VLSI Layout Algorithms

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X(ABCD) (AD)(A(BC)) Y (A(BC)AC
DA(BCD))
VLSI Layout Algorithms CSE 6404
Dr. Md. Saidur Rahman
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Is it possible to lay the circuit on a single
layered PCB?
ELECTRONIC CIRCUIT
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Is it possible to lay the circuit on a single
layered PCB?
POSSIBLE !
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Is it possible to lay the circuit on a single
layered PCB?
NOT POSSIBLE
ELECTRONIC CIRCUIT
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A planar graph
non-planar graph
planar graph
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Kuratowskis Theorem
A graph is planar if and only if it contains
neither a subdivision of K5 nor a subdiision of
K3,3.
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Kuratowskis 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 Kuratowskis
theorem.
Time complexity exponential.
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A polynomial time algorithm
AP 61, Gol63, Shi69
Idea decompose a graph with respect to a cycle.
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Pieces
C
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Pieces
attachment
C
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Connected components after deleting C
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attachment
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P3
P2
P1
P4
P5
P6
Six pieces
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Separating cycle and nonseparating cycle
Nonseparating cycle
Separating cycle
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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.
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Interlace
Two pieces of G, with respect to C, interlace if
they cannot be drawn on the same side of C
without violating planarity.
P1
P2
P1 and P2 interlace.
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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.
P6
P5
P4
P1
P3
P2
interlacement graph
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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.
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Time complexity
Computing pieces
O(n)
O(n2)
Construction of Interlacement graph
Cheking bipartite graph
O(n2)
O(n)
Depth of recursion
O(n3)
Overall
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Linear Algorithms
Hopcroft and tarjan, 1974
Planarity testing
Booth and Lueker, 1976
Chiba et al, 1985
Finding planar embedding
Shih and Hsu, 1992-1999
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The circuit is not planar.
How many PCBs are required?
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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.
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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
For a planar graph
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For a planar graph
Thickness
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Thank you.