On the Code Length of TCAM Coding Schemes

Ori Rottenstreich (Technion, Israel)

Joint work with Isaac Keslassy (Technion, Israel)

1

Network Structure

Switch

2

Packet Classification

Action

--------

---- ----

--------

Rule ActionPolicy Database (classifier)

Packet Classification

Forwarding Engine

Incoming Packet

HEADER

3

Switch

The TCAM Architecture

Enc

oder

Match lines

0

1

2

3

4

6

5

7

8

9

deny

accept

accept

denydeny

deny

denyaccept

deny

accept

00111011111110000001101001011001000

00111011100100

1110010010011100101011111111111

0011101000011100

0

0

0

1

0

1

0

1

0

1

row 3

Each entry is a word in {0,1,}W

Packet Header

TCAM Array

Source Port

4Width W

Range Rules

Rule Source address

Source port

Dest-address

Dest-port

Protocol

Action

Rule 1 123.25.0.0/16 80 255.2.3.4/32 80 TCP Accept

Rule 2 13.24.35.0/24 >1023 255.2.127.4/31 5556 TCP Deny

Rule 3 16.32.223.14 20-50 255.2.3.4/31 50-70 UDP Accept

Rule 4 22.2.3.4 1-6 255.2.3.0/21 20-22 TCP Limit

Rule 5 255.2.3.4 12-809 255.2.3.4 17-190 ICMP Log

Range rule = rule that contains range field Usually source-port or dest-port

5

How many TCAM entries are required for representing one range field?

Range Rule Representation in TCAM

Coding objective: minimize the number of TCAM entries needed to code the rangeMore TCAM entries represent more power

consumption

Some ranges are easy to representExample: W=3:

[4, 7] = {100,101,110,111} = 1

But what about [1,6]?

6

Range [1,6] in tree of all elements with W=3 bits:

(Internal) Coding of [1,6]

010 011001 110100 101

10* 11001*001

111000

Known result: expansion in 2W-2 TCAM entries Here: 2W-2=4 TCAM entries

0 1 2 3 4 5 6 7

7

Upper Bounds on the TCAM Range Expansion

2W-2 (Srinivasan et al., 1998)

2W-4 (Bremler-Barr et al., 2007)

2W-5 (Roth, 2009)

W (Rottenstreich et al., 2010; Cohen et al., 2010)

???

8

Is W also the lower bound?

External Coding

010 011001 110100 101

***

111000

111000

Reduced number of TCAM entries: exploiting TCAM entry order by coding range complimentary as well

Recent result: expansion in W TCAM entries Here: W=3 TCAM entries (instead of 4) Is this new coding scheme

optimal? 0 1 2 3 4 5 6 7

9

Basic Definitions

Let be the number of TCAM entries required for representing the range R using the coding scheme .

We define the range code length f(W) as the best achievable code length for W-bit ranges given all coding schemes

Likewise, we define fp(W) as the minimal code length given all prefix coding schemes.

[0,2 1]

( ) min max ( )][

Wcode range R

f W n R

10

Basic Definitions (2)

We define the range code length g(W) as the best achievable code length for extremal W-bit ranges given all coding schemes

{[0, ] | [0,2 ]}1( ) min max ( )][

Wcode range R y y

g W n R

010 011001 110100 101 111000

0 1 2 3 4 5 6 7

R=[0,0]R=[0,1]R=[0,2]R=[0,3]R=[0,7]

11

Results

Known Our Contribution

1( )

2

Wg W

1

( )2

Wg W

( ) ( )pf W f W W ( )pf W WOptimality over all coding schemes

12

Optimality over prefix coding schemeswhich are the most common coding

schemes

Optimality Proof Intuition

a1 a2a3

Assume W = 2 and R = [0,2].

If a1(01), a3(10) R are encoded in the same TCAM

entry, it must be **.

Therefore, a2(11) R also matches this entry.

00 01 10 11

R

13

The range R = [0,2] cannot be coded in a single TCAM entry.

11

1

( ,..., ) { ( ,..., ) {0,1} |

[1, ], { ,..., }}

n WW

nj j j

H a a x x x

j W x a a

Hull Property

We define the hull of a set of W -bit strings in the W -dimensional string space:

Definition (Hull): Let (n, W) N2, and consider n strings a1, …, an of W bits each, with ai = (ai

1, …, aiW) for each

i [1,n]. Then the hull of (a1, …, an), denoted H(a1, …, an), is the

smallest cuboid containing a1, …, an in the W-dimensional string space, and is defined as

14

H(a1, a3) = H(10, 01) = }00 ,01 ,10 ,11{

00 01 10 11

R

a1 a2a3

Alternating Path and Independent Set

Let n and W be positive integers and let α:{0,1}W→{0,1} be

a classifier function.

An = (a1, …, a2n-1) is an alternating path if it satisfies the following conditions:

Alternation: For i [1,2n-1],

α(a1) = α(a3) = … = α(a2n-1) = 1

α(a2) = α(a4) = … = α(a2n-2) = 0

Hull: For any i1, i2, i3 such that i1 ≤ i2 ≤ i3 ≤2n-1, ai2 H(ai1, ai3)

15

In such an alternating path )a1, a3, a5, … , a2n-3, a2n-1 (is an

independent set

Example

a1 a2a3

W = 2 and R = [0,2].

Alternation: α(a1) = α(a3) = 1, α(a2)=0.

Hull: a2 H(a1, a3).

)a1, a2, a3 (is an alternating path and )a1, a3 (is an independent set

00 01 10 11

R

16

Alternating Path and Independent Set

Theorem: A classifier function with an independent set of size n cannot be coded in less than n TCAM entries.

17

Extremal Range Code-Length Optimality

Assume W=4.

Let R = [0, c] = [0, {1010}] .

Define: a1 = 0101

a2 = 1101

a3 = 1001,

a4 = 1011

a5 = c = 1010. Observe: a1, a3, a5 ≤ c = 1010,

a2, a4 > c = 1010. Alternation: α(a1) = α(a3) = α(a5)= 1, α(a2) = α(a4)= 0.

Hull: ai2 H(ai1, ai3) for i1 ≤ i2 ≤ i3 .

)a1, a3, a5 (is an independent set.The range R = [0,{1010}] cannot be coded in less than 3 TCAM entries.

18

Extremal Range Code-Length Optimality

We showed optimality for g(W) with W=2 and W=4, let’s show it for any even W.

Let R = [0, c] = [0, {10}W] = [0,2/3(2W-1)].

Define: a1 = {01}W/2, a2 = 11{01}W/2-1, a3 = 10{01}W/2-1,

a4 = 1011{01}W/2-2, a5 = 1010{01}W/2-2,

…, aW+1 = c = {10}W/2

Observe: a2k+1 ≤ c

a2k ≥ c. Alternation: α(a2k+1) = 1, α(a2k) = 0.

Hull: ai2 H(ai1, ai3) for i1 ≤ i2 ≤ i3. (a1, a2, …, aW+1) is an alternating path and (a1, a3 , … ,

aW+1) is an independent set.

The upper bound of g(W) is tight.

19

General Range Code-Length Optimality

In this paper, we present a range R that cannot be coded in less than W prefix TCAM entries, whether the first TCAM entry encodes the range R (internal coding) or its complementary (external coding).

Future Direction:

• Coding scheme optimality over all coding schemes

20

Summary

TCAM coding optimality• g(W) (Extremal ranges, all coding schemes)

• fp(W) (All ranges, prefix coding schemes)

New tools for TCAM: Independent sets and alternating paths

21

Thank You

22