the university of north carolina at chapel hill cache-oblivious mesh layouts sung-eui yoon, peter...

The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL

Cache-Oblivious Mesh Layouts

Sung-Eui Yoon, Peter LindstromValerio Pascucci, Dinesh Manocha1: University of North Carolina - Chapel Hill2: Lawrence Livermore National Laboratory

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1

2

2

http://gamma.cs.unc.edu/COL

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Goal

• Compute cache-coherent layouts of polygonal meshes ♦ For geometric processing and

visualization♦ Handle any kinds of polygonal

models (e.g., irregular geometry)

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Motivation

• High growth rate of computational power of CPUs and GPUs

Growth rateduring 1993 – 2004

Courtesy: http://www.hcibook.com/e3/online/moores-law/

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Memory Hierarchies and Caches

CPU or GPU

Fast memory or cache

Slow memory

Blocktransfer

Disk

106nsAccess time: 102ns100ns

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Cache-Coherent Layouts

• Cache-Aware♦ Optimized for particular cache

parameters (e.g., block size)

• Cache-Oblivious♦ Minimizes data access time without

any knowledge of cache parameters♦ Directly applicable to various

hardware and memory hierarchies

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82 million trianglesIrregular distribution of geometry

CAD Model – Double Eagle Tanker Model

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Isosurface and Scanned Models

Isosurface100M triangles

St. Matthew372M triangles

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Main Contribution

• Algorithm to compute cache-oblivious layouts of polygonal meshes

Cache-oblivious metric

Multilevel optimization framework

Applicable to hierarchical representations

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Live Demo – View-Dependent Rendering (VDR)

GeForce Go 6800 Ultra

• Based on multiresolution hierarchy♦ Dynamically computes simplification♦ Cache-oblivious layout is used to

minimize GPU vertex cache misses

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Related Work

• Cache-coherent algorithms• Mesh layouts

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Cache-Coherent Algorithms

• Cache-aware [Coleman and McKinley 95, Vitter 01, Sen et al. 02]

• Cache-oblivious [Frigo et al. 99, Arge et al. 04]

Focus on specific problems such as sorting and linear algebra computations

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Mesh Layouts

• Rendering sequences♦ Triangle strips♦ [Deering 95, Hoppe 99, Bogomjakov

and Gotsman 02]

• Processing sequences♦ [Isenburg and Gumhold 03, Isenburg

and Lindstrom 04]

Assume that access patternglobally follows the layout order!

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Mesh Layouts

• Space-filling curves♦ [Sagan 94, Velho and Gomes 91,

Pascucci and Frank 01, Lindstrom and Pascucci 01, Gopi and Eppstein 04]

Assume geometric regularity!

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Outline

• Overview• Cache-oblivious metric• Results

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Outline

• Overview• Cache-oblivious metric• Results

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Overview

Multilevel optimizationCache-oblivious metric

Local permutations

va

vb vd

vc

Input graph

va vb vd vc

Result 1D layout

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Graph-based Representation

• Undirected graph, G = (V, E)♦ Represents access patterns of

applications

• Vertex♦ Data element ♦ (e.g., mesh vertex or mesh triangle)

• Edge♦ Connects two vertices if they are

likely to be accessed sequentially

va

vb vd

vc

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Problem Statement

• Vertex layout of G = (V, E)♦ One-to-one mapping of vertices to

indices in the 1D layout

• Compute a that minimizes the expected number of cache misses

: |}|, ... ,1{ VVva

vb vd

vc

va vb vd vc

1 2 3 4

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Local Permutation

Vertex layout

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Terminology

• Edge span of (va, vb)|)()(| ba vv

Layout mapping

1)( av

5)( cv

4|)()(| ca vv

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Terminology

• ♦ Set of edges having edge span i in

the layout

iE

4),( Evv ca 4

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Terminology

• Edge span distribution ♦ where i is in [1, n]|| iE

1|| 3 E1|| 2 E

1|| 4 E

4|| 1 E

Edge span1

Number of edges

2 3 4

1

1

1

1

4

2

3

4

1

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Cache Miss Ratio Function (CMRF),

