Abstract
Partitioning divides the circuit into several subcircuits (partitions or blocks) while minimizing the number of connections between partitions. Such partitioning enables each subcircuit to be processed with some degree of independence and parallelism, in stages that follow. Netlist partitioning (Sects. 2.1–2.4) can handle large netlists and can redefine a physical hierarchy of an electronic system, ranging from boards to chips, and from chips to blocks. Traditional netlist partitioning can be extended to multilevel partitioning (Sect. 2.5), which can be used to handle large-scale circuits and systems.
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Notes
- 1.
The empirical observation known as Rent’s rule suggests a power-law relationship between the number of cells nG and the number of external connections nP = t⋅ nGr, for any subcircuit of a “well-designed” system. Here, t is the number of pins per cell and r, referred to as the Rent’s exponent or the Rent parameter, is a constant < 1. In particular, Rent’s rule quantifies the prevalence of short wires in ICs, which is consistent with a hierarchical organization.
- 2.
For convenience, hyperedges may be referred to as edges. However, graph edges are formally defined as node pairs.
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Exercises
Exercises
Exercise 1: KL Algorithm
The graph to the right (nodes a–f) can be optimally partitioned using the Kernighan–Lin algorithm. Perform the first pass of the algorithm. The dotted line represents the initial partitioning. Assume that all nodes have the same area and all edges have the same weight.
Note: Clearly describe each step of the algorithm. Also, show the resulting partitioning (after one pass) in graphical form.
Exercise 2: Critical Nets and Gain During the FM Algorithm
-
(a)
For cells a–i, determine the critical nets connected to these cells and which critical nets remain after partitioning. For the first iteration of the FM algorithm, determine which cells would need to have their gains updated due to a move. Hint: It may be helpful to prepare a table with one row per move that records (1) the cell moved, (2) critical nets before the move, (3) critical nets after the move, and (4) which cells require a gain update.
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(b)
Determine Δg(c) for each cell c ∈ V.
Exercise 3: FM Algorithm
Perform Pass 2 of the FM algorithm example given in Sect. 2.4.3. Clearly describe each step. Show the result of each iteration in both numerical and graphical forms.
Exercise 4: Multilevel FM Partitioning
List and explain the advantages that a multilevel framework offers compared to the FM algorithm alone.
Exercise 5: Clustering
Consider a partitioned netlist. Clustering algorithms covered in this chapter do not take a given partitioning into account. Explain how these algorithms can be modified such that each new cluster is consistent with one of the initial partitions.
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Kahng, A.B., Lienig, J., Markov, I.L., Hu, J. (2022). Netlist and System Partitioning. In: VLSI Physical Design: From Graph Partitioning to Timing Closure. Springer, Cham. https://doi.org/10.1007/978-3-030-96415-3_2
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