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Basic Data Structures for IP lookups and Packet Classification

Basic Data Structures for IP lookups and Packet Classification. Routing table examples. 2001:0200:0136::/48 2001:0200:0900::/40 2001:0200:0905::/48 2001:0200:0c00::/40 2001:0200::/32 2001:0200:c000::/35 2001:0200:e000::/35 2001:0208::/32 2001:0218::/32 2001:0218:6002::/48

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Basic Data Structures for IP lookups and Packet Classification

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  1. Basic Data Structures for IP lookups and Packet Classification

  2. Routing table examples • 2001:0200:0136::/48 • 2001:0200:0900::/40 • 2001:0200:0905::/48 • 2001:0200:0c00::/40 • 2001:0200::/32 • 2001:0200:c000::/35 • 2001:0200:e000::/35 • 2001:0208::/32 • 2001:0218::/32 • 2001:0218:6002::/48 • 2001:0220::/35 • 2001:0238::/32 • 2001:0240::/32 • 2001:0250:0204::/48 • 2001:0250::/32 • 2001:0250:e000::/36 • 4.0.0.0/8 • 6.0.0.0/8 • 9.2.0.0/16 • 9.20.0.0/17 • 12.0.0.0/8 • 13.0.0.0/8 • 15.0.0.0/8 • 16.0.0.0/8 • 17.0.0.0/8 • 18.0.0.0/8 • 20.0.0.0/8 • 24.0.0.0/18 • 24.0.0.0/14 • 24.1.0.0/17 • 24.4.0.0/17 • 24.48.0.0/18

  3. oix-route-views - Route Views Archive • http://archive.routeviews.org/oix-route-views/ • * 3.0.0.0 203.181.248.233 0 7660 1 7018 80 i • * 4.0.0.0 203.194.0.5 0 9942 1 i • * 6.1.0.0/16 203.194.0.5 0 9942 1 7170 1455 i • * 6.2.0.0/22 203.194.0.5 0 9942 1 7170 1455 i • * 6.3.0.0/18 203.194.0.5 0 9942 1 7170 1455 i • * 6.4.0.0/16 203.194.0.5 0 9942 1 7170 1455 i • * 6.5.0.0/19 203.194.0.5 0 9942 1 7170 1455 i • * 6.8.0.0/20 203.194.0.5 0 9942 1 7170 1455 i • * 6.9.0.0/20 203.194.0.5 0 9942 1 7170 1455 i • * 6.10.0.0/15 203.194.0.5 0 9942 1 7170 1455 i • * 6.14.0.0/15 203.194.0.5 0 9942 1 7170 1455 i • * 9.2.0.0/16 203.194.0.5 0 9942 1239 701 i • * 9.184.112.0/20 203.194.0.5 0 9942 3786 i • * 9.186.144.0/20 203.194.0.5 0 9942 3786 i • * 12.0.0.0 203.194.0.5 0 9942 1239 7018 i • * 12.0.48.0/20 203.194.0.5 0 9942 16631 16631 16631 1742 i

  4. Routing table format (1/3) • Destination: IP address of the packet's final destination • Next hop: The IP address to which the packet is forwarded • Interface: The outgoing network interface the device should use when forwarding the packet to the next hop or final destination • Metric: Assigns a cost to each available route so that the most cost-effective path can be chosen

  5. Routing table format (2/3) • Routes: Includes (1) directly-attached subnets, (2) indirect subnets that are not attached to the device but can be accessed through one or more hops, and (3) default routes to use for certain types of traffic or when information is lacking.

  6. Routing table format (3/3) • Routing tables can be maintained manually or dynamically. • Tables for static network devices do not change unless a administrator manually changes them. • In dynamic routing, devices build and maintain their routing tables automatically by using routing protocols to exchange information about the surrounding network topology. • Dynamic routing tables allow devices to "listen" to the network and respond to occurrences like device failures and network congestion.

  7. Prefix • Prefix Length Distribution

  8. Prefix • Length format: bn-1…b0/l (l is prefix length) • In IPv4, d3.d2.d1.d0/l can also be used. • Mask format: bn-1…b0/mn-1…m0 (prefix length is l) • mj = 1 for all n – 1  j  n – l+1, and mj =0 otherwise. • d3.d2.d1.d0/ m3.m2.m1.m0 for IPv4. • Ternary format: bn-1…bn-l+1*…* (prefix length is l) • bj = 0 or 1 for n – 1  j  n – l + 1. • If tk is *, then tj must also be * for all j < k. • A single don’t care bit can be used to denote a series of don’t care bits, e.g., 1* denotes 1**** in the 5-bit address space.

