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Codes over Trees Lev Yohananov and Eitan Yaakobi Technion Israel - PowerPoint PPT Presentation

Codes over Trees Lev Yohananov and Eitan Yaakobi Technion Israel Institute of Technology 2020 IEEE International Symposium on Information Theory Motivation Trees and their properties are very beneficial in numerous applications. In


  1. Codes over Trees Lev Yohananov and Eitan Yaakobi Technion – Israel Institute of Technology 2020 IEEE International Symposium on Information Theory

  2. Motivation • Trees and their properties are very beneficial in numerous applications. • In biology. • In chemistry. • In programming languages. • In cyber applications. • etc. • In coding theory , a novel family of codes is presented. 1

  3. Trees • A finite undirected tree over 𝑜 nodes is a connected undirected graph with 𝑜 − 1 edges. • 𝑈 𝑜 : the set of all trees over 𝑜 nodes. 9 1 2 3 4 5 6 7 8 • By Cayley ’ s formula 1 it holds that |𝑈 𝑜 | = 𝑜 𝑜−2 . 1 M. Aigner and G. M. Ziegler, Proofs from THE BOOK, pp. 141 – 146,Springer-Verlag, New York, 1998. 2

  4. Codes (over Trees) • 𝐷 𝑈 : a code (over trees) denoted by 𝑜, 𝑁 𝑈 , such that • 𝑜: the number of nodes in a tree. • 𝑁: the size of 𝐷 𝑈 . • Example 𝑜 = 5, 𝑁 = 8 4 4 1 4 3 0 1 3 1 0 3 2 0 2 3 0 2 2 4 1 4 4 4 4 3 1 0 3 1 0 3 2 1 2 3 1 2 2 0 0 3

  5. Edge Erasure • An edge erasure is a removal of an edge from a tree. 4

  6. Tree Distance • Given two trees 𝑈 1 = (𝑊 𝑜 , 𝐹 1 ) and 𝑈 2 = 𝑊 𝑜 , 𝐹 2 . • 𝑒 𝑈 𝑈 1 , 𝑈 2 : the tree distance (or distance ) between 𝑈 1 and 𝑈 2 is 𝑒 𝑈 𝑈 1 , 𝑈 2 = 𝑜 − 1 − |𝐹 1 ∩ 𝐹 2 |. 9 9 1 2 3 4 5 1 2 3 4 5 6 7 6 7 8 8 𝑒 𝑈 𝑈 1 , 𝑈 2 = 8 − 7 = 1. • This distance is a metric. 5

  7. Codes with Minimal Distance 𝑒 • 𝑜, 𝑁, 𝑒 𝑈 : a code over trees of minimal distance 𝒆. • 𝑠: the redundancy of the code 𝐷 𝑈 . • 𝑠 = 𝑜 − 2 log 𝑜 − log(𝑁) . • 𝐵 𝑜, 𝑒 : the largest size of a code of distance 𝑒 . • 𝑠 𝑜, 𝑒 : the minimal redundancy of a code of distance 𝑒 . • Theorem: A 𝑜, 𝑁 𝑈 code over trees 𝐷 𝑈 is of tree distance 𝒆 if and only if it can correct any 𝒆 − 𝟐 edges . 6

  8. Forests • An undirected graph that consists of only disjoint union of trees is called a forest . 9 1 2 3 4 5 6 7 8 • ℱ 𝑜, 𝑢 : the set of all forests over 𝑜 nodes with exactly 𝒖 trees . • Note that ℱ(𝑜, 1) = 𝑈(𝑜) . 7

  9. Number of Forest with Exactly 𝑢 Trees • The value of |ℱ 𝑜, 𝑢 | was shown to be 𝑗 𝑢 𝑢 − 1 (𝑢 + 𝑗) 𝑜 − 𝑢 ! 𝐺 𝑜, 𝑢 = 𝑜 𝑢 𝑜 𝑜−𝑢−1 ෍ 𝑜 𝑗 𝑜 − 𝑢 − 𝑗 ! . 𝑗 2 𝑗=0 J. Moon 2 1970. • Another representation of it 𝑗 𝑢 𝑢 − 1 𝑢 + 𝑗 ! 𝑜 − 1 𝐺 𝑜, 𝑢 = 𝑜 𝑜−𝑢 ෍ . 𝑗 𝑜 𝑗 𝑢! 𝑢 − 1 + 𝑗 2 𝑗=0 B. Bollobas 3 1979. 2 J. W. Moon, Counting labeled trees, 1970. 3 B. Bollobas, Graph Theory: An Introductory Course, Springer-Verlag, New York, 1979. 8