• Probability of a cache miss for a given edge span i

ip

0

1Cache miss ratio =Probability to have

a cache miss

Edge span

ip

1 n-1i

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Number of Cache Misses at Runtime

• Estimated by multiplying two factors♦ Runtime edge span distribution♦ CMRF

1D Layout:

Edge span 2 Edge span 4 Edge span 2

2p 2p4p+ + ( 2 1, () 2p 4p, )( )

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Number of Cache Misses at Runtime

1D Layout:

Edge span 2 Edge span 4 Edge span 2

2p 2p4p+ + ( 2 1, () 2p 4p, )

Runtime edge span distribution CMRF

( )

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Expected Number of Cache Misses

♦ Approximate runtime edge span distribution with one of the layout

1

1

||n

iii pE

Edge span distribution of the layout

The number of vertices

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Outline

• Overview• Cache-oblivious metric• Results

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Cache-Oblivious Metric

• Decides if a local permutation reduces number of cache misses♦ Probabilistic formulation♦ Reduces to geometric volume

computation

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Does a Local Permutation Decrease Cache Misses?

1

1

||n

iii pE

1

1

|)||(|n

iiii pEE

|||| ii EE || iE

?

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Does a Local Permutation Decrease Cache Misses?

1

1

||n

iii pE

1

1

|)||(|n

iiii pEE

0||1

1

n

iii pE

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Monotonocity of CMRF,ip

• Assume CMRF is a monotonically increasing function of edge span

0

1Cache miss

ratio

Edge span

ip

1 ∞i

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Exact Cache-Oblivious Metric

0||1

1

n

iii pE

where

All the possible cache configurations

1...0 1221 nn pppp

Monotonicity of CMRF

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Geometric Formulation

where

0||1

1

n

iii pE

1...0 1221 nn pppp

Half hyperspacep2

p10

Closed hyperspace1 n

p2

p10

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Geometric Volume Computation

• Assume each CMRF to be equally likely

• Half hyperspace (blue area)♦ Space of CMRFs that reduce cache misses

p2

p10where

0||1

1

n

iii pE

1...0 1221 nn pppp

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Geometric Volume Computation

Time complexity♦ Exact: [Lasserre and Zeron

01]♦ Approximate: [Kannan et al. 97]

)( 1nnO)( 5nO

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p2

p10

Fast and Approximate Volume Comparison

• Define a top polytope in closed hyperspace

• Compute the centroid, C, of the top polytope

Top polytope Centroid, C

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p2

p10

Fast and Approximate Volume Comparison

• Use the centroid for approximate volume comparison♦ The volume containing the centroid is

likely to be larger

Centroid, C

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Bound of Approximation

• 0.1% ~ 0.3% compared to the exact metric

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Final Approximate Metric

0||1

)(

m

jjl jE

Centroid

Pack non-zero to 1,…, m || iE

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Layout Optimization

• Find an optimal layout that minimizes our metric♦ Combinatorial optimization problem

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Multilevel Minimization

Step 1: Coarsening

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Multilevel Minimization

Step 2: Ordering of coarsest graph

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Multilevel Minimization

Step 3: Refinement and

local optimization

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Outline

• Overview• Cache-oblivious layouts• Results

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Layout Computation Time

• Process 70 million vertices per hour♦ Takes 2.6 hours to lay out St.

