VLSI Layout AlgorithmsCSE 6404
A 46
B 65
C 11D 56
E
23
F 8
H 37
G 19
I 12 J 14
K 27
X=(AB*CD)+ (A+D)+(A(B+C))Y = (A(B+C)+AC+ D+A(BC+D))
Dr. Md. Saidur Rahman
AB
F
E
C
D
ELECTRONIC CIRCUIT
Is it possible to lay the circuit on a single layered PCB?
AB
C
DE
F
Is it possible to lay the circuit on a single layered PCB?
POSSIBLE !
AB
F
E
C
D
ELECTRONIC CIRCUIT
Is it possible to lay the circuit on a single layered PCB?
NOT POSSIBLE
A planar graph
planar graph non-planar graph
Kuratowski’s Theorem
A graph is planar if and only if it contains neither a subdivision of K5 nor a subdiision of K3,3.
Kuratowski’s Theorem
A graph is planar if and only if it contains neither a subdivision of K5 nor a subdiision of K3,3.
Planarity testing algorithm based on Kuratowski’s theorem.
Time complexity : exponential.
A polynomial time algorithm
Idea: decompose a graph with respect to a cycle.
C
[AP 61, Gol63, Shi69]
Pieces
C
attachmentPieces
C
Connected components after deleting C
attachment
P1
P2P3
P4 P5 P6
Six pieces
Separating cycle and nonseparating cycle
Separating cycle Nonseparating cycle
Lemma
Let G be a biconnected graph and let C be a nonseparating cycle of G with piece P. If P is not a path, the G has a separating cycle C’ consisting of a subpath of C plus a path of P between two attachmet.
Interlace
Two pieces of G, with respect to C, interlace if they cannot be drawn on the same side of C without violating planarity.
P2P1
P1 and P2 interlace.
Interlacement Graph
The interlacement graph of the pieces of G, with respect to C, is the graph whose vertices are the pieces of G and whose edges are the pairs of pieces that interlace.
P2 P1
P3
P4
P6
P5P1
P2P3
P4
P5
P6
interlacement graph
Theorem
A biconnected graph G with a cycle C is planar if and only if the following two conditions hold.
For each piece P of G with respect to C, the graph obtained by adding P to C is planar.
The interlacement graph of the pieces of G, with respect to C, is bipartite.
The theorem above leads to a recursive algorithm.
Time complexity
Computing pieces O(n)
Construction of Interlacement graph O(n2)
Cheking bipartite graph O(n2)
Depth of recursion O(n)
Overall O(n3)
Linear Algorithms
Hopcroft and tarjan, 1974
Booth and Lueker, 1976Planarity testing
Finding planar embeddingChiba et al, 1985
Shih and Hsu, 1992-1999
The circuit is not planar.
How many PCB’s are required?
Thickness of a graph t(G)
The thickness t(G) of a graph G is defined to be the smallest number of planar graphs that can be superimposed to form G.
Euler Theorem
Let G be a connected plane graph, and let n, m, f denote respectively the number of vertices, edges and faces of G. Then
2 fmn
For a planar graph 63 nm
For a planar graph 63 nm
63)(
n
mGtThickness
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