  9. Prefix • (n+1)-bit format: bn-1…bn-l+110…0 (l is prefix len) • for the prefix bn-1…bn-l+1* of length l in ternary format, there is one trailing ‘1’ followed by n – l 0’s. • or • (n+1)-bit format: bn-1…bn-l+101…1 • for the prefix bn-1…bn-l+1* of length l in ternary format, there is one trailing ‘0’ followed by n – l 1’s.

  10. 5-bit Prefixes: bn-1…bn-l+110…0 ***** 0**** 00*** 11*** 1 1 1 * * 0 0 0 * * 0 0 0 0 * 0 0 0 1 * 1 1 1 0 * 1 1 1 1 * 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 1 1 1 1 0 0 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 1 1 1 0 0 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 1 1 1 1 0 1 0 1 1 1 0 1 1 1 1 1 1 0 0 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 6-bit binary address space 000000 is not used

  11. 5-bit Prefixes:bn-1…bn-l+101…1 ***** 0**** 00*** 11*** 1 1 1 * * 0 0 0 * * 0 0 0 0 * 0 0 0 1 * 1 1 1 0 * 1 1 1 1 * 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 1 1 1 1 0 0 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 1 1 1 1 0 1 0 1 1 1 0 1 1 1 1 1 1 0 0 1 1 1 1 0 1 1 1 1 1 1 0 1 1 0 1 1 1 6-bit binary address space 111111 is not used

  12. Prefix properties • Disjoint prefixes: • Two prefixes are said to be disjoint if they do not share any address. • Prefix enclosure: • A = bn-1…bj…bi* and B = bn-1…bj* and j > i. • Prefix A is enclosed by B (B  A) since the IP address space covered by A is a subset of that covered by B, where  is the enclosure operator. • A special case of overlapping. • Prefix comparison • The inequality 0 < * < 1 is used to compare two prefixes in the ternary representation of prefixes.

  13. 5 1 3 2 1 1 2 2 1 1 3 2 1 1 1 1 1 1 3 2 1 2 1 1 1 3 2 1 2 4 4 Prefix properties • The most specific prefixes (MSP): • The prefixes that do not cover any others. • Disjoint, so can be put in an array for binary search • Grouping prefixes in layers based on MSP. • Six layers at most for IPv4 tables

  14. Prefix properties

  15. Prefix properties Number Prefix length

  16. Prefix Forwarding table example • P1 is disjoint from the other three prefixes. • P2  P3  P4 • Longest prefix match(LPM), not exact match • enclosure makes (1) sorting prefixes and (2) binary searching prefixes difficult

  17. Example Forwarding Table • Longest prefix match(LPM), not exact match • Prefix enclosure makes (1) sorting prefixes and (2) binary searching prefixes difficult. • So, trie based schemes emerge naturally

  18. Add P5=1110* 0 P5 I Binary Trie (Radix Trie) Trie node Lookup 10111 A next-hop-ptr (if prefix) 1 B right-ptr left-ptr 1 C D 0 P2 1 1 F E P1 0 G P3 1 H P4

  19. Binary prefix search • Definition 1 (Prefix comparison): The inequality 0 < * < 1 is used to compare two prefixes in the ternary format. 成功大學資訊工程系 CIAL 實驗室

  20. Binary prefix search • Directly performing a binary search on the list of sorted prefixes may encounter a failure: Dst = 01011000 2 3 1 4 Correct match Failed match 成功大學資訊工程系 CIAL 實驗室

  21. Binary prefix search • Enclosure relationship between prefixes results in the search failure • Generate some auxiliary prefixes that inherit the routing information of the original LPM (e.g., F) and put them where the binary search operations can find them. ex. auxiliary prefix 01011000. • Therefore, it is feasible to split prefix F into two parts such that both sides of prefix O are covered. 成功大學資訊工程系 CIAL 實驗室

  22. Binary prefix search • The full tree expansion. The full tree expansion splits the enclosure prefixes into many longer prefixes (leaf pushing). • Auxiliary prefix merges Many auxiliary prefixes may inherit the same routing information of a common enclosure prefix. These prefixes can be merged into one. The merge operation is defined as follows. • Prefix merge: The prefix obtained by merging a set of consecutive prefixes is the longest common ancestor (LCA) of these consecutive prefixes in the binary trie. 成功大學資訊工程系 CIAL 實驗室

  23. Binary prefix search • The full tree expansion F3=01011000 成功大學資訊工程系 CIAL 實驗室

  24. Binary prefix search • The full tree after the merge operations F3=01011000 成功大學資訊工程系 CIAL 實驗室

  25. Binomial spanning tree 1111 1110 1100 2 1 0 3 0000 1000 0000 3 1000 2 1100 1 1110 0 1111 A 4-cube and its corresponding binomial spanning tree.

