IP-Lookup and Packet Classification

Computational Geometry, WS 2007/08Lecture 14

Prof. Dr. Thomas Ottmann

Algorithmen & Datenstrukturen, Institut für InformatikFakultät für Angewandte WissenschaftenAlbert-Ludwigs-Universität Freiburg

Computational Geometry, WS 2007/08Prof. Dr. Thomas Ottmann 2

Overview

• The IP-Lookup problem

• Generalisations of IP-Lookup:

Dynamisation, conflict-free ranges

• Packet classification

• Category 1 solutions: Structure the universe

• Category 2 solutions: Structure the prefix set

• Priority Search Trees

Computational Geometry, WS 2007/08Prof. Dr. Thomas Ottmann 3

Dstn Addr

--------

---- ----

--------

Dstn-prefix Next Hop

Forwarding Table

Forwarding EngineHEADER

Lookup in an IP Router

Next-Hop-Computation

Computational Geometry, WS 2007/08Prof. Dr. Thomas Ottmann 4

0 224

232-1

128.9.0.0/16

65.0.0.0

142.12.0.0/19

65.0.0.0/8

65.255.255.255

Destination IP Prefix

Outgoing Port

(Next-Hop)

65.0.0.0/ 8 3

128.9.0.0/16 1

142.12.0.0/19 7

IP prefix: 0-32 bits

128.9.16.14

Example Forwarding Table

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128.9.16.0/21128.9.172.0/21

128.9.176.0/24

0 232-1

128.9.0.0/16142.12.0.0/19

65.0.0.0/8

Nested intervalls

Sets of intervalls corresponding to sets of prefixes of IP-addressesare nested:Any two intervalls are either disjoint or one is contained in the other!

Overlaps are impossible!

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128.9.16.0/21128.9.172.0/21

128.9.176.0/24

Routing lookup: Find the longest matching prefix (the most specific route) among all prefixes that match the destination address.

0 232-1

128.9.0.0/16142.12.0.0/19

65.0.0.0/8

128.9.16.14

LMP-Matching

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Example of an IP-Lookup Table

Prefix Next-hop

P1 111* H1

P2 10* H2

P3 101* H3

P4 10101 H4

P5 * H5

Prefixes may be considered as bitstrings of a maximal length W.

Example: W = 5, p = 10111

Computational Geometry, WS 2007/08Prof. Dr. Thomas Ottmann 8

Generalisations of IP-Lookup

• Dynamisation: Insertion, deletion of prefixes (intervalls)• Conflict free sets of intervalls (not nested)

• Priority matching instead of LMP lookup:

Prefix Next-hop

R1 10* H3

R2 1001* H2

R3 01100 H5

R4 0* H4

R5 010* H1

R6 * H6

Determine the prefix with highestpriority matching a given bitstring

10011

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Extension to higher dimensions

Classify incoming packages into different flows according to different services.

Service Example

Traffic Shaping

Ensure that ISP3 does not inject more than 50Mbps of total traffic on interface X, of which no more than 10Mbps is email traffic

Packet Filtering

Deny all traffic from ISP2 (on interface X) destined to E2

Policy Routing

Send all voice-over-IP traffic arriving from E1 (on interface Y) and destined to E2 via a separate ATM network

PAYLOADL4-SP16b

L4-DP16b

PROTO8b

L3-SA32b

L3-DA32b

L3-Proto8b

L2-SA48b

L2-DA48b

Computational Geometry, WS 2007/08Prof. Dr. Thomas Ottmann 10

Example Classifier

Rule Destination Address

Source Address

R1 0* 10*

R2 0* 01*

R3 0* 1*

R4 00* 1*

R5 00* 11*

R6 10* 1*

R7 * 00*

Field 1 Field 2

IP-Lookup is a special case of the packet classification problem.

Example: (00101, 11011)

Computational Geometry, WS 2007/08Prof. Dr. Thomas Ottmann 11

R1

R2

R3R4

2-dim Classifier

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R1

R2

R3

{R1, R2}

{R2, R3}

{R1, R2, R3}

7 regions

For a given point p: Find the rule (region) with highest priority containing p.

2-dim Classifier

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Solution stategies

• Structure the universe of all bitstrings of length W:

Trie based methods

+ Lookup becomes dependent only on the length of the given prefix

- Updates expensive

• Structure the current set of n prefixes:

Geometry based methods

+ Efficient updates

- Lookup dependent on n

• Hardware-based solutions

• Hybrid methods

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Binary Tries

0

0

1

1 10

00

A full binary tree of height W is a possible raster for storing all prefixes and bitstrings of length ≤ W.

