Bounds on the Geometric Mean of Arc Lengths for Bounded-Degree Planar Graphs

Bounds on the Geometric Mean of Arc Lengths for Bounded-Degree Planar

Graphs

M. K. HasanSung-eui Yoon

Kyung-Yong Chwa

KAIST, Republic of Korea

Image Rendering

Image Rendering

Image Rendering

Image Rendering

Image Rendering

Two Level I/O Memory Model

a

c

d

b

f

e

g

h

2-block cache

Memory with infinite capacity

Memory Layout

a

c

d

b

f

e

g

h

2-block cache

Memory with infinite capacity

a b c d e f gh

…

Performance of the Layout

a

c

d

b

f

e

g

h

2-block cache

Memory with infinite capacity

Not AvailableIn Cache

h a b c d e f g …

Performance of the Layout

a

c

d

b

f

e

g

h

2-block cache

Memory with infinite capacityNumber of misses: 1

h a b c d e f g …a b c

Performance of the Layout

a

c

d

b

f

e

g

h

2-block cache

Memory with infinite capacityNumber of misses: 1

h

a b c

d e f g …

Hit !

a b c

Performance of the Layout

a

c

d

b

f

e

g

h

2-block cache

Memory with infinite capacityNumber of misses: 1

h

a b c

d e f g …

Hit !

a b c

Performance of the Layout

a

c

d

b

f

e

g

h

2-block cache

Memory with infinite capacityNumber of misses: 1

h

a b c

d e f g …

Not AvailableIn Cache

a b c

Performance of the Layout

a

c

d

b

f

e

g

h

2-block cache

Memory with infinite capacityNumber of misses: 2

h

a b c

d e f g …a b c d e f

Performance of the Layout

a

c

d

b

f

e

g

h

2-block cache

Memory with infinite capacityNumber of misses: 2

h

a b c

d e f g …a b c

d e f

Performance of the Layout

a

c

d

b

f

e

g

h

2-block cache

Memory with infinite capacityNumber of misses: 2

h

a b c

d e f g …a b c

d e fHit !

Performance of the Layout

a

c

d

b

f

e

g

h

2-block cache

Memory with infinite capacityNumber of misses: 2

h

a b c

d e f g …a b c

d e fHit !

Performance of the Layout

a

c

d

b

f

e

g

h

2-block cache

Memory with infinite capacityNumber of misses: 2

h

a b c

d e f g …a b c

d e f

Not AvailableIn Cache

Performance of the Layout

a

c

d

b

f

e

g

h

2-block cache

Memory with infinite capacityNumber of misses: 3

h

a b c

d e f g …a b c

d e f

hg …

Performance of the Layout

a

c

d

b

f

e

g

h

2-block cache

Memory with infinite capacityNumber of misses: 3

h d e f g …a b c

d e fhg …

Performance of the Layout

a

c

d

b

f

e

g

h

2-block cache

Memory with infinite capacityNumber of misses: 3

h d e f g …a b c

d e fhg …Hit !

Example of a “Bad Layout”

a

c

d

b

f

e

g

h a h g c b f d e …

2-block cache

Memory with infinite capacityNumber of misses = 4

Cache Parameters

• In reality we do not know the block size and cache size

• In the Cache-Oblivious environment our objective is to develop efficient algorithm and/or layout without knowing the cache parameters

• Reducing the number of cache misses is the only goal. It does not matter how bad the algorithm is (in terms of complexity) as long as it is polynomial.

Cache-Oblivious Mesh Layout

• Optimization problem: Find the layout of a mesh which minimizes the number of cache misses for a set of applications:

– Rendering – Collision Detection – Ray Tracing

We Need a Metric

• Given a layout

– We want to know the number of cache misses for a particular class of applications

– How can we get that if we do not know the cache parameters (cache size, block size)?

– We need a metric which reflects the expected number of cache misses for the application class

– The metric must be related to the layout and must not be dependent on the cache parameters

Cache-Obliviousness

Efficient Layout

• Suppose we have chosen a metric which reflects the expected number of cache misses for our target application.

