New successor rules for constructing de Bruijn sequences

Transcription

1 New successor rules for constructing de Bruijn sequences Dennis Wong Northwest Missouri State University Daniel Gabric and Joe Sawada University of Guelph, CAN Aaron Williams Simon s Rock, USA Southeastern / 13

2 Outline 1. What is a de Bruijn sequence? 2. Construction methods Euler cycles in a de Bruijn graph Greedy approach Linear feedback shift registers Successor rules 3. Necklaces and co-necklaces 4. New successor rules Southeastern / 13

3 Definition What is a de Bruijn sequence? A de Bruijn sequence is a circular string of length 2 n where every binary string of length n occurs as a substring (exactly once). Example is a de Bruijn sequence for n = 4 The 16 unique substrings of length 4 are: 0000, 0001, 0010, 0101, 1011, 0110, 1101, 1010, 0100, 1001, 0011, 0111, 1111, 1110, 1100, Southeastern / 13

4 Definition What is a de Bruijn sequence? A de Bruijn sequence is a circular string of length 2 n where every binary string of length n occurs as a substring (exactly once). Example is a de Bruijn sequence for n = 4 The 16 unique substrings of length 4 are: 0000, 0001, 0010, 0101, 1011, 0110, 1101, 1010, 0100, 1001, 0011, 0111, 1111, 1110, 1100, PROBLEM: How to efficiently construct a de Bruijn sequence? Southeastern / 13

5 Construction method #1: Graph model Construct the de Bruijn graph with 2 n 1 vertices and 2 n edges: A de Bruijn sequence is in 1-1 correspondence with an Euler cycle in a de Bruijn graph. Southeastern / 13

6 Construction method #1: Graph model Construct the de Bruijn graph with 2 n 1 vertices and 2 n edges: A de Bruijn sequence is in 1-1 correspondence with an Euler cycle in a de Bruijn graph. By counting Euler cycles, the number of de Bruijn sequences is: 2 2n 1 n Southeastern / 13

7 Construction method #1: Graph model Construct the de Bruijn graph with 2 n 1 vertices and 2 n edges: A de Bruijn sequence is in 1-1 correspondence with an Euler cycle in a de Bruijn graph. By counting Euler cycles, the number of de Bruijn sequences is: 2 2n 1 n this method requires exponential space to store the graph Southeastern / 13

8 Construction method #2: Greedy approach Prefer-larger greedy algorithm (Martin 1934) 1. Start with 0 n (very important!) 2. Repeat: append the largest bit that does not create a duplicate length n substring Southeastern / 13

9 Construction method #2: Greedy approach Prefer-larger greedy algorithm (Martin 1934) 1. Start with 0 n (very important!) 2. Repeat: append the largest bit that does not create a duplicate length n substring Example n = Southeastern / 13

10 Construction method #2: Greedy approach Prefer-larger greedy algorithm (Martin 1934) 1. Start with 0 n (very important!) 2. Repeat: append the largest bit that does not create a duplicate length n substring Example n = Southeastern / 13

11 Construction method #2: Greedy approach Prefer-larger greedy algorithm (Martin 1934) 1. Start with 0 n (very important!) 2. Repeat: append the largest bit that does not create a duplicate length n substring Example n = Southeastern / 13

12 Construction method #2: Greedy approach Prefer-larger greedy algorithm (Martin 1934) 1. Start with 0 n (very important!) 2. Repeat: append the largest bit that does not create a duplicate length n substring Example n = Southeastern / 13

13 Construction method #2: Greedy approach Prefer-larger greedy algorithm (Martin 1934) 1. Start with 0 n (very important!) 2. Repeat: append the largest bit that does not create a duplicate length n substring Example n = Southeastern / 13

14 Construction method #2: Greedy approach Prefer-larger greedy algorithm (Martin 1934) 1. Start with 0 n (very important!) 2. Repeat: append the largest bit that does not create a duplicate length n substring Example n = Southeastern / 13

15 Construction method #2: Greedy approach Prefer-larger greedy algorithm (Martin 1934) 1. Start with 0 n (very important!) 2. Repeat: append the largest bit that does not create a duplicate length n substring Example n = Southeastern / 13

16 Construction method #2: Greedy approach Prefer-larger greedy algorithm (Martin 1934) 1. Start with 0 n (very important!) 2. Repeat: append the largest bit that does not create a duplicate length n substring Example n = Southeastern / 13

17 Construction method #2: Greedy approach Prefer-larger greedy algorithm (Martin 1934) 1. Start with 0 n (very important!) 2. Repeat: append the largest bit that does not create a duplicate length n substring Example n = Southeastern / 13