Matthew model (372 million triangles)

♦ 2.4GHz of Pentium 4 PC with 1 GB main memory

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Edge Span Distributions of Different Layouts

Cache-oblivious layout

Spectral layout

Original layout

Edge span

Nu

mb

er o

f ed

ges

>

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Applications

• View-dependent rendering• Collision detection• Isocontour extraction

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View-Dependent Rendering

• Layout vertices and triangles of CHPM [Yoon et al. 04]♦ Reduce misses of GPU vertex cache

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View-Dependent Rendering

Models # of Tri.Our

layout

Simplification layout

[Yoon et al. 04]

St. Matthew

372M 106 M/s 23 M/s

Isosurface 100M 90 M/s 20 M/s

Double Eagle

Tanker82M 47 M/s 22 M/s

4.5X

2.1X

Peak performance: 145 M tri / s on GeForce 6800 Ultra

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Realtime Captured Video – St. Matthew Model

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Comparison with Other Rendering Sequences

Our layout

Universal rendering sequences[Bogomjakov and Gotsman 2002]

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Comparison with Other Rendering Sequences

Our layout

[Hoppe 99]

Optimized for 16 vertex cache sizewith FIFO replacement

Optimized for no particular cache size

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Performance during View-Dependent Rendering

Our layout

[Hoppe 99]

Optimized for various resolutions

Optimized for full resolution

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Comparison with Space Filling Curve on Power Plant Model

Our layout

Space filling curve (Z-curve)

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Collision Detection

• Bounding volume hierarchies♦ Widely used to accelerate the

performance of collision detection♦ Traversed to find contacting area♦ Uses pre-computed layouts of OBB

trees [Gottschalk et al. 96]

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Rigid Body Simulation

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Collision Detection Time

2X on average

Depth-first layout

Cache-oblivious layout

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Isocontour Extraction

• Contour tree [van Kreveld et al. 97]

• Use mesh as the input graph

• Extract an isocontour that is orthogonal to z-axis

Puget sound, 134 M triangles

Isocontourz(x,y) = 500m

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Comparison – FirstExtraction of Z(x,y) = 500m

Relative Performance

overZ-axis sorted

layout

Nearly optimized for particular isocontour

2

21

13

1

Disk access time is bottleneck

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Comparison – Second Extraction of Z(x,y) = 500m

Relative Performance

overZ-axis sorted

layout

2

21

13

379

212

10.8

Memory and L1/L2 cache access times are bottleneck

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Limitations

• Assumptions on CMRF♦ May not work well for all applications

• Does not compute global optimum♦ Greedy solution

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Advantages

• General ♦ Applicable to all kinds of polygonal

models♦ Works well for various applications

• Cache-oblivious♦ Can have benefit from CPU/GPU

cache to memory and disk

• No modification of runtime application♦ Only layout computation

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OpenCCL: Cache-Coherent Layouts of Graphs and Meshes• Source codes for computing a

cache-coherent layout • Easy to use

CLayoutGraph Graph (NumVertex);

0

1 2

Graph.AddEdge (0, 1);Graph.AddEdge (0, 2);Graph.AddEdge (1, 2);

int Order [NumVertex];Graph.ComputeOrdering (Order);

Google “Cache Oblivious Mesh Layout” or

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Conclusion

• Novel algorithm for computing cache-oblivious mesh layouts♦ Cast the problem as an optimization♦ Probabilistically compute the

expected number of caches misses♦ Achieve significant improvements (2

to 20X) without modifying runtime applications

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Ongoing and Future Work

• Apply to other applications ♦ Simplification and approximate

collision detection [Yoon et al. 04]♦ Shortest path computation, etc.

• Investigate optimality

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Ongoing and Future Work

• Cache-Oblivious Layouts of Bounding Volume Hierarchies [Yoon and Manocha 05] ♦ Tech. Report, University of North

Carolina at Chapel Hill

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Acknowledgements

• Anonymous donor ♦ Power plant model

• Digital Michelangelo Project♦ St. Matthew model at Stanford

University

• LLNL ASCI VIEWS♦ Isosurface model

• Newport news shipbuilding♦ Double eagle tanker

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Acknowledgements

• Army Research Office• DARPA• Intel Corporation• Lawrence Livermore Nat’l Lab.• National Science Foundation• Office of Naval Research• RDECOM

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• Martin Isenburg• Dawoon Jung• Brandon Lloyd• Elise London• Brian Salomon• Avneesh Sud

Acknowledgements

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Questions?

Project URLhttp://gamma.cs.unc.edu/COL