4 Polygon and VLSI design By Madhu Reddy Enugu

Outline Real World Problem. Introduction to 4 polygon graph. Properties of 4 polygon graphs. Graph construction. Problem solution. References.

Real World Problem Recent days many tools are being used for solving many routing and layout problems in the design of VLSI (very large scale integrated) circuits. Prominent among such tools are algorithms for solving the maximum independent set problem. In many such problems, the designers model the boundary of a given routing region (usually made of silicon) as a circle and represent a set of 2-terminal wires between points on the boundary by a set of chords. The problem asks for routing the given set of wires using the minimum possible number of vias.

4 Polygon Graphs An undirected graph G = (V, E) is a k-polygon graph if it is the intersection graph of a set of chords inside a convex k-gon, for some k. G has a vertex U i for each chord (i, i’) and an edge (vi, uj) if and only if (i, i’) intersects (j, j’). We denote such a class by P.

Properties of 4 polygon Graphs Connected Eulerian Hamiltonian 4 polygon graphs are proper subset of circle graphs.

Graph Construction 1’

Graph Construction Input: A circle representation of an n-vertex circle graph. Output: The minimum number of sides, s, required to embed the given circle representation in a polygon. s:=2n; for each of the 2n segments (i, j) of the circle do label all 2n endpoints 0; z:=l; for each endpoint k encountered in a clockwise traversal from corner i do let k’ be the other endpoint of the chord with endpoint k; if k’ has label z then z = z+ 1 end if; label[k] := z; end for; s:=min{s,z}; end for.

Graph Construction

Problem Solution Problem solution can be done in two ways. One class of algorithms to solve the problem is based on maximizing the number of wires that can be routed on the two layers without using any vias. The resulting problem is thus to find a maximum r- independent subset of vertices. The other class of Algorithms is based on using the wires but using the minimum number of vias. The resulting solution is thus to find the minimum Dominating set.

Problem Solution Finding the maximum independent set from the r-graph by using the algorithm gives the problem solution. Algorithm: Input: A polygon diagram of G. Output: A maximum independent set MIS(G) of G. 1.for every possible subgraph H’ of R(G) do 2. for every possible independent type-l cast 9V of H’ do 3. Set MIS( $V’) to a maximum independent set of chords in SV’; 4. end for; 5. end for; 6. Set MIS(G) to a maximum cardinality independent set among all possible sets obtained in step 3.

Problem solution Algorithm for 2-independent set: Input: A polygon diagram of G. Output: A maximum 2-independent set of G. 1.for every possible sequence of r outer plane sub graphs H’, H’’ of R(G) do 2.for every possible sequence of independent type-l casts H’=(H’,H’’) of (H’,H’’) do 3. Set MIS( H’), MIS(H’’)to a maximum independent set of chords in H’ and H’’; 4.End for; 5.End for; 6. Set MIS(G) to a maximum cardinality independent set among all possible sets obtained in step 3

Problem Solution Another solution to the problem is obtained by finding the minimum dominating set from the r-graph that is obtained.

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