1. Encoding Short Ranges in TCAM Without Expansion: Efficient Algorithm and Applications
Yotam Harchol
The Hebrew University of Jerusalem, Israel
Joint work with:
Anat Bremler-Barr
David Hay
Yacov Hel-Or
IDC Herzliya,
Israel
Hebrew University,
ACM SPAA 2016
This research was supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/ )/ERC Grant agreement no

2. TCAM – Ternary Content Addressable Memory
TCAM is an associative memory module
Useful for parallel multidimensional prefix lookup
Each entry may consist of 0, 1, and * bits
Widely used in high-speed networking devices
What about ranges? (e.g., port numbers)
Example: 2-dimensional IP lookup:
Source IP
Dest. IP P1 P2
Drop P3
SRAM
00110****110001***
0111***** **
00*******110******
01*******11000****
TCAM 1 2 3 4 5

3. Range Encoding in Binary Representation
Can ternary encode only certain intervals
Or we need more than one entry (expansion)
Exponential space blow-up when using multiple range fields
0000 1
0001 2
0010 3
0011 4
0100 5
0101 6
0110 7
0111 8
1000 9
1001 10
1010 11
1011 12
1100 13
1101 14
1110 15
1111
00**
011*
100*
0***
01**
????
????
0101
011*
100*
1010

4. Encoding Short Ranges
Ranges of arbitrary length – infeasible without expansion
Ranges with bounded length – feasible!
Useful for:
Short TCP/UDP ranges (>60% of the ranges in real life)
QoS mechanisms (IP ToS/DSCP)
Packet size classification (by categories)
Timestamps and counters
IP spoofing detection (using IP TTL)
SDX – AS numbering
Also useful for applications outside the networking domain

5. Range Encoding on TCAM: Two Approaches
Database-independent encoding:
[Lakshminarayanan et al., 2005; Bremler Barr et al., 2007]
fenc([x, y]) =
*1**1001**10
**1*10**101*
*1**100***1*
Easy to update
Lower bound for w-bits field without expansion: 2w-1 bits per range [Lakshminarayanan et al., 2005]
Database-dependent encoding:
[Liu, 2002; Van Lunteren & Engbersen, 2003; Chang & Su, 2007; Che et al., 2008; Bremler Barr et al., 2009; Rottenstreich & Keslassy, 2010; Rottenstreich et al., 2013; Kogan et al., 2014] DB
fenc( , [x, y]) =
*1**100*10
**1*10*01*
Compact codes
Both schemes require expansion for a feasible code length

6. RENÉ – Range Encoding with No Expansion
Encoding function for short ranges:
fenc([x, y]) =
*1**1001**10
Database independent - easy to update
No row expansion
Near-optimal TCAM space usage
Useful for packet classification applications
Useful for high dimensional nearest neighbor search

7. Binary-Reflected Gray Code (BRGC)
Build recursively by reflecting binary code: 1
0000
0001 2
0011 3
0010 4
0110 5
0111 6
0101 7
0100 8
1100 9
1101 10
1111 11
1110 12
1010 13
1011 14
1001 15
1000 1
000
001 2
011 3
010 4
110 5
111 6
101 7
100 1 00 01 2 11 3 10 1
Hamming distance between each two adjacent points = 1

8. Range Encoding with Binary-Reflected Gray Code
With ternary BRGC we can encode some of the ranges of length h=2k (k∈N)
What about other (red) ranges?
00**
0000 1
0001 2
0011 3
0010 4
0110 5
0111 6
0101 7
0100 8
1100 9
1101 10
1111 11
1110 12
1010 13
1011 14
1001 15
1000
01**
11**
10**
0*1*
*10*
1*1*
*00*
*00**

9. Layers Layer: a set of disjoint consecutive ranges
Two (blue) layers can be encoded with a BRGC-based ternary word
Other (red) layers need more bits
Wait with the k-1 bits. Not here. Also don’t color bits in gray
Not necessarily a power of 2
0000 1
0001 2
0011 3
0010 4
0110 5
0111 6
0101 7
0100 8
1100 9
1101 10
1111 11
1110 12
1010 13
1011 14
1001 15
1000
00**
01**
11**
10**
0*1*
*10*
1*1*
*00*
*00**

10. Cover Ranges
Cover range of R: the smallest blue range that fully contains R
In red layers: No two consecutive ranges in the same layer are fully contained in a cover range
Given the cover range, one bit is enough to differentiate ranges in the same layer
Take a red interval from previous slide with its extra bit
Show it’s the only one with this bit value in the cover
0000 1
0001 2
0011 3
0010 4
0110 5
0111 6
0101 7
0100 8
1100 9
1101 10
1111 11
1110 12
1010 13
1011 14
1001 15
1000 R 1
0***
cover(R)

