A d-step Approach for Distinct Squares in Strings

Transcription

1 A -step Approach for Distinct Squares in Strings Mei Jiang Joint work with Antoine Deza an Frantisek Franek Department of Computing an Software McMaster University LSD & LAW, Feb 2011

2 Outline 1 Introuction 2 (, n ) Table 3 Conjecture Reformulations 4 Relatively Short Square-Maximal Strings Structure 5 Conclusions

3 Outline 1 Introuction 2 (, n ) Table 3 Conjecture Reformulations 4 Relatively Short Square-Maximal Strings Structure 5 Conclusions

4 Backgroun In 1998 Fraenkel an Simpson showe the number of istinct squares in a string of length n is boune from above by 2n an gave a lower boun asymptomatically approaching n from below; for primitively roote squares, they showe that an upper boun of n o(n) is vali for infinitely many values of n. In 2005 Ilie provie a simpler proof of Fraenkel an Simpson s main lemma an slightly improve the upper boun to 2n Θ(log n) in It is believe, that the number of istinct squares is boune by the length of the string.

5 -step Approach We investigate the problem of istinct squares in relationship to the alphabet of the string. We construct a table whose rows are inexe by an columns are inexe by n with entries of σ (n). We conjecture that the upper boun for the maximum number of primitively roote istinct squares is n. -step approach was inspire by the techniques use for investigating the Hirsch boun for the maximum possible iameter over all -imensional polytopes with n facets.

6 Basic Notation A square is a repetition with power of 2, istinct squares means only the types of the squares are counte, primitively roote istinct squares means the generator itself is not a repetition. A run, a maximal fractional primitively roote repetition, is forme by a maximal repetition followe by a tail. s(x) enotes the number of primitively roote istinct squares in a string