  10. Forest Ball of a Tree • 𝒬 𝑈 𝑜, 𝑢 : the forest ball of a tree 𝑈 of radius 𝑢. • Note that 𝒬 𝑈 𝑜, 𝑢 ⊆ ℱ(𝑜, 𝑢 + 1) . 4 4 1 0 2 1 0 2 4 3 3 𝑈 = 𝒬 𝑈 𝑜 = 5, 𝑢 = 1 = Regular 1 0 2 4 4 3 1 0 2 1 0 2 3 3 Note that |𝒬 𝑈 𝑜, 𝑢 | = 𝑜 − 1 𝑢 9

  11. Sphere Packing Bound • Theorem: For all 𝑜 ≥ 1 and 1 ≤ 𝑒 ≤ 𝑜 , it holds that 𝐵 𝑜, 𝑒 ≤ 𝐺(𝑜, 𝑒)/ 𝑜 − 1 𝑒 − 1 . 𝐺 𝑈 2 𝑈 1 𝑒 − 1 10

  12. Corollary • It was also proved 2 that for any fixed 𝑒 , 𝐺 𝑜, 𝑒 1 lim = 2 𝑒−1 𝑒 − 1 ! . 𝑜 𝑜−2 𝑜→∞ • Thus, 𝐵 𝑜, 𝑒 ≤ 𝐺 𝑜, 𝑒 = 𝑃 𝑜 𝑜−1−𝑒 . 𝑜 − 1 𝑒 − 1 2 J. W. Moon, Counting labeled trees, 1970. 11

  13. Results from Sphere Packing Bound 𝑜 • Correcting 𝒐 − 𝟑 erasures : 𝐵 𝑜, 𝑜 − 1 ≤ 2 . • Correcting 𝒐 − 𝟒 erasures : 𝐵 𝑜, 𝑜 − 2 = 𝑃 𝑜 2 . • Correcting 𝒐 − 𝟓 erasures : 𝐵 𝑜, 𝑜 − 3 = 𝑃 𝑜 3 . 12

  14. The Results of this Work 𝐵 𝑜, 𝑜 − 1 ≤ 𝑜 • Correcting 𝒐 − 𝟑 erasures : 𝐵 𝑜, 𝑜 − 1 = 𝑜/2 . 2 . 𝐵 𝑜, 𝑜 − 2 = 𝑃 𝑜 2 . • Correcting 𝒐 − 𝟒 erasures : 𝐵 𝑜, 𝑜 − 2 = 𝑜. • Correcting 𝒐 − 𝟓 erasures : 𝐵 𝑜, 𝑜 − 3 = 𝑃 𝑜 2 . 𝐵 𝑜, 𝑜 − 3 = 𝑃 𝑜 3 . • For fixed 𝑒 and 𝑜 ≥ 2𝑒 , Ω 𝑜 𝑜−2𝑒 ≤ 𝐵 𝑜, 𝑒 ≤ 𝑃 𝑜 𝑜−1−𝑒 . 13

  15. Lower Bound on 𝐵(𝑜, 𝑜 − 1) • A line tree 𝑈 : 1 0 2 3 4 • Our code will be constructed from 𝑜/2 line trees as follows: 3 2 2 4 1 3 1 5 0 4 0 6 7 5 7 6 • Thus, this code is a set of 𝑜/2 disjoint Hamiltonian paths, see Lucas 4 . 𝐵 𝑜, 𝑜 − 1 = 𝑜/2 4 E. Lucas, “ Les rondes enfantines, ” Recreations mathematiques, vol. 2, Paris, 1894. 14

  16. Lower Bound on 𝐵(𝑜, 𝑜 − 2) 4 • A star tree 𝑈 : 1 0 2 • Our code will be constructed from 𝑜 star trees as follows: 3 4 4 4 4 0 1 0 2 0 1 2 1 2 0 1 3 2 1 4 2 3 3 3 0 3 • Every two trees have exactly one edge in common. 𝐵 𝑜, 𝑜 − 2 = 𝑃 𝑜 2 𝐵 𝑜, 𝑜 − 2 ≥ 𝑜 15

  17. Upper Bound on 𝐵(𝑜, 𝑜 − 2) egree = 𝒐 − 𝟐 deg 𝐵 𝑜, 𝑜 − 2 = 𝑃 𝑜 2 𝑓 1 𝑈 𝐵 𝑜, 𝑜 − 2 ≤ 𝑜 1 𝑓 2 𝑈 2 • Let 𝐻 be a bipartite graph as in example: ⋮ ⋮ 𝑓 𝑜 𝑈 𝑁 2 • Any code of tree distance 𝑜 − 2 yields a bipartite 𝐻 of girth of at least 6 . • Using Reiman’s inequality 5 1958, it is shown that every such bipartite 𝐻 holds 𝑁 ≤ 𝑜. 5 I. Reiman, “Uber ein Problem von K. Zarankiewicz,” Acta mathematica hungarica, vol. 9, issue 3 – 4, pp. 269 – 273, 16 Hungary, Budapest, Sep. 1958.