  26. Perfect code: Hamming code (7, 4) • 7-cube example: 0000000 1000000 0100000 0010000 0001000 0000100 0000010 0000001 = 7-cube 24(16) one-level binomial spanning trees

  27. 1 0 0 0 1 1 0 1 1 0 1 1 0 0 H7 = G7 = 0 1 0 0 1 0 1 1 0 1 1 0 1 0 0 0 1 0 0 1 1 0 1 1 1 0 0 1 0 0 0 1 1 1 1 Perfect code: Hamming code (7, 4) (a) Parity-check and generator matrices of Hamming code (7, 4). Syndrome ErrorPattern Inner product Transpose 000 0000-000 001 0000-001 010 0000-010 011 0010-000 100 0000-100 101 0100-000 110 1000-000 111 0001-000 r = received code Syndromes = (s2 s1 s0) = r．H7T Corrected code = r+ ErrorPattern[s] (c) Decoding table

  28. Perfect code: Hamming code (7, 4) uCodeword 0000 0000-000 0001 0001-111 0010 0010-011 0011 0011-100 0100 0100-101 0101 0101-010 0110 0110-110 0111 0111-001 1000 1000-110 1001 1001-001 1010 1010-101 1011 1011-010 1100 1100-011 1101 1101-100 1110 1110-000 1111 1111-111 Generate 16 Codewords u．G7 16 codewords

  29. Perfect code: Golay code (23, 12) • 212 3-level binomial spanning trees • C(23,0)+C(23, 1)+C(23,2)+C(23,3) = 1 + 23 + 23*22/2 +3*22*21/(3*2) = 24 + 23*11 + 23*11*7 = 24 + 253*8 = 24 + 2024 = 2048 = 211

  30. Ranges • Why ranges? • Prefixes can also be represented by ranges. • The source/destination port fields of rule tables for packet classification are ranges. • Prefixes are special cases of ranges. • Prefix bn-1…bn-l+1* of length l is the range of addresses from bn-1…bn-l+10…0 to bn-1…bn-l+11…0, denoted as [bn-1…bn-l+10…0, bn-1…bn-l+11…0]. • Overlapping: • Two ranges are overlapping if they are not disjoint. • Partially overlapping: • Two ranges are partially overlapping if they are neither disjoint nor enclosing.

  31. Elementary Intervals for Ranges • Definition: Let the set of k elementary intervals constructed from a set R of ranges in the address space of 0 … N – 1 be X = {Xi | Xi = [ei, fi], for i = 1 to k}. • X must satisfy the following: • e1 = 0 and fk = N – 1, • fi = ei+1 – 1 for i = 1 to k – 1, • all addresses in Xi are covered by the same subset of R (called the range matching set of Xi) denoted by EIi, and • EIiEIi+1, for i = 1 to k – 1.

  32. Elementary Intervals for Ranges ID Prefix Range Minus-1 Traditional start finish start finish P1 000000/2 [0, 15] - 15 0 15 P2 010000/2 [16, 31] 15 31 16 31 P3 000100/4 [4, 7] 3 7 4 7 P4 100000/1 [32, 63] 31 - 32 63 P5 010110/5 [22, 23] 21 23 22 23 P6 110000/2 [48, 63] 47 - 48 63 P7 110000/4 [48, 51] 47 51 48 51 P8 110111/6 [55, 55] 54 55 55 55 P9 100000/3 [32, 39] 31 39 32 39

  33. Elementary Intervals for Ranges • Graphical view EI1 {P1} X1 [0, 3] EI2 {P1,P3} X2 [4, 7] EI3 {P1} X3 [8, 15] EI4 {P2} X4 [16, 21] EI5 {P2,P5} X5 [22, 23] EI6 {P2} X6 [24, 31] EI7 {P4,P9} X7 [32, 39] EI8 {P4} X8 [40, 47] EI9 {P4,P6,P7} X9 [48, 51] EI10 {P4,P6} X10 [52, 54] EI11 {P4,P6,P8} X11 [55, 55] EI12 {P4,P6} X12 [56, 63]

  34. Segment Tree w 23 y z 7 47 P1 P4P6 u v g q 15 3 54 31 15 P1 P3 P2 X3 [8,15] X1 [0,3] X2 [4,7] X6 [24,31] h s r P2 P4 t 21 39 51 55 leaf node P5 P9 P7 P8 X4 [16,21] X5 [22,23] X7 [32,39] X8 [40,47] X9 [48,51] X10 [52,54] X11 [55,55] X12 [56,63]

