Cache Oblivious Algorithms

Zhang JiaHuiNeel Kamal

Introduction

Cache Oblivious vs Cache Aware (Z,L) Idea-Cache-Model Large Integer Multiplication & RSA Dynamic Programming

- Floyd All-Pair Shortest Paths- Longest Common Sequence

Cache-Behavior Simulator Experimental Results

2LZ

Large Integer Multiplication(1)

We have two large Integer x and y. x has m digits and y has n digits

If m>n, append zeros to the left side of nIf n>m, append zeros to the left side of mSuppose m>n, we now have two large

Integers both of length m

Large Integer Multiplication(1)

A B

C D

B x CA x C

B x D

A x D

Final Result

Large Integer Multiplication(1)

m > nCASE I

After k steps, m

4Zm

otherwiseOmQ

ZZmifLm

mQ,)1(

24

)4

,8

(,4

)(

4,

82ZZm

k

LZm

L

mmQmQ

kk

kk

21624

42

4)(

Large Integer Multiplication(1)

m>nCASE II

If n>m, in CASE I, and

So, combine all the cases, we have

4Zm

LmmQ 4)(

LZnnQ

216)(

LnnQ 4)(

Ln

Lm

LZm

LZnnQ

22

)(

Large Integer Multiplication(2)

We do not append zero to the left hand side of the shorter Integer

CASE I

otherwiseOnmQ

nmifotherwiseOnmQ

ZZnmifLnm

Ln

Lm

nmQ

,)1(2

,2

)(,,)1(,2

2

),2

,(,

),(

Znm ,

Large Integer Multiplication(2)

After k1 steps m

After k2 steps n

ZZm

k ,22 1

ZZn

k ,22 2

LZmn

Lnm

L

nmnmQnmQ

kkkkkk

kkkk

122121

2121 22222

222

2,

222),(

Large Integer Multiplication(2)

CASE II

If

Zm

otherwiseOnmQ

ZZnifLnm

Ln

Lm

nmQ,)1(

2,2

),2

(,),(

Ln

LZmnnmQ ),(

Zn

Lm

LZmnnmQ ),(

Large Integer Multiplication(2)

CASE III

Combine all the cases:

Total work

Znm ,

Lnm

Ln

LmnmQ ),(

LZmn

Lnm

Ln

LmnmQ ),(

mnO

Large Integer Multiplication & RSA

Summary of RSA n = pqn = pq where p and q are distinct

primes. phi, φ = (p-1)(q-1)φ = (p-1)(q-1) e < n such that gcd(e, phi)=1 d = e^-1 mod phi. c = m^e mod n. m = c^d mod n.

All-pair shortest Paths Floyd for k=1 to n for i=1 to n for j=1 to n d[i][j][k]=min(d[i][j][k-1],d[i][k][k-1]+d[k][j][k-1]

For each k (1..n)

3nO

All-pair shortest Paths

For each iteration of k CASE I

After k steps

Zn

otherwiseOnQ

ZZnifLn

nQ

,)1(2

4

),2

(,)(

2

Ln

L

nnQnQ

kk

kk

2

2

242

4)(

ZZnn k ,

22

All-pair shortest Paths

CASE II

Combine the cases:

We have n iterations,

Zn

)()( nnQ

LnnnQ

2

)(

LnnnQtotal

32)(

Longest Common Sequence

We have 2 long sequences x and y, x is of length m, and y is of length n.

Try to find the Longest Common Sequence of x and y.

