Intrinsic Shortest Path Length: A New, Accurate A Priori Wirelength Estimator

Intrinsic Shortest Path Length: A New, Accurate A Priori Wirelength Estimator Andrew B. Kahng Sherief Reda abk@ucsd.edu sreda@ucsd.edu VLSI CAD Laboratory University of CA, San Diego http://vlsicad.ucsd.edu/~sreda

Outline • Previous work and motivation • Intrinsic Shortest Path Length (ISPL) definition • Validation of ISPL as wirelength estimator • Practical Applications: • A Priori Total wirelength estimation • A Priori Global interconnect prediction • Relationship to Rent parameter

Definition and Applications • A priori wirelength estimation is the process of estimating and predicting the wirelength characteristics of VLSI netlists without knowledge of the netlist placement or floorplanning. • Applications that benefit from a priori wirelength estimation: • Physical driven synthesis  Faster timing convergence • Early system planning • Determining amount of necessary whitespace • VLSI netlist characterization/reverse engineering/creation

Previous Work • Previous approaches: • Correlators: • If some measure e correlates with net length l then e can be used in relevant applications, e.g., clustering. Typically no analytical modeling between l and e. • Examples: mutual contraction and edge separability. • Average wirelength estimators: • Rent parameter-based. • Predict aggregate wirelength characteristics, e.g., wirelength distribution and total wirelength.

Motivation • Wanted: • Estimator has intuitive physical meaning • Handles hypergraphs transparently • Individual net length estimator: If l1> l2 then e1> e2 • Analytical modeling between li and ei, e.g., li = f(ei) • Estimator and wirelength have similar distributions • Total wirelength estimation • Practical runtime for calculation

A Motivating Observation Input Netlist b a • Nodes a and b are directly connected by an edge. • Does this mean a and b will be placed spatially close in a good placement? Observation: Unlikely. Despite edge {a, b}, a and b are “structurally” far from each other

Intrinsic Shortest Path Length (ISPL) Input Netlist Will edge {a, b} be short? b • analyze the “structural proximity” of a and b • “structural proximity”  shortest path • shortest path between nodes a and b that does not include {a, b}. a To estimate the Intrinsic Shortest Path Length ISPL of edge {a, b} : delete {a, b} and calculate the shortest path length (number of edges) between a and b • Example: ISPL of {a, b} = 8. {a, b} and its ISP form a cycle • BUT: Netlists are hypergraphs  a transparent mechanism is needed

The ISPL of a k-pin hyperedge, h = {a, b, c}, is calculated as follows: • 1. Delete h • 2. Calculate the ISPL for every pair of nodes that belong to h: • {a, b}, {b, c}, and {a, c} • 3. The ISPL of h is maximum among all values calculated in Step 2 n: number of nodes, m number of edges Runtime requirement: ISPL in Hypergraphs • Set the “distance” or weight of a k-pin hyperedge by k/2 h a b u c v ISPL of {u, v} = 1+1.5+1 = 3.5

Validation of ISPL as Wirelength Estimator • To validate our observation: • Correlation between the placed net length and net ISPL? • Correlation between the effect of net pin count on average net length and average net ISPL? • 3. Correlation between the average/total netlist wirelength and the average/total ISPL over a range of benchmarks? • 4. Given two individual nets of some netlist, can we predict which individual net will be placed with greater wirelength? • 5. Relationship between the distribution, or profile, of ISPL and the wirelength distribution?

100 buckets 30 buckets • Reduce data clutter by dividing data into buckets and averaging the results within each bucket Validation: 1. ISPL and Net Length Objective: validation of the relationship between ISPL and net length: Given a netlist (ibm01): 1. Calculate the ISPL of every hyperedge 2. Place the netlist using some placer (Dragon) 3. Plot ISPL versus Half-Perimeter Wirelength (HPWL) of every net • As ISPL increases, HPWL increases • Correlation coefficient 0.91

Validation: 1. ISPL and Net Length • Calculate correlation coefficients between ISPL and wirelength • For comparison, calculate correlation coefficient between: • Mutual Contraction (UCSB) and HPWL • Edge Separability (UCLA) and HPWL |correlation coefficients| MCis mutual connectivity ES is edge separability

Average length of k-pin nets Average ISPL of k-pin nets Validation: 2. Effect of Pin Count Objective: Test the effect of pin count on both average wirelength and ISPL • For every k (2…) on ibm01: • Calculate the average ISPL of all k-pin net • Run a placer and calculate the average placed wirelength of all k-pin nets • Correlation coefficient of 0.95 between average HPWL and average ISPL (typical result)

