1. Worst-Case TCAM Rule Expansion
AUTHOR：
ORI ROTTENSTREICH AND ISSAC KESLASSY
PUBLISHER：
IEEE INFOCOM 2010
PRESENTER：
ZONG-LIN SIE
DATE：
2010/09/15

2. Outline Introduction Background Range Expansion Guarantees
Experimental Result
Suggested Architectures

3. Introduction
In this paper , given several type of rules , provide new upper bounds on the TCAM worst-case rule expansions.
For a W-bit range：
previously known upon bound：2W-5 TCAMs.
this paper’s method provide：W TCAMs.
Also propose a modified TCAM architecture that uses additional logic to significantly reduce the rule expansion.

4. Background (1/3) What is TCAM？ Ternary Content Addressable Memory.
Packet classification is the core function behind many network application.(routing、filtering…)
Hardware-based TCAMs are the standard devices
for high-speed packet classification.
1.講出TCAM high-speed的原因(parallel)。

5. Background (2/3) What is rule expansion？ two types of rules：
(1) simple rules：need single entry per rule.
(2) range rules：might need many entries per rule.
Thus cause rule expansion.
for example：
range[2,4] means 01* and 100
range[1,14] means 0001,001*,01**,10**,11**,1110

6. Background (3/3)
However , power consumption constitutes a bottleneck for TCAM scaling.
Given same access rate , a TCAM chip consumes up to 30 times more power than equivalent SRAM chip with a software-based solution.
Goal of this paper is to gain a more fundamental understanding of the worst-case number of TCAM entries needed to encode a rule . Provide upper bounds.

7. Related Work (1/3) Internal expansion ：
only uses entries from within the range . W-bit field can be encoded in 2W-2 entries for W ≥ 2 .
Rule with d fields , can be encoded up to
entries .
For instance ：
W=3 , single field R = [1,6]
needs 2W-2 = 4 entries.
(see 111 as default)

8. Related Work (2/3)
The same way but rely on Gray codes instead of binary codes reduce the worst-case internal range expansion from 2W-2 to 2W-4 for sufficient large W.
Using more complex coding , improve to 2W-5 .

9. Related Work (3/3) External encoding ：
the same instance , W=3 , R = [1,6] .
we encode the range exterior first , and then the range itself is encoded indirectly.
So , only 3 entries!

10. Range Expansion Guarantees (Model and Notations)
Definition 1(Header) ：
is a W-bit string defined on the d fields Each sub-string of length
corresponds to field with
Definition 2(Range Rule) ：
The range rule in field is defined as an integer range[r1,r2] . r1≤r2 ,both W-bit integers.
if xi match ,
Match if r1 <= xi <= r2

11. Range Expansion Guarantees (Model and Notations)
Definition 3(TCAM entry) ：A TCAM entry is
composed of a TCAM rule S = ,
where {0,1} are bit values and * stands for don’t-care ,
And an action , where A is a set of actions . A W-
bit string b = matches S , denoted as , if
for all

12. Range Expansion Guarantees (Model and Notations)
Ex: R = [1,6] A = {accept ,deny}, ,

13. Optimal Range Expansion Problem
We want to find a TCAM encoding scheme that minimizes the worst-case TCAM expansion
over all possible ranges R .

14. Optimal Range Expansion Problem
Then the range expansion f(W) is defined as the best-achievable range expansion for W-bit ranges given all encoding schemes , i.e.
To find the range expansion ,we first characterize the extremal range expansion g(W) , which is defined ：
A range R over is called extremal if R = [0,y] or R = for some arbitrary value of y.

15. Range Expansion Guarantees
Theorem 1 ： for all W ≥ 1 , the extremal range expansion satisfies the following upper-bound ：
(by induction on W)
example ：for W=3 , the extremal range R = [0,6] can encoded with only 2 TCAMs .(111 , ***)
This is better than the best internal encoding (0**,10* ,110) ,which use W=3 TCAMs entries.

16. Range Expansion Guarantees
Theorem 2： for all W≥ 1 , the worst-case range
expansion satisfies the following upper-bound：
proof： by induction on W ≥ 1.
(1) for W = 1 , all non-empty ranges are extremal ,
therefore , the result follows by Theorem 1.
for W = 2 , all non-empty and non-extremal ran-
ges are either single points , or[1,2] can be enco-
ded in 2 TCAMs.

17. Range Expansion Guarantees
(2) induction step：Now let W ≥ 3 ,and assume the
claim true until W – 1. Consider any range
, and cut it into 4 sub-ranges.
, which
and distinguish the following cases：

18. Range Expansion Guarantees
case(1)：if ,
then by induction R can be encoded in at most
W-1 entries .
case(2)：Else if ,
then this is just a shifted version of the previo-
us case and , by lemma 2 ,R can be encoded in
at most W-1 entries .

19. Range Expansion Guarantees
case(3) ：
(a) If is a
right –extremal range on , and
by lemma 1,can be encoded in g(W-1) TCAM
entries . Further by lemma 2 can be
encoded in g(W-2) TCAM entries .
(b) if can be
encoded in g(W-2) TCAM entries , and
in g(W-1) TCAM entries .

20. Range Expansion Guarantees
Therefore , in both sub-cases , by theorem 1 ,
R can be encoded in up to
TCAM
entries .
case(4) ：
we use 2 different techniques according to
the parity of W .

21. Range Expansion Guarantees

22. Range Expansion Guarantees

23. Range Expansion Guarantees
Exponential Number of TCAM Entries：
Theorem 4：
For any classification rule R =
of d fields , there is an encoding scheme
Linear Number of TCAM Entries
Theorem 5：Any classification rule R on d fields can
be encoded in at most d．W TCAM entries with-
out any additional logic.

24. Range Expansion Guarantees
Example of linear number entries：
we first negatively encode the four striped regions ,
using an encoding of four one-dimensional extremal
intervals , 4W . And a default positive entry . Total 4W+1 .

25. Experimental Result Internal binary-prefix approach 2W-2 = 14.
Suggested scheme W = 8.

26. Suggested Architecture (1/3)
Standard INTERNAL –RRODUCT
Assume d=2 ,W=4 ,k=2
R1 = [1,14] ×[5,14]
R2 = [7,10] ×[2,3]
use 6×5 to encode R1
use 3×1 to encode R2
total 33 entries

27. Suggested Architecture (2/3)
Proposed COMBINED-PRODUCT
(using complementary to encode , upper bound W)
Assume d=2 ,W=4 ,k=2
R1 = [1,14] ×[5,14]
R2 = [7,10] ×[2,3]
use 12 to encode R1
use 3 to encode R2
total 15 entries

28. Suggested Architecture (3/3)
Proposed COMBINED-SUM
(each field of each range is encoded separately)
Assume d=2 ,W=4 ,k=2
R1 = [1,14] ×[5,14]
R2 = [7,10] ×[2,3]
use 3+4 to encode R1
use 3+1 to encode R2
total 11 entries

29. Suggested Architecture