Dynamic Programming

Longest Common Sequencey1 y2 y3 y4 y5 y6

B D C A B A

x1 A 0 0 0 1 1 1

x2 B 1 1 1 1 2 2

x3 C 1 1 2 2 2 2

x4 B 1 1 2 2 3 3

x5 D 1 2 2 2 3 3

x6 A 1 2 2 3 3 4

x7 B 1 2 2 3 4 4

If x[i] == y[j]c[i][j] = c[i -1][j -1]+1;

elsec[i][j] = max{c[i -1][j]; c[i][j -1];

Longest Common Sequencey1 y2 y3 y4 y5 y6

B D C A B A

x1 A 0 0 0 1 1 1

x2 B 1 1 1 1 2 2

x3 C 1 1 2 2 2 2

x4 B 1 1 2 2 3 3

x5 D 1 2 2 2 3 3

x6 A 1 2 2 3 3 4

x7 B 1 2 2 3 4 4

B D C A B A

A 0 0 0 1 1 1

B 1 1 1 1 2 2

C 1 1 2 2 2 2

B 1 1 2 2 3 3

Longest Common Sequence

CASE I Znm ,

otherwiseOnmQ

nmifotherwiseOnmQ

ZZnmifLmn

nmQ

,)1(2

,2

)(,,)1(,2

2

),2

,(,

),(

Longest Common Sequence

Suppose:After k1 steps, m

After k2 steps, n

ZZm

k ,22 1

ZZn

k ,22 2

Lmn

L

nmnmQnmQ

kkkk

kkkk

2121

2121 2222

2,

222),(

Longest Common Sequence

CASE II Zm

otherwiseOnmQ

ZZnifmnmQ

,)1(2

,2

),2

(,1),(

ZmnnmQ ),(

In the case when Zn

otherwiseOnmQ

ZZmifmnmQ

,)1(,2

2

),2

(,1),(

mnmQ ),(

Longest Common Sequence

CASE III

Combine all 3 cases:

Total Work

Znm ,

mnmQ 1),(

Lmn

ZmnmnmQ 21),(

mnO

Cache Oblivious approaches for Dynamic Programming

Dynamic Programming to find an optimal solution Sub-problems overlap

Approaches bottom up (by recursion usually) top down but with a table to memorize earlier solutions

Divide and Conquer method to build the table recursively to make the approach cache oblivious?

Cache Simulator

With a tall cache assumption

A fully associative cache

2LZ

Other Assumptions • No temporary variable put into the cache• All input data is assumed to be already present in cache.

ResultsSummarizing the theoretical Results:

LZmn

Lnm

Ln

LmnmQ ),(Large Integer Multiplication

All-pair shortest Paths

Longest Common Sequence

LnnnQ

2

)(

Lmn

ZmnmnmQ 21),(

Results from Simulation

Target Machine: arch : IA-64 family : Itanium 2 CPU MHz : 896.262997 Cache size : 303312 KB OS Linux version 2.4.22 gcc version 2.96 20000731

We will see that there is a very close match between the theoretical results and the

simulation result.

Results

Cache Oblivious Large Integer Multiplication

248115

289047097 2712 1089 465 270 101 89 810

50000100000150000200000250000300000

10 20 30 40 50 60 70 80 90 100

Size of cache Line

Num

ber o

f Cac

he M

isse

s

Size of Integer: M = 1000 N = 1000

3),(

LmnnmQ

Comparing Results

Case 1: L = 20, Z = 400Theoretical Result = Θ (1000/8)Simulator Result = 28904ratio = 0.0041

Case 2: L=30, Z = 900Theoretical Result = Θ (1000/27)Simulator Result = 7097ratio = 0.0044

Case 3: L=40, Z = 1600Theoretical Result = Θ (1000/64)Simulator Result = 2712ratio = 0.0052

More Results

Cahe Oblivious Longest Common Sequence

0

500000

1000000

1500000

2000000

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

Size of Cache Line

Num

ber o

f Cac

he M

isse

s

Size of Sequence: M = 1000 N = 1000

LmnnmQ ),(

More ResultsCache Oblivious Floyd Algorithm

0100000200000300000400000500000600000700000

20 30 40 50 60 70 80 90 100

Size of Cache Line

Num

ber o

f Cac

he M

isse

s

Number of Vertices N = 100

LnnnQ

2

)(

Some More Work

We also implemented Parallel solutions to each of these problems. We had test results of their performance on CILK.

Thank You