Detailed Routing مرتضي صاحب الزماني

Left-Edge Algorithm Vertical constraint may occur here. Assumptions: One horizontal routing layer No vertical constraint, e.g., VHV model Always gives a solution with channel width equal to channel density, i.e., optimal solution. 2 1 Vertical constraint may occur here. 2 1 1 2 مرتضي صاحب الزماني

Left-Edge Algorithm All the wires are sorted in increasing order of their left edge Scan the sorted list of wires the first unplaced wire is placed in the first track and that wire is deleted from the list Search the sorted list to find the first wire which does not overlap the wire which was just placed If there are no other non-overlapping wires then move to the next track مرتضي صاحب الزماني

Example LEA example: 1 3 5 6 3 1 2 4 2 4 1 5 7 7 6 Step One: The sorted list of wires (by left edge) is: (1,2,3,4,5,6,7) مرتضي صاحب الزماني

Sorted List 1 3 5 6 3 1 2 4 2 4 1 5 7 7 6 1 2 3 4 5 6 7 مرتضي صاحب الزماني

Step Two The first wire in the list is 1, so place it on track 1 the first non overlapping wire is 6, so place it on track 1 as well 1 3 5 6 3 1 2 4 2 4 1 5 7 7 6 7 1 2 3 4 5 6 مرتضي صاحب الزماني

Step Three There are no other non-overlapping wires so move to the second track and start over with the revised sorted list (2,3,4,5,7) place wire 2, followed by 5, followed by 7 1 3 5 6 3 1 2 4 2 4 1 5 7 7 6 7 2 3 4 5 مرتضي صاحب الزماني

Example - Final Result The next two cycles will place wire 3 on track 3 and wire 4 on track 4 1 3 5 6 3 1 2 4 2 4 1 5 7 7 6 3 4 مرتضي صاحب الزماني

Problem Short circuits are possible on the vertical segments: 1 3 5 6 3 1 2 4 2 4 1 5 7 7 6 We need to account for the vertical wire constraints مرتضي صاحب الزماني

Constrained LE Algorithm The Constrained Left Edge Algorithm uses a vertical constraint graph, VCG(V,E), to document cases where one wire must be placed above another in a routing VCG is a directed graph The nodes, V, are the wires There is a directed link from node i to node j if there is a vertical column in which node i has a pin on top and node j has a pin on the bottom مرتضي صاحب الزماني

VG Example For the example problem the VG graph looks like: 6 3 4 5 7 1 3 5 6 3 1 2 4 2 4 1 5 7 7 6 6 3 4 5 7 1 2 مرتضي صاحب الزماني

CLE Algorithm Modify the Left Edge Algorithm in two ways RESULT: Only place a wire if it has no descendants in VG When a wire is placed it is removed from the VG RESULT: 1 3 5 6 3 1 2 4 2 4 1 5 7 7 6 مرتضي صاحب الزماني

Solution Steps 1 3 5 6 3 1 2 4 2 4 1 5 7 7 6 7 1 2 3 4 5 6 6 6 3 3 5 5 4 1 4 2 7 7 1 2 مرتضي صاحب الزماني

Vertical Constraint Loop This algorithm will fail if there is a cycle in the vertical constraint graph - vertical constraint loop occurs when a constraint in one column requires wire A to be placed above wire B while a constraint in another column requires wire B to be placed above wire A EXAMPLE 1 1 2 2 1 Solution: use a dogleg 1 1 2 2 1 1 2 مرتضي صاحب الزماني

Deutch’s Dogleg Algorithm “A Dogleg Channel Router”, D. Deutch, DAC, pages 425-433, 1976. مرتضي صاحب الزماني

Reduce Channel Width by Dogleg Even without cycle in the VCG, Dogleg is useful because it can reduce channel width. Without Dogleg With Dogleg 1 1 2 2 ??? 2 3 3 مرتضي صاحب الزماني

Deutch’s Dogleg Algorithm Split each multi-terminal net into several horizontal segments. Split only at columns that contain a pin (i.e. using Restricted Dogleg only). Simply apply the Constrained Left-edge algorithm. Restricted Dogleg Unrestricted Dogleg 1 1 2 2 1 1 2 2 3 3 2 3 3 مرتضي صاحب الزماني

Deutch’s Dogleg Algorithm 1 1 2 2 3 Vertical Constraint Graph 1 1 2a 2b 3a 3b 2 3 4 4 2 3 3 4 4 Without Dogleg 1 1 2 2 3 2 3 3 4 4 مرتضي صاحب الزماني

Deutch’s Dogleg Algorithm 1 1 2 2 3 1 Vertical Constraint Graph after splitting 1 2a 3a 2a 2b 3a 3b 2b 3b 4 4 2 3 3 4 4 By Deutch’s Dogleg Algorithm 2a, 3a, 4 1 1 2 2 3 3a, 4 3a, 2b, 3b 1, 2b, 3b 1, 2b 2 3 3 4 4 مرتضي صاحب الزماني

Drawbacks of CLEA Vertical Constraint Graph By Constrained Left- 1 3 3 Vertical Constraint Graph 1 3 2 1 2 2 1 3 2 By Constrained Left- edge algorithm There is a better solution... مرتضي صاحب الزماني

Drawbacks of CLEA By Constrained Left- edge algorithm A better solution 1 3 3 1 3 3 1 2 2 1 2 2 What’s wrong with the CLEA? It takes care of the vertical constraints, but it does not have a global view to optimize the length of the longest path in the VCG. مرتضي صاحب الزماني

Constrained LEA Cannot resolve VCG cycle problem. Routing Sequence: The following sequence gives better results: Switch between up and down tracks. Switch between “left” and “right” edge. مرتضي صاحب الزماني