LMP: For a given bitstring p find the deepest node representing a prefixon the search path to leaf p!Time: O(W)

{*, 00*, 010*, 10*,100*}

0000 1111

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Compression (1)

0

0

1

1 0

00

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Compression (2)

0

0

10

10

0

The number of nodes in a contracted binary trie storing n prefixes is ≤ 2n.CBT are similar to Ukkonen‘s implicit suffix trees.Lookup, insertion and deletion of prefixes in time O(W).

Many possible variations:Increase branching factor in order to decrease height.Adapt branching factor on each level to number of prefixes.……

Computational Geometry, WS 2007/08Prof. Dr. Thomas Ottmann 17

Increasing the branching factor

00 01 10 11

0000 1111

Trie based structures are set- and order-unique, hence,oblivious structures!

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Geometric view of IP-table lookup

Find a representation of a (dynamic) set of intervals such that the following operations can be carried out efficiently:

• Insertion of an interval• Deletion of an interval• For a given point p: Find the smallest interval stabbed by p (LMP-

query)

Assumptions:

Set of intervals is nested (but not necessarily a prefix set).

Every newly inserted interval does not overlap with any interval in the set.

Conflict test: For a given interval test whether or not it overlaps (is in conflict) with any interval already in the set.

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0 5 10 15 0 5 10 15

A

BC

D

A

B

C

D

p = (d, d)

10

5d

0 5 10 15 0 5 10 15

A

BC

D

A

B

C

D

p = (d, d)

10

5d

Mapping intervals to points

Assumption: All points have pairwise distinct x-coordinates Map point (x, y) to ((x, – y), y)

Interval (l, r) is mapped to point (r, l) below the main diagonal.

Computational Geometry, WS 2007/08Prof. Dr. Thomas Ottmann 20

B

A

B

A

X and Y represent disjoint intervals⇔ There is no point p on the diagonal, s.t. p ≼ X und p ≼ Y

Disjoint intervals

Computational Geometry, WS 2007/08Prof. Dr. Thomas Ottmann 21

A

B

A

B

A ⊒ B Interval A = (lA, rA) includes interval B = (lB, rB) iff. point A is right below point B, A B ⋟

B ⊑ A ⇔ A ⊒ B ⇔ point B left above point A ⇔ B ≼ A

Geometric interpretation of interval-inclusion

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C

p

D

C

D

X and Y represent overlapping intervals⇔ There is a point p on the diagonal, s.t. p ≼ X and p ≼ Y , but X and Y are incomparable with resp. to ≼

Overlapping intervals

Computational Geometry, WS 2007/08Prof. Dr. Thomas Ottmann 23

BB

p

A

D

C

X is smallest intervall, that contains point p⇔ p ≼ X and for all Y with p ≼ Y : X ≼ Y

Observation: Sets of nested intervals stabbed by a point are totally ordered with resp. to ≼ Points closer to the diagonal represent smaller intervals.

Nested intervals

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Equivalent Query

For a given p, find the leftmost point whose x-value is greater than or equal to p and whose y-value is less than or equal to p.

p

p

Since the original intervals are nested, the topmost-leftmost point is also the leftmost one!

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Geometric version of the IP-lookup-problem

Maintain a set of points in the plane (below the main diagonal) s.t. the following operations can be carried out efficiently:

• insertion of points• deletion of points• minXinRectagle queries (corresponds to topmost-leftmost queries)

minXinRectangle(l, r, y0)

Data structures used:

• Priority-search-trees (Lu/Sahni, IEEE TC, 2004)• Min-augmented-search-trees• Priority-search-pennats

(Lauer, Ottmann, Datta: Update efficient data structures for dynamic IP router tables, International Journal of Foundations of Computer Science, 18(1):139--161, Feb. 2007. World Scientific Publishing Company. )

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Priority Search Trees

Priority Search trees are a 1.5-dim structure for the storage of points, they support the following operations :

Insertion of a point Deletion of a point South-grounded range queries

Computational Geometry, WS 2007/08Prof. Dr. Thomas Ottmann 27

83, 8

32 4 56, 9

6 71

2, 41

4, 53

5, 45 7

1, 22

8, 36

7, 14

Priority search trees are - binary leaf search trees for the x-coordinates of the points. - min heaps for the y-coordinates of the points.

Priority Search Tree

Computational Geometry, WS 2007/08Prof. Dr. Thomas Ottmann 28

l

y0

r

Executable in O(log n) time.

Find topmost leftmost (orleftmost topmost) point inthe range (l, r , y0)

p

l

y0

Well defined, if intervalls are nested,otherwise not!

r

minXinRectangle(l, r, y0)

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15,1

40

4,2

9

33,4

28

1

1,3

1

6,10

1317,35

302,4

2

17 28

17,9

17

Finding topmost leftmost points in PST

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Finding leftmost topmost points in PST