Compressing Forwarding Tables

Ori Rottenstreich (Technion, Israel)

Joint work with Marat Radan, Yuval Cassuto, Isaac Keslassy (Technion, Israel)

Carmi Arad, Tal Mizrahi, Yoram Revah (Marvell Israel)

Avinatan Hassidim (Google Israel & Bar-Ilan Univ.)

• Forwarding information in datacenter networks is stored in switch forwarding tables• Example:

o Lookup key: MAC Address + VLANo Forwarding Information: Target Port

Forwarding Tables in Datacenters

2

?

• Typical table size: h∙d·b bits

• Host virtualization in datacenter networks results in a rapid growth of forwarding tables.

o More hosts ( h↑ )o More attributes ( d↑ )o More attribute values ( b↑ )

Growth of Forwarding Tables

3

d fields, each of b bits (on average)

1

h

h rows

Field 1

Field 2

Field d

This table architecture doesn’t scale!

hash-based

matching

Packet

header

• Direct access to all rows regardless of their indiceso High packet rate

• Storing each row in a fixed-width memory wordo We would like to minimize the required memory width

Required Properties of a Solution

4

width

row 1

row h

i

P bits (with d fields)

• Illustration of access to the encoded forwarding table. • A single encoded row of at most P bits is retrieved. • Then, d pipelined accesses to d dictionaries are performed.

• Goal: Find d codes that minimize the maximal width P.

Solution Overview

σ1

σ2σd

1

h

hash-based

matching

h rows

5

001 1 10

Packet

header

• Table A with d=3 columns and its encoding scheme

• Obtained maximal width of bits

A Forwarding Table Example

Today

d=3 dictionaries

Suggested Solution

6

Encoded table

Original table

3+1+1=5

2+1+2=5

2+1+1=4

2+1+2=5

2+1+2=5

3+1+1=5

2+1+2=5

width

• Huffman encoding:

o Minimizes the average width but not the maximal widtho Existence of long rows

• Fixed uniform encoding:

o Without long rows, but not efficient

• Our goal: Find an encoding scheme that minimizes the maximal width:

Traditional Solutions

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• Definition 1 (Encoding Scheme): An encoding scheme of a FIB table with d columns is composed of d prefix codes, one for each column. •Definition 2 (Encoding Width): The encoding width of an encoding scheme is defined as the maximal sum of the lengths of the d codewords in the encoding of a row of A, i.e.

Problem Definition

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• Definition 3 (Optimal Encoding Width): For a table A, we denote by the smallest possible width of an encoding scheme of A such that

• Our goal: Find an encoding scheme that minimizes the encoding width

• Constraint on the encoding scheme (Kraft’s Inequality): There exists a prefix encoding of the elements in a set with codeword lengths iff

Problem Definition (2)

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• For a table A with h rows , we define the following optimization problem:

• We would like to minimize the maximal width P while satisfying the Kraft’s inequality for the d prefix codes The codeword lengths should be positive integers.

Optimization Problem

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• Theorem: Finding the optimal encoding is NP-hard in the general case ( ). • Convex Algorithm:

o Consider the relaxed problem: codeword lengths are not necessarily integers. (The relaxed problem is convex.)o Take the ceiling of the codeword lengths in its solutiono Example: Solution: (2.2, 3.71) (3, 4)

• Theorem: By taking the ceiling of the codeword lengths in an optimal solution to the relaxed problem, we achieve a legal encoding scheme that satisfies

Approximation of the Optimal Encoding

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Outline

Introduction and Problem Definition General Properties Two Column FIB Tables

Graph Representation Optimal Conditional Encoding Bounds on the Optimal Encoding Width

Experimental Results Summary

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• Motivation: L2 MAC tables with two fields: Target Port + An additional attribute.

• Assumption: The table A has two columns with distinct elements in each column. •Theorem: The optimal encoding width of A satisfies

Two-Column FIB Tables

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• A two-column FIB table A can be represented as a bipartite graph. •Example: Two-column FIB table with h=6 rows and (for W=2) distinct elements in each column.

Graph Representation

Forwarding table Bipartite graph

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• Given an encoding of one column, we would like to find an encoding of the second column that minimizes the encoding width P.

• For an element , we should consider the maximal codeword length of an element in that shares a row with .

Notation:

Optimal Conditional Encoding

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u a

y

w b

y

y

P bits

v

z

u

• Theorem: The optimal conditional encoding width satisfies

• Notes:• We should encode in bits. • Linear complexity

Optimal Conditional Encoding

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• Reminder (Edge Cover): In an undirected graph

a set of edges is an edge cover of if .

• Example: nodes at each side ( )

• Intuition:

Smaller covering number More independent edges

Constraints on more elements Larger optimal width

Lower Bound on the Optimal Encoding Width

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(A) (B)

Edge covering number: 8 Edge covering number: 14

• Theorem: The optimal encoding width of a table A with an edge covering number satisfies

• Example: ( )

Lower Bound on the Optimal Encoding Width

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(A) (B)

Edge covering number: 8 Edge covering number: 14

33333333

33333333

14444444

14444444

Effectiveness of the encoding algorithm on synthetic FIB tables with d=2 columns and (W=8) elements, as a function of Zipf dist. parameter.

Simulation details: 1000 rows, 10 iterations.

Experimental Results

A lower bound of

.

Lower parameter Closer to the

uniform distribution

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8 8 1 9

9 1

• Table: 903 rows. Collected in the enterprise network of a large company. • Without encoding: row width of 29 bits.

• With convex encoding: width of only 11 bits.

Real-Life FIB Table

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improvement

Concluding Remarks

• New architecture for compression of forwarding tables in datacenters

• Approximation of the optimal solution

• Analysis of the optimal encoding width

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Thank You