Validation: 3. Average ISPL and Total Wirelength Objective: Is the average ISPL correlated with the total wirelength? • Synthesize netlists (10k nodes/nets) with varying Rent parameter with GNL • A higher rent parameter  more global communication  larger wirelength • Calculate the average ISPL of each netlist • Place the netlists using mPL and measure the HPWL • Perfect correlation between average ISPL and total wirelength

{, <} Predictor i j Yes/No Success? Oracle {, <} • Dragon gives the best placement and will be used as an Oracle • Performance of the Predictor: • Lower bound is 50% • Upper bound is the performance of any other placer • What is performance if we use ISPL for the predictor? Validation: 4. Individual Net Length Prediction Objective: Given two arbitrary nets i and j with the same number of pins, can we a priori decide which net will be longer?

Validation: 4. Individual Net Length Prediction The success of prediction in percentage MC: Mutual Contraction ISPL: Intrinsic Shortest Path Length

HPWL ISPL Validation: 5. ISPL and Net Length Distribution • Objective: Examine the relationship between ISPL and HPWL profiles • Sort all nets according to their ISPL and their HPWL • Plot all sorted HPWL normalized to the maximum HPWL value • Plot all sorted ISPL normalized to the maximum ISPL value • ISPL and HPWL have roughly similar profiles

Exponential fitting Actual results Applications: 1. A Priori Wirelength Total Estimation • Devise an analytical model between ISPL and HPWL • Using empirical data, we find an exponential relationship between ISPL and wirelength

Applications: 1. A Priori Total Wirelength Estimation • How to determine ak and gk? • Ideal modeling (not a priori): based on the netlist characteristics from the placement (only useful for model validation and calibration) • Static modeling (a priori): fixed values for all netlists based on values typically encountered

Applications: 1. A Priori Total Wirelength Estimation Objective: Given m ideal models, how to calculate an approximate static model ? m ideal exponential fits (from typical netlists) linearize An estimate function that minimizes the total square error calculate exp model

Applications: 1. A Priori Total Wirelength Estimation • Calculate the total wirelength of the IBM (version 1) benchmarks (unit size cells) using ideal model • Calculate typical values and use it for a priori static modeling. • On the average, ideal modeling is 3.61% accurate compared to actual HPWL. Static modeling is 16.60% accurate

If we declare all nets with ISPL  15 then we declare 10% of all nets global, actually capturing the 60% of the future global interconnects Global nets All nets Applications: 2. A Priori Global Interconnect Prediction Global interconnects hurt performance and are typically buffered Definition: a net is global (long) if it is in the top 5% of the longest nets in the final placement • Objective: • Can we a priori decide which nets are going to be “long” before placement? Given a netlist: 1. Calculate the ISPL of all nets 2. Sort all nets based on their ISPL 3. Plot net count vs ISPL

Relationship to Rent Parameter • We develop a characterization, Range Parameter, of VLSI netlists Definition 1: The range of a node u is the average ISPL of all nets incident to it. • The larger a node’s range , the more wirelength it needs to communicate with its neighbor Definition 2: The Range of a netlist is the average range of all nodes V.  A large Range parameter predicts that a netlist would require a large amount of global communication.

Range Rent Relationship to Rent Parameter Intuitive connection to Rent parameter:a netlist with large Rent parameter  requires more global communication in any good placement Correlation coefficient of 0.701

Rent Parameter Range Parameter • Calculated in top-down fashion • Calculated in bottom-up fashion • Useful for complete netlist characterization • Useful for complete netlist characterization • Useless for individual net prediction • Useful for individual net prediction • Unstable value • Stable value (same topology) • works on graphs or hypergraphs • Hypergraph to graph transformation  Runtime normalized with respect to FengShui

Conclusions • Developed the new concept of Intrinsic Shortest Path Length (ISPL) • Demonstrated strong correlation between ISPL and HPWL • Used it for individual net length predictor • Correlated average ISPL with total wirelength • Studied the relationship between ISPL and HPWL distributions • Developed a characterization to VLSI netlists and studied its relation to Rent parameter • Used ISPL for two practical applications: • Total wirelength estimation • Global interconnect prediction

Future Work • Runtime improvement • Studying the effect of different net weights on ISPL performance • Better wirelength models • Synthetic benchmark generation based on ISPL • Analytical relationship between Range and Rent parameters • Fixed blocks/white space effects • Deducing wirelength distribution, pin-effect count from the analytical models • Estimating RSMT by using weighting coefficients

Thank you

Intrinsic Shortest Path Length: A New, Accurate A Priori Wirelength Estimator