• Now, our objective is to develop a layout of the input mesh such that the metric value is minimized

Related Work : Derivation of Metrics [Yoon and Lindstrom(2006)]

• Consider two types of block size progressions– 1, 2, 3, …– 20, 21, 22, …

• The second one is more reasonable choice, e.g., 32 and 128 bytes for L1 and L2 caches and 4KB for the virtual page

• Based on these progressions we find two metrics

– Arithmetic mean of edge spans:

– Logarithm of geometric mean of edge spans:

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c

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Ajiija l

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logA

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Mesh Layout

Related Work : Comparison of Two Metrics

[Yoon and Lindstrom(2006)]

Related Work :Multi-level Construction Algorithm

[Yoon and Lindstrom(2006)]

Algorithm:• Partition the input graph using

graph partitioning tool.

• Make the layout of each small component using Mg metric

• Marge the layout of each component to get the layout of the whole graph

Average query times for collision detection between the Lucy model and Dragon Model

Result:• Experimental results show nice

performance of multi-level construction algorithm

• However there is no theoretical bound for the performance and the complexity of the algorithm

Our Contribution

• We have given an algorithm which gives a layout for bounded-degree planar graph such that the Mg value of the layout is within a constant factor of the Mg value of the optimal layout (optimal in terms of Mg value).

• Our algorithm is simple and easy to implement. The time complexity of our algorithm is O(nⅹlog2n)

Separator Theorem For Planar Graphs [Lipton and

Tarjan(1979)]

BA

|C| 2ⅹ21/2ⅹn1/2

No edge between A and B

|A|≤2ⅹn/3 |B|≤2ⅹn/3C

Planar Graph with n vertices

Algorithm: Given Bounded-Degree Planar Graph G

with n (a power of 2 ) Vertices

G

Algorithm: Split G Into Three Parts A, B and C

BA

C

|C|=O(n1/2)

|B|≤n/2|A|≤n/2

Number of external edges = O(n1/2)Number of internal edges = O(n1/2)

No edge between A and B

Algorithm: Split C into C1 and C2

BA

C1

|A|+|C1|=n/2

C2

|B|+|C2|=n/2

Each edge is either

an external or an internal edge crossing

Number of crossing edges = O(n1/2)

Algorithm: Combine A,C1 and B,C2

|A|+|C1|=n/2 |B|+|C2|=n/2

BA

C1 C2

Algorithm: Recurse on A ’ and B ’

Make Layout of A ’ with 1…n/2 and B ’ with n/2+1…n recursively

A ’ = A∪C1 B ’ = B∪C2

1 2 3 4 5 6 7 8

Cost of The Split of a n Vertices Bounded Degree Planar Graph

Layout with values [1…n./2]

A ’ = A∪C1 B ’ = B∪C2

Number of crossing edges = O(n1/2)

Cost of crossing edges = O(n1/2 log n)

Aji

ijg lA

M),(

)log(||

1Cost

Layout with values [n/2…n]

i j

nijlij log|)log(|)log( i j

1i nj

Bounding MG Value

……………………………………

l=0

l=1........

lll

l

nncCost

2log

22

2

1

…………………………

l

lCost Cost

Number of subgraphs

Number of crossing edges

Cost of each crossing edge

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nl

2 is levelat subgraph each of Size

)1(),(|| Assuming OMnA g

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Results Summary• Value of Mg metric of the layout generated by the recursive

algorithm is O(1), where,

• Lower bound of Mg of any layout is Ω (1) under suitable assumptions

• The recursive algorithm gives a constant approximation factor for Mg

• The upper bound of Mg value for any layout is O (log n)

• The time complexity of our algorithm is O(nⅹlog2n)

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Future Works

• We can extend the algorithm to handle three dimensional mesh

• For applications like streaming

mesh and mesh smoothing, we can develop good metrics and cache-oblivious layouts based on those metrics