18 Construction method #2: Greedy approach Prefer-larger greedy algorithm (Martin 1934) 1. Start with 0 n (very important!) 2. Repeat: append the largest bit that does not create a duplicate length n substring Example n = Southeastern / 13

19 Construction method #2: Greedy approach Prefer-larger greedy algorithm (Martin 1934) 1. Start with 0 n (very important!) 2. Repeat: append the largest bit that does not create a duplicate length n substring Example n = Southeastern / 13

20 Construction method #2: Greedy approach Prefer-larger greedy algorithm (Martin 1934) 1. Start with 0 n (very important!) 2. Repeat: append the largest bit that does not create a duplicate length n substring Example n = Other greedy approaches: Prefer-same (Eldert 1958) Prefer-opposite (Alhakim 2010) these methods require exponential space to remember the sequence Southeastern / 13

21 Construction method #3: Linear feedback shift registers Construct and iterate an LFSR 1. Find a primitive polynomial polynomial of order n 2. Construct a linear feedback shift register from the polynomial 3. Iterate through register 2 n 1 times time and space efficient misses the string 0 n requires a different primitive polynomial for each n Southeastern / 13

22 Construction method #4: Successor rules Successor rule constructions A successor rule for a given de Bruijn sequence is a function { b1 if conditions NEXT (b 1 b 2 b n ) = otherwise that returns the bit following the substring b 1 b 2 b n. b 1 Southeastern / 13

23 Construction method #4: Successor rules Successor rule constructions A successor rule for a given de Bruijn sequence is a function { b1 if conditions NEXT (b 1 b 2 b n ) = otherwise that returns the bit following the substring b 1 b 2 b n. b 1 For the n = 4 de Bruijn sequence , its successor rule would define NEXT (1010) = 0. Southeastern / 13

24 Necklaces and co-necklaces Strings from cycling register (rotate left) Strings from Complementing Cycling Register (complement first bit then rotate left) The lexicographically smallest element from each equivalence class is called a necklace and co-necklace respectively. Southeastern / 13

25 Necklace based successor rules Wong Successor (SWW 2016) { b1 if b NEXT (b 1 b 2 b n ) = 2 b 3 b n 1 is a necklace b 1 otherwise. Southeastern / 13

26 Necklace based successor rules Wong Successor (SWW 2016) { b1 if b NEXT (b 1 b 2 b n ) = 2 b 3 b n 1 is a necklace b 1 otherwise. Williams Successor *NEW* { b1 if 0b NEXT (b 1 b 2 b n ) = 2 b 3 b n is a necklace b 1 otherwise. Fredricksen (1972) gave the only other necklace based conditions Southeastern / 13

27 Wong successor rule (necklace spanning tree n = 5) Southeastern / 13

28 Wong successor rule (necklace spanning tree n = 5) Southeastern / 13

29 Wong successor rule (necklace spanning tree n = 5) Southeastern / 13

30 Wong successor rule (necklace spanning tree n = 5) Southeastern / 13

31 Wong successor rule (necklace spanning tree n = 5) Southeastern / 13

32 Wong successor rule (necklace spanning tree n = 5) Southeastern / 13

33 Wong successor rule (necklace spanning tree n = 5) Southeastern / 13

34 Wong successor rule (necklace spanning tree n = 5) Southeastern / 13

35 Wong successor rule (necklace spanning tree n = 5) Southeastern / 13

36 Co-necklace based successor rules Huang (1990) gave the only other co-necklace based conditions (quite complex) Gabric Successor *NEW* Let β = b 1b 2 b n and let γ = b 2b 3 b n0 { b1 if b 2b 3 b n0 0 n is a co-necklace NEXT (b 1b 2 b n) = otherwise. b 1 Southeastern / 13

37 Gabric successor rule (co-necklace spanning tree n = 5) Southeastern / 13

38 Gabric successor rule (co-necklace spanning tree n = 5) Southeastern / 13

39 Summary of results So far we have found 6 new de Bruijn successors where each bit can be computed in O(n)-time O(n)-memory is required Related techniques allow us to construct an exponential number of de Bruijn sequences where each bit can be computed in O(n)-time O(n 2 )-memory is required Southeastern / 13

40 Summary of results So far we have found 6 new de Bruijn successors where each bit can be computed in O(n)-time O(n)-memory is required Related techniques allow us to construct an exponential number of de Bruijn sequences where each bit can be computed in O(n)-time O(n 2 )-memory is required -Gracias- Southeastern / 13