each square unit-cost edge to 4 neighboring nodes w+s 1 1 1 1 1 1 1 1 Maze Routing

3. Shortest path search (Lee-Moore) maintain leaf nodes of expansion tree at each step add unexplored neighbors to list with accumulated cost stop when target vertex reached backtrace for route Guaranteed to find shortest path large memory requirements - the grid long search time - number of vertices marked local optimization - only routing one net at a time 3 3 2 3 3 2 1 2 3 3 2 1 0 1 2 3 3 2 1 2 3 3 2 3 3 16 15 14 13 12 11 10 9 8 7 6 5 4 16 15 14 13 5 4 3 16 15 14 8 7 6 5 4 3 2 9 8 7 6 5 2 1 16 15 14 13 12 11 10 8 7 1 0 13 12 11 9 8 2 1 16 15 14 10 8 7 3 2 15 9 8 7 6 5 4 3 16 Maze Routing

4. add source node S into list costS = 0 for all nodes i on list if i == destination D pathcost = costD break for j=N,S,E,W neighbors that are not visited and not occupied add node j to list costj = costi+1 mark i as visited remove node i from list while i != S do mark i as path i = min cost N,S,E,W visited neighbor Algorithm

5. Avoiding blockage wire at a time => can block wires Solutions rip-up and re-route remove wire(s) causing blockage route blocked wire route ripped up wires shove-aside add dummy grid lines squeeze wire through A A A B B A B A B A B Maze Routing

6. Gridless - store list of lines and obstructions sorted lists of vertical and horizontal lines Algorithm from source and destination project 4 horizontal/vertical rays (probes) if probes intersect, done - route wires from nodes to intersection if probes blocked, choose escape point and send new probes a point just past obstruction Line Probe Routing E E B E A E

7. Advantages small memory requirements - no grid store sorted lists of vertical and horizontal segments fast binary search of segments for blockage and escape points # probes << # grid points higher precision coordinates - no grid Disadvantages serial wire at a time - blockages similar rip-up and shove-aside approaches to cope basic algorithm may not find route when one exists need to try more escape points degenerates to grid search Line Probe Routing

8. Channel terminals on two sides of rectangle are fixed horizontal wires on layer1 - trunks, which run in tracks vertical wires on layer2 - branches, which run in columns Routing Problem minimize number of tracks used => minimizes channel height minimize wire lengths, vias all wires routed at same time - better overall optimization Channel Routing A B A A track via B A branch trunk dogleg

9. Left edge algorithm one trunk per net trunks on one layer, branches on other sort trunks by left edge scan trunks, put all that fit into first track allocate new tracks as necessary repeat scanning until done add vertical branches ignores vertical constraints can add doglegs later may need more tracks advantage minimum wire length, vias not very optimal 31 tracks on “Deutsch’s difficult example” density is 19 tracks Channel Routing B A B A B A B A

10. Greedy algorithm route column by column rather than track by track scan left-right by column at each track apply heuristics to bring new nets to an existing trunk, and move nets between tracks greedy - at each column try to minimize tracks and minimize distance to terminals can often use exhaustive search number of tracks in a column is small - < 100 good results, but lots of vias lots of track to track movement for each net Channel Routing

11. for each column for each terminal bring in branch connect to trunk if it exists create trunk at first empty track otherwise create new track move between tracks using heuristics collapse nets to fewer trunks with spare branch space reduce distance between net trunks move nets towards next terminal connect multiple trunks for a net at end of channel Algorithm A B B A

12. Slide window along channel recognize patterns of terminals and occupiedGet more optimal results doglegs Challenge developing rule set Pattern Channel Routers Problem: Vertical constraint between A and B. C and D occupy surrounding tracks and columns Solution: Shift contacts down and to right, doglegs on each layer to reach them A C D B