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