  35. Interval Tree • Each node in an interval tree is associated with a key which must be covered by at least one range. • Depending on whether a node can store 1 or 1+ range, • fat interval tree • each node is allowed to store more than one range. • The number of nodes in the interval tree is O(N). • To insert a range R = [e, f], if R covers root’s key, R is stored in the root. Otherwise, R is inserted in the left (right) subtree of the root when f is smaller (e is larger) than the key of the root. • When R does not cover the key of any node which is traversed, a new node with the key selected from addresses e to f is created and inserted as the left or right child of the node which was last visited. • O(logN + k) time, k is # of prefixes that match the given address. • Prefix insertion and deletion are very expensive because ranges in some nodes may need relocations after tree rotations.

  36. Interval Tree • thin interval tree: • each node of the interval tree stores exactly one range. • Since ranges may overlap, two comparison rules are used to compare if a range is smaller or larger than another range. For two ranges R1 = [e1, f1] and R2 = [e2, f2], • R1 < R2 if e1 < e2. If tie, the second rule applies. • R1 < R2 if R2 is a subrange of R1 (i.e. e1 = e2 and f2 < f1). • Also, a node stores a max value, Max(the finish endpoints of all ranges) stored in the subtree rooted at that node. • In contrast with the fat interval tree, prefix insertion and deletion take O(logN) time. However, O(min{N, klogN}) time is needed to find the longest matching prefix as well as the highest-priority matching prefix, where k is the number of matched prefixes for a given address.

  37. Hash Table • Narrowing down the search space. • Index = Hash_function(key)%m, where key may be the first k bits of IP addresses and m is the size of the hash table. • Perfect hash: no collision • Minimal perfect hash: A perfect hash, where the size of its hash table is k for k different hashing keys.

  38. Hash Table • Difficulties: prefixes and ranges can not be used as the keys of the hash functions directly. Array of m elements H(k1)%m k2 k1 H(k2)%m collision

  39. Hash Table: 8-bit Segmentation table • A 8-bit segmentation table is usually used for IPv4 forwarding tables because there is no prefix of length shorter than 8. Array of 256 elements 0 Prefix: 0.x.y.z H(prefix)%256 (MSB 8 bits of prefix) 1 Prefixes with the same first 8 MSB bits Maybe empty set 255

  40. Hash Table: 16-bit Segmentation table • Prefixes of length <= 16 must be stored properly. • For example, duplicate 0.0.b.c/15 into buckets 0 and 1 or store the port of 0.0.b.c/15 into elements 0 and 1. • Put them into another set (good for update but need to search two sets in the worst case). Array of 216 elements 0 Prefix: 0.0.y.z H(prefix)%216 (MSB 16 bits of prefix) 1 Prefixes with the same first 16 MSB bits Maybe empty set 216-1 Prefixes of length  16

  41. Hash Table: Compression • Since there are many empty elements in the segmentation table, we can use bitmap to compress the segmentation table. 216-Bitmap containing M 1’s Array of M elements 0 Prefix: 0.0.y.z 1 1 0 0 . . . 0 1 1 0 0 1 1 Prefix: 0.1.y.z Prefixes with the same first 16 MSB bits Must be non-empty M-1

  42. Bloom filter • H1(key) = P1 • H2(key) = P2 • H3(key) = P3 • H4(key) = P4 • … • Hk(key) = Pk • Hi() is a hash function, e.g. MD5 Bit vector of m bits 1 1 m bits 1 1

  43. Bloom filter • After inserting n keys (kn bits), the probability that a particular bit is still 0 is (1-1/m)kn • So, the probability of a false positive is • p for the right-hand side is minimized when k = ln2m/n • m/n = 6, k = 4: p = 0.0561 • m/n = 8, k = 6: p = 0.0215 • m/n=12, k = 8: p =0.00314 • m/n=16, k=11: p =0.000458

  44. Bloom filter • Update: • Update whole SC • Threshold: when the digests differ beyond a threshold, say, 5% or 10%, • Regular time intervals: every say 5 mins,

  45. Counting Bloom filter • Deletion operation for local digest: • For each bit in the m-bit vector, use an l-bit counter to record the number of times that a particular bit is turned on by different URLs • l = 4 by experience • If deletion is not supported, cache summary must be rebuilt from scratch on a periodic basis to erase stale bits and prevent bit pollution

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