constructed in NNS only shows the nearest neighbor sink for a given sink.
3.2 Overlap graph construction using agglomera-tive clustering
The slack value for a flip-flop can be treated as a distance budget for a flip-flop. The farther away flip-flop moved from its original location, the less slack value a flip-flop has. In this regard, the slack value for a flip-flop can be modeled as a form of wire delay. The distance in this context refers to Manhattan distance, thus the region a flip-flop can be relocated without violating the given slack constraint forms a tilted rectangle region (TRR). The TRR for flip-flop merging takes the same concept of TRR in clock tree except that TRR in flip-flop merging has radius defined by its
Algorithm 1 Segment-Overlap
slack value while the radius for TRR in clock tree is defined by its capacitance load.
If two flip-flop’s TRR overlaps, it implies that two flip-flop can safely be merged in the overlapping region without violating slack constraint for both flip-flops. In graph representation, flip-flop is represented by a node and overlapping region of two flip-flops is represented by an edge. An edge is created between two flip-flops if and only if there exists an overlapping region between the two flip-flops.
The naive implementation of agglomerative clustering has time complexity O(N2) which is to compare every node with every other node. However, not every node has an edge connecting to every other node. A more efficient approach is to sort all the rectangles on X-coordinate and find overlapping segment in Y-coordinate which can reduce number of comparisons. This approach is derived from Line Sweep Method[6]. Red-black tree is also used so that each insertion and deletion of TRR is guaranteed to be bounded in O(lgN ) rather than O(N ).
We formulate the flip-flop merging problem as a graph problem. In graph, node represents a flip-flop movable zone which is a rotated rectangle and edge represents there exists overlapping region between two nodes it connects. To identify all the overlapping rectangles, the straight forward method is to compare one rectangle with every other rectangle. However, it is rare that every rectangle overlaps with every other rectangle. A more efficient approach is to sort all the rectangles on X-coordinate and find overlapping segment in Y-X-coordinate which can reduce number of comparisons. This approach is commonly known as Line Sweep Method [6],
Figure 3.2: The sweep line method first sorts the rectangle based on their X’-coordinate. When the sweep line touches a rectangle, the rectangle will compare with all the rectangles stored in the RB-Tree. When a sweep line leaves a rectangle, that rectangle is removed from the RB-Tree.
which is described in Algorithm 2. Red-black tree is used so that each insertion and deletion of rectangles is guaranteed to be bounded in O(lgV ) rather than O(V ).
The algorithm first takes all rectangle’s minimum and maximum X-coordinates and store them in an array. The X-coordinates will then be sorted in non-decreasing order using heap sort which corresponds to line 1-4 in Algorithm 2. After the array is sorted, an imaginary line starts to sweep from the beginning of the array. If the X value is the starting point of a rectangle, the rectangle is stored into a red-black tree and compare with all the rectangles stored in the red-black tree to check whether if there exists overlapping region between the two rectangles. Given that rectangles are already sorted by their X-coordinate, if two rectangles were to be compared, then it implies that two rectangles must be overlapped in X-coordinate. Hence, checking whether if two rectangles exist overlapping region only needs to check whether the segments of two rectangle are overlapped in Y-coordinate, which reduces the problem
Algorithm 2 Line Sweep Method to Identify Overlapping Rectangle
1: for ni ∈ Q do
2: X ← X ∪ minX[ni] ∪ maxX[ni]
3: end for
4: Sort X in non-decreasing order
5: for x ∈ X do
6: if x = lowX then
7: for nj ∈ P do
8: if Segment-Overlap(ni, nj)=true then
9: E ← E ∪ E(ni, nj)
from 2-dimensional space to 1-dimensional space. The algorithm to check whether two segments overlap is described in Algorithm 1. If the X value is the ending point of a rectangle, the rectangle is removed from the red-black tree. When the sweep line reaches to the end of the list, all the edges can be generated.
Fig. 3.2 demonstrates a simple example using Line Sweep Method. In Fig.
3.2(a), sweep line first enters rectangle A and rectangle A is stored in the red-black tree. In Fig. 3.2(b), sweep line enters rectangle D. Rectangle D are then compared with rectangle A and B to check whether if there exist overlapping region. In Fig. 3.2(c), sweep line leaves rectangle D which is removed from the red-black tree.
Finally, in Fig. 3.2(d), the sweep line leaves rectangle E and rectangle E is removed.
In worst case scenario in which every rectangle overlaps with every other rectan-gles, the graph representation of such circumstance is a complete graph. The time complexity for Line Sweep Method in a complete graph is O(N2) since no rectangles will be removed from red-black tree and every inserted rectangle must compare with every other rectangles stored in the red-black tree.
Algorithm 3 Hierarchical Agglomerative Clustering
After graph is constructed, the edge with minimum cost will be selected and checked whether if it is mergeable. For two flip-flop to be mergeable, there must exist corresponding type in given flip-flop library. For example, if FF-A is 1-bit, FF-B is 2-bit and there is no 3-bit flip-flop available, FF-A and FF-B can not be merged.
In each iteration, each merge will select two flip-flops with least edge cost. If two flip-flops is successfully merged, new node representing the merged flip-flop will be created and added to the graph with corresponding edges. Edges connecting to two original flip-flops will be removed from the graph and new edges connecting to merged flip-flop will be added.
The algorithm will terminate until there is no more edge left in the graph. In Fig. 3.3(a), node A and B are picked to be merged, edges connected to node A and node B are removed. In Fig. 3.3(b), new node MAB is created with two new edges added to the graph. Fig. 3.3(c) illustrates the same concept of merging node MAB
and node MCD to a 4-bit node MABCD. The algorithm terminates in Fig. 3.3(d) since there is no more edge in the graph.