Intrinsic Shortest Path Length: A New, Accurate A Priori Wirelength Estimator Andrew B. KahngSherief Reda VLSI CAD Laboratory University of CA, San Diego

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 Outline

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 l 1 > l 2 then e 1 > e 2 Analytical modeling between l i and e i, e.g., l i = f(e i ) Estimator and wirelength have similar distributions Total wirelength estimation Practical runtime for calculation

A Motivating Observation a b 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? Input Netlist Observation: Unlikely. Despite edge {a, b}, a and b are “structurally” far from each other

Intrinsic Shortest Path Length (ISPL) a b Input Netlist “structural proximity”  shortest path shortest path between nodes a and b that does not include {a, b}. Will edge {a, b} be short? analyze the “structural proximity” of 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 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

ISPL in Hypergraphs Set the “distance” or weight of a k-pin hyperedge by k/2 u v ISPL of {u, v} = = 3.5 ab c h 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 Runtime requirement: n: number of nodes, m number of edges

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 Outline

Validation of ISPL as Wirelength Estimator To validate our observation: 1.Correlation between the placed net length and net ISPL? 2.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?

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 100 buckets 30 buckets As ISPL increases, HPWL increases Correlation coefficient 0.91 Reduce data clutter by dividing data into buckets and averaging the results within each bucket

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

Validation: 2. Effect of Pin Count Average length of k-pin nets Average ISPL of k-pin nets 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? CircuitRent parameter WirelengthAv. ISPL GNL GNL GNL GNL GNL 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

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? Predictor Oracle i j Success? { , <} Yes/No 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 benchCapo9mPL4FS2.6MCISPL ibm ibm ibm ibm ibm ibm ibm ibm ibm Average MC: Mutual Contraction ISPL: Intrinsic Shortest Path Length The success of prediction in percentage

Validation: 5. ISPL and Net Length Distribution 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 HPWLISPL Objective: Examine the relationship between ISPL and HPWL profiles

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 Outline

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 symbolMeaning kNumber of pins on a net sISPL akak constant of netlist and k gkgk HPWL of k-pin net with ISPL=s Exponential fitting Actual results

Applications: 1. A Priori Total Wirelength Estimation kakak gkgk ……… 20…… How to determine a k and g k ? 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 An estimate function that minimizes the total square error Objective: Given m ideal models, how to calculate an approximate static model ? m ideal exponential fits (from typical netlists) linearize calculate exp model

Applications: 1. A Priori Total Wirelength Estimation circuitstaticidealcircuitstaticideal ibm %-3.52%ibm %0.64% ibm107.36%9.39%ibm %3.14% ibm %-1.45%ibm %0.44% ibm %-1.56%ibm %-0.34% ibm %0.22%ibm %-1.11% 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

Applications: 2. A Priori Global Interconnect Prediction Global interconnects hurt performance and are typically buffered Objective: Can we a priori decide which nets are going to be “long” before placement? Definition: a net is global (long) if it is in the top 5% of the longest nets in the final 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 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 All nets Global nets

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 Outline

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. Definition 2: The Range of a netlist is the average range of all nodes V.  The larger a node’s range, the more wirelength it needs to communicate with its neighbor  A large Range parameter predicts that a netlist would require a large amount of global communication.

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

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

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

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