  18. Upper Bound on 𝐵(𝑜, 𝑜 − 3) 𝐵 𝑜, 𝑜 − 3 = 𝑃(𝑜 2 ) 𝐵 𝑜, 𝑜 − 3 = 𝑃 𝑜 3 • Also done using Reiman’s inequality. (short in time) = Ω(𝑜 2 ). • In the longest version 6 : 𝐵 𝑜, 3𝑜/4 • 𝑜 is a prime number. 6 L. Yohananov and E. Yaakobi, “ Codes over trees, ” arXiv:2001.01791,Jan. 2020. 17

  19. Lower Bound on General 𝐵(𝑜, 𝑒) • Theorem: for fixed 𝑒 and 𝑜 ≥ 2𝑒 it holds 𝐵 𝑜, 𝑒 = 𝛻 𝑜 𝑜−2𝑒 . • Construction: Let (𝑓 1 , 𝑓 2 , … , 𝑓 𝑜 ) be some order of all the edges of 2 the complete graph over 𝒐 nodes . • Each tree 𝑈 will be represented as a characteristic vector of length 𝑜 2 and weight 𝒐 − 𝟐 as in the example: 1 0 2 1 1 1 0 0 0 {0,1} {0,2} {0,3} {1,2} {1,3} {2,3} 3 18

  20. The Proof • A linear binary code of length 𝑂 = 𝑜 2 and Hamming distance 𝐸 = 2𝑒 − 1 can correct at most 𝑒 − 1 substitution. • Corresponding to 𝑒 − 1 edge erasures . • Applying BCH codes , we pay the redundancy of 𝑠 = 𝑒 − 1 log 𝑜 2 + 𝑃 1 = 2 𝑒 − 1 log 𝑜 + 𝑃 1 . • The 2 𝑠 cosets of such code are also ( 𝑜 2 , 𝐿, 2𝑒 − 1) codes. • Thus, by the pigeonhole principle there is a code of cardinality at least 𝑜 𝑜−2 2 2 𝑒−1 log 𝑜 = Ω 𝑜 𝑜−2𝑒 . 19

  21. Tree Balls of Trees 20

  22. Tree Ball of Trees • 𝐶 𝑈 𝑜, 𝑢 : tree ball of trees: 𝒬 𝑈 𝑜, 𝑢 4 4 ⋯ 1 0 2 1 0 2 4 3 3 Not Regular 1 0 2 Regular 4 4 3 ⋯ 1 0 2 1 0 2 |𝐶 𝑈 𝑜, 𝑢 | =? |𝒬 𝑈 𝑜, 𝑢 | = 𝑜 − 1 3 3 𝑢 ⋯ ⋯ 21

  23. Radius One Arbitrary 𝑈 𝑈 𝑈 𝑈 Θ(𝑜 2 ) Average ball size: Θ(𝑜 2.5 ) Θ(𝑜 3 ) Explicit formulas. 22

  24. Radius 𝑢 (fixed) Arbitrary 𝑈 𝑈 𝑈 𝑈 Θ(𝑜 2𝑢 ) Average ball size: Θ(𝑜 2.5𝑢 ) Θ(𝑜 3𝑢 ) Recursive formulas. 23

  25. The Results of this Work • Correcting 𝒐 − 𝟑 erasures : 𝐵 𝑜, 𝑜 − 1 = 𝑜/2 . • Correcting 𝒐 − 𝟒 erasures : 𝐵 𝑜, 𝑜 − 2 = 𝑜. • Correcting 𝒐 − 𝟓 erasures : 𝐵 𝑜, 𝑜 − 3 = 𝑃 𝑜 2 . • For fixed 𝑒 and 𝑜 ≥ 2𝑒 , Ω 𝑜 𝑜−2𝑒 ≤ 𝐵 𝑜, 𝑒 ≤ 𝑃 𝑜 𝑜−1−𝑒 . • Studying tree balls of trees. 24

  26. Conclusions and Future Work • Improve the lower and upper bounds on 𝐵(𝑜, 𝑒) . • Study codes over trees under different metrics such as the tree edit distance. • Study the problem of reconstructing trees based upon several forests in the forest ball of trees. Thank You! 25

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