1. Range Trees with Variable Length Comparisons (RT-VLC)
Author:
Ioannis Sourdis, Ruben de Smet, and Georgi N. Gaydadjiev
Publisher:
15th IEEE Workshop on High Performance Switching and Routing (HPSR), 2009
Presenter:
Chin-Chung Pan
Date: 2009/09/08

2. Outline
Introduction
RT-VLC Rules
Constructing a RT-VLC
Performance

3. Introduction-Range Tree
L: less, E:equal, GE: greater or equal, G: greater.

4. Introduction-Range Tree
Range Trees avoid the length dimension performing comparisons on the expanded prefixes (full addresses).
Range Trees need to store the complete addresses to be compared at each stage and therefore consume considerable memory size.

5. Introduction-Range Trees with Variable Length Comparisons(RT-VLC)
CP: Common Prefix, ‘-’: bit not necessary to be processed, L: less, E:equal, GE: greater or equal, G: greater.

6. RT-VLC Rules
The address range [Na,Nb) and divides it in k+1 subranges R1, R2, ..., Rk+1 defined by the unique addresses (of width W) Ai ∈ [Na,Nb), ∀i ∈ N, i ≤ k, such that R1 = [Na,A1), .., Ri = [Ai−1,Ai), ... Rk+1 = [Ak,Nb).

7. RT-VLC Rules (cont.) Rule 1：Omit the common prefix of the node border.
Rule 2：Omit address suffix of value ’0’.
Rule 3：Share addresses’ common prefix.
Rule 4：Share addresses’ common suffix.

8. Rule 1：Omit the common prefix of the node border
When there is a common prefix of length L (L < W) of the node borders Na and Nb then the L most significant bits of the addresses Ai can be omitted from the comparisons at the node.
Na ≦ AIN＜ Nb
RT-VLC Node Na Nb A1 A2
00000
00011
00101
01000
(00111)
--011
--101

9. Rule 2：Omit address suffix of value ’0’
Ai have a suffix of length L, where L < W, that is zero. Then, this suffix of Ai does not need to be compared against the L last bits of AIN.
Na ≦ AIN＜ Nb
RT-VLC Node Na Nb A1 A2
00100
01010
001--
0101-

10. Rule 3：Share addresses’ common prefix
The Common Prefix ACPi of the addresses Ai of length L (L < W) can be shared among concurrent comparisons and processed separately.
Na ≦ AIN＜ Nb
1110- G E
11100
11101 L
----1 L GE L GE Na Nb L GE A1 A2

11. Rule 4：Share addresses’ common suffix
The common suffix ACSi of the addresses Ai, of length L (L < W) can be shared among concurrent comparisons and processed separately
011--
110--
Na ≦ AIN＜ Nb G L E E L G
---01
01101
11001 L GE L GE L GE Na Nb A1 A2

12. Constructing a RT-VLC
Top-Down Heuristic
Bottom-Up Heuristic

13. Constructing a RT-VLC - Top-Down Heuristic

14. Constructing a RT-VLC - Top-Down Heuristic
01---
1----
--11-
--01-
--10-
-1100
CP: -110-
-1101
----1 R1 R2 R3 R4 R5 R6 R7 R6

15. Constructing a RT-VLC - Top-Down Heuristic
Step1:Apply the RT-VLC rules 1 and 2. Step2:Select a comparator length that maximizes the possible number of branches, i.e., 8-bits. Step3:Consider all addresses in the set to be processed with the above comparison length. Omit address suffixes that cannot be compared assuming they are equal to zero. Step4:Based on the above create the defined from the comparisons groups/intervals.

16. Constructing a RT-VLC - Top-Down Heuristic
Step5:If necessary merge neighboring groups until the number of comparisons are reduced to the available comparator resources.
Step6:Recursively repeat for the created children nodes.
Step7:Terminate when each node branch points to a single address range.

17. Constructing a RT-VLC - Bottom-Up Heuristic

18. Constructing a RT-VLC - Bottom-Up Heuristic
Na 00000 A A A A A A A
Nb 11111
01---
1----
CP:-1100
--11-
--01-
--10-
-1100
-1101
-1101 A1
00110 A2
01000 A3
01010 A4
01100 A5
10000 A6
11100 A7
11101 Nb Na
N’a
00000
N’a
01000
N’b
01000
N’a
10000
N’b
10000
N’b
11111

19. Constructing a RT-VLC - Bottom-Up Heuristic
Step1:Select the first b addresses Ai > Na (where Na initially is 0) that can be compared in one node after applying the RT- VLC rules (e.g., addresses Ai,Ai+1, ...Ab). Step2:Set node’s upper bound Nb a point in the address space where Nb < Ab+1 that Na has a long suffix of 0’s2. The resulted node maps to an address range [Na,Nb). Step3:Repeat the above starting from the upper bound of the previous group (Nnewa = Nprevb ) until all addresses Ai in the address space are grouped.

20. Constructing a RT-VLC - Bottom-Up Heuristic
Step4:Recursively repeat the above steps for the next upper tree level L−1 using as new set of addresses AL−1i the borders NLi of the previous level. Step5:Terminate when all addresses in the list are processed in a single (root) node.

21. Performance-Top-Down

22. Performance-Bottom-Up

23. Performance-IPv6