11. Single Bit Range Index Red ranges require different encoding
Add one bit for each red layer
Bit value alternates between ranges in the same layer
Bits of other red layers are set to *
For blue layers, these bits are always *
Only one red layer and one bit
One bit is enough because I know where I am to some extent
0000 1
0001 2
0011 3
0010 4
0110 5
0111 6
0101 7
0100 8
1100 9
1101 10
1111 11
1110 12
1010 13
1011 14
1001 15
1000 ** 0* 1* 0* 1* ** *1 *0 *1 *0

12. Encoding Scheme for Ranges
Encoding of a single range of length h=2k is:
Value encoding is now:
Blue ranges:
BRGC(R)
Red ranges: BRGC(cover(R))
Ternary BRGC
Bit of L1
. . .
Lh/2-1
w bits
h-2 bits
Lh-1
Lh/2+1
BRGC
Bit of L1
. . .
Lh/2-1
w bits
h-2 bits
Lh-1
Lh/2+1

13. Toy Example

14. Optimization
For ranges of length h=2k k-1 LSBs of the BRGC part are always *
Code can be shorten to w-log(h)+h-1 bits
0000|00 1
0001|10 2
0011|10 3
0010|11 4
0110|11 5
0111|01 6
0101|01 7
0100|00 8
1100|00 9
1101|10 10
1111|10 11
1110|11 12
1010|11 13
1011|01 14
1001|01 15
1000|00
00**|**
01**|**
11**|**
10**|**
0***|1*
*1**|0*
1***|1*
*0**|0*
0*1*|**
*10*|**
1*1*|**
*00*|**
0***|*1
*1**|*0
1***|*1
*0**|*0
0000|00 1
0001|10 2
0011|10 3
0010|11 4
0110|11 5
0111|01 6
0101|01 7
0100|00 8
1100|00 9
1101|10 10
1111|10 11
1110|11 12
1010|11 13
1011|01 14
1001|01 15
1000|00
00**|**
01**|**
11**|**
10**|**
0***|1*
*1**|0*
1***|1*
*0**|0*
0*1*|**
*10*|**
1*1*|**
*00*|**
0***|*1
*1**|*0
1***|*1
*0**|*0

15. Encoding Multiple Range Lengths
For ranges of length 3, intersect two ranges of length 4:
00*|**
000|00 1
000|10 2
001|10 3
001|11 4
011|11 5
011|01 6
010|01 7
010|00 8
110|00 9
110|10 10
111|10 11
111|11 12
101|11 13
101|01 14
100|01 15
100|00
01*|**
11*|**
10*|**
0**|1*
*1*|0*
1**|1*
0*1|**
*10|**
1*1|**
0**|*1
*1*|*0
1**|*1
*0*|0*
*00|**
*0*|*0
We present the conjunction operator Π:
For two ranges R1, R2: tcode(v) ≈ tcode(R1) Π tcode(R2) if and only if v∈R1 ∩ R2
(also: tcode(R1) Π tcode(R2) = ⊥  R1 ∩ R2 = ∅)
(⊥ means ‘undefined’, and if ai Π bi = ⊥ then a Π b = ⊥)
Remove gray bits away 4 7
01*|** 5 8
*1*|0* 5 7
01*|** Π *1*|0* = 01*|0*

16. TCAM bits used per range field
Theoretical Bound
TCAM bits used per range field
(log2 scale)
Maximal range length
(log2 scale)
For w=16

17. The Nearest Neighbor Search Problem
Input: - A set of data points - A query point (or a series of those) (points are in a discrete space)
Output: Data point closest to the query point
The Curse of Dimensionality: Hard problem for high dimensions (even over 10 or so)

18. The Nearest Neighbor Search Problem
Encode cubes on data points
Given a query point, single TCAM lookup returns the smallest cube that contains it
No TCAM entry expansion in high dimension
Add legend
Bullets
Data
Query

19. The Nearest Neighbor Search Problem
Or instead, save TCAM space
Encode cubes on the query point
Query TCAM with growing cubes
Find first data point to match
Goal – small space expansion, one TCAM lookup OR
Multiple queries, less entries
Data
Query

20. Experiment on a Real TCAM
Currently - no evaluation board for TCAM
Instead, we used a commodity network switch with a TCAM
Switch has 48 ports of 1Gbps each (1.5 Million packet per second)
TCAM is 192 bits wide
First cut the ternary string into dimensions
Then ignore it and cut by header fields
d-Dimensional Cube Representation:
00*1**001***011***01*1**00*1**00*1**00****0***1*0****1
1st dimension
Src. MAC
2nd dimension
Dest. MAC
3rd dimension
Src. IP
4th dimension
5th dimension
Dest. IP
6th dimension
7th dimension
OpenFlow flow_mod
Single port: 1.5 Million Queries Per Second!
Queries as packets
OpenFlow
Counters

21. fenc([x, y]) = Conclusions Encoding function for short ranges:
*1**1001**10
Database independent - easy to update
No row expansion
Near-optimal TCAM space usage
Useful for packet classification applications
Useful for high dimensional nearest neighbor search

22. Questions?
Thank you.