a data structures´ view at the ip-lookup and packet classification problem
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A Data Structures´ View at the IP-Lookup and Packet Classification Problem. (Lecture 11: The IP-LookUp & Packet Classification Problem, Part I) Advanced Algorithms & Data Structures. Prof. Dr. Th. Ottmann Albert-Ludwigs-Universität Freiburg, Germany [email protected]. - PowerPoint PPT PresentationTRANSCRIPT
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A Data Structures´ View at theIP-Lookup and Packet Classification Problem
Prof. Dr. Th. OttmannAlbert-Ludwigs-Universität Freiburg, Germany
(Lecture 11: The IP-LookUp & Packet Classification Problem, Part I)Advanced Algorithms & Data Structures
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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
Inverted Next Lists
Priority Search Trees
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Dstn Addr
--------
---- ----
--------
Dstn-prefix Next Hop
Forwarding Table
Forwarding EngineHEADER
Lookup in an IP Router
Next-Hop-Computation
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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
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Generalisations of the IP-Lookup Problem
• 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: Classification
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
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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)
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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 for IP-Lookup and Packet Classification
• 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.……
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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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1 8 92 3 4 5 6 7 10
A
F
B
G
H
C D E
Structure the set of intervals
Set of n nested intervals, among them the interval *representing the whole universe.
n intervals partition the universe into ≤ 2n fragments.
Fragments
Intervals
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1 8 92 3 4 5 6 7 10
A
F
B
G
H
C D E
LMP
For a given point p:Find the fragment f, into which p falls, and determine the smallest interval X containing f.
Fragments
Intervals
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LMP and updates
P1 P2 P3 P4 P5 P6
P0
Storing the smallest interval I containing fragment f at the fragments:
LMP in time O(log n)Update: (n)
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1 8 92 3 4 5 6 7 10
A
F
B
G
H
C D E
Let x be an arbitrary fragment or interval, define:
next(x) = the smallest interval containing x
For any given set of nested intervals:For each fragment and each interval x (except *) next(x) existsand is well defined!
Fragments
Definition next(x)
Intervals
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1 8 92 3 4 5 6 7 10
A
F
B
G
H
C D E
Fragments
Inverted-Next-Lists
Inverted-Next-List of interval X = {Y; next(Y) = X}, sorted in l.t.r.o.
A: B, 3, C, 7, D, 9, EB: 1, FC: 4, HD: 8E: 10F: 2G: 5H: G, 6
Intervals
For each point p in fragment Y:If next(Y) = X, then X is the smallestinterval containing p!
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1 8 92 3 4 5 6 7 10
A
F
B
G
H
C D E
Fragments
Inverted-Next-Lists
A: B, 3, C, 7, D, 9, EB: 1, FC: 4, HD: 8E: 10F: 2G: 5H: G, 6
Intervals
Fragment i has direct link to the position,where i occurs in an INL.Each fragment and intervall in an INL is specified by ist start- and endpoint.Sum of lengths of all INL is at most 2 n.
Inverted-Next-List of interval X = {Y; next(Y) = X}, sorted in l.t.r.o.
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Longest Prefix Matching for point p
1 8 92 3 4 5 6 7 10
A
F
B
G
H
C D E
Determine fragment Y, into which p falls,follow the link from Y to the INL, in which Y occurs,proceed from Y to the head of this INL, report the interval X at the header of this list!
A: B, 3, C, 7, D, 9, EB: 1, FC: 4, HD: 8E: 10F: 2G: 5H: G, 6
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1 8 92 3 4 5 6 7 10
A
F
B
G
H
C D E
Insertion of a new interval without expansion of the fragment raster(2)
A: B, 3, C, 7, D, 9, EB: 1, FC: 4, HD: 8E: 10F: 2G: 5H: G, 6
I
A: B, I, 9. EI: 3, C, 7, D
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1 8 92
3a
4 5 6
7a
10
A
F
B
G
H
C D E
Fragments
Insertion of a new interval with expansion of the fragment raster(1)
A: B, 3, C, 7, D, 9, EB: 1, FC: 4, HD: 8E: 10F: 2G: 5H: G, 6
3b7b
I
A: B, 3a, I, 7b, D, 9, EI: 3b, C, 7a
Intervals
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1 8 92 3 4 5 6 7 10
A
F
B
G
H
C D E
Fragments
Deletion of an interval without contraction of the fragment raster(1)
A: B, 3, C, 7, D, 9, EB: 1, FC: 4, HD: 8E: 10F: 2G: 5H: G, 6
A: B, 3, 4, H, 7, D, 9, E
Intervals
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Implementation
List of fragments: Balanced leaf-search-tree, Operations: Search, insert, deleteInverted Next Lists: Balanced leaf-search-tree Operations: Find root (marking),
splitting a tree according to a given x-value, join of 2 or 3 trees to a new one.
All operations can be carried out in time O(log n), n = #(intervals)
Alternative implementations may use:
Biased Skip Lists (for bursty access patterns): Sahni/Kim 2003Splay Trees (self adjusting to the access pattern)……..
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Summary
Trie-based solutions: (1-dim)static: LMP in time O(W)dynamic: insertion, deletion in time O(W) (Doeringer et al, 1996)Dynamic programming algorithm for the construction of space optimal tries of a predetermined height (optimize the strides of tries)(Sahni et al, 2003)
Geometry-based solutions: (dynamic)Dimension: 1-dim, 2-dim, higher …Interval sets: nested, conflict free, generalPriorities: length of intervals (LMP), generalStructures: inverted lists, (balaced) search trees, skip lists
segment trees, interval trees, PSTEfficiency: worst case, amortized w.c., expected case
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1-dim, nested intervals, LMP-priority, Sahni et al. 2003search, insertion, deletion in time O(log n), inverted lists, balanced trees, skip lists
1-dim, nested intervals, general priorities, Kaplan et al. 2003search, insertion, deletion in time O(log n) and space O(n)
worst case: balanced dynamic treesamortized: self-adjusting dynamic trees
1-dim, general intervals, general priorities, Kaplan et al. 2003search in worst-case time O(log n)insertion in amortized time O(log n)deletion in amortized time O(log n log log n)space O(n)interval trees, BB[] trees
2-dim, general rectangles, general prioritiessearch, insert, delete in time O(log2 n) and space O(n log n)dynamic segment trees
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2-dim, nested rectangles, general priorities, Kaplan et al. 2003search in amortized time O(log n)insertion in amortized time O(log2 n)space O(n log n)splay trees
2-dim, general rectangles over fixed universe U, general priorities, Eppstein et al. 2001
Static case (no updates)search in time O(log |U|)conflict detection in time O(n3/2)k-d-trees,
Open problems:
Simple, efficient solution for the general, higher-dimensional case.Bursty and clustered updates.Using other data structures, like
fully dynamic segment trees,relaxed balanced search trees, relaxed balanced PST, ….