Question

3. Given the following grid: a. Perform Lee Algorithm to solve the maze and connect S to T. b. Re-apply Lee Algorithm but with time reducing techniques (discuss all given in the book). C. Re-apply Lee Algorithm but with memory reducing techniques (discuss all given in the book). d. Perform Hadlock Algorithm to solve the maze and connect Sto T.

S. Sait and H. Youssef, VLSI Physical Design Automation, McGraw Hill, 1995.

Answer 1

Maze Router: Lee Algorithm • Lee, "An algorithm for path connection and its application," • Discussion mainly on single-layer routing • Strengths - Guarantee to find connection between 2 terminals if it exists.- Guarantee minimum path. • Weaknesses-Requires large memory for dense layout - Slow • Applications: global routing, detailed routing

817 18 1817161718 7 6 5 6 7 8 T7 6 5 4 5 6 1876543 4 5 6 7 87654323456 7 6 5 4 3 2 1 6 5 4 3 2 1 5 .6 5 4 3 2 1 8 7 6 5 43 876545 81765 81716 81718 8 87678 765678 T 7 6 5 4 5 6 7 8 87654345678 87654312345678 7 6 5 4 3 2 1 6 7 8 654 321 6543211 76 543 876545 8765 8|76|7 1878 8 Filing Retrace • Time & space complexity for an M x N grid: O(MN) (huge!)

Reducing Memory Requirement Adjacent labels for k are either k-1 ork+1. - Want a labelling scheme such that each label has its preceding label different from its succeeding label. • Way 1: coding sequence 1,2,3, 1, 2, 3,...; states: 1, 2, 3, empty, blocked (3 bits required) • Way 2: coding sequence 1, 1, 2, 2, 1, 1, 2, 2, ...; states: 1, 2, empty, blocked (need only 2 bits)

2 1/2 21312 1 3 2 3 1 2 T 1 3 2 1 2 3 1 2 2 132 1312312 213 21312312312 213 211321 T2H 321321s 321321 213 213 2|1|32|12 2132 2131 21112 2212 221221 2 1 1 1 2 21 T211 21122|| 2211221211221 221 122 122112 2 2 2 1122 11 1 22 1 122 115 1122 11 2211 220 2 2 1122 2 2 11 2212 2122 Sequence: 1, 2, 3, 1, 2, 3, ... Sequence: 1, 1, 2, 2, 1, 1, 2, 2, ...

Reducing Running Time . Starting point selection: Choose the point farthest from the centre of the grid as the starting point. • Double fan-out: Propagate waves from both the source and the target cells. • Framing: Search inside a rectangle area 10-20% larger than the bounding box containing the source and target. - Need to enlarge the rectangle and redo if the search fails.

starting point selection double fan-out framing

Hadlock's Algorithm • Hadlock, "A shortest path algorithm for grid graphs," Networks, 1977. • Uses detour number(instead of labelling wave front in Lee's router) - Detour number, d(P): # of grid cells directed away from its target on path P.- MD(ST): the Manhattan distance between S and T. - Path length of P, I(P):(P)= MD(S,T) + 2d(P).- MD (S,T) fixed! Minimize d(P) to find the shortest path. - For any cell labelledi, label its adjacent unblocked cells away from 1 į+1; labeli otherwise. • Time and space complexities: O(MN), but substantially reduces the # of searched cells. Finds the shortest path between Sandt. d(P): # of grid cells directed away from its target on path P. MD (ST): the Manhattan distance between Sand T. • Path length of P, I(P):1(P)= MD (5,1)+2d(P). MD (S, T) fixed! =Minimized(P) to find the shortest path. . For any cell labelled i, label its adjacent unblocked cells away from 1 i+1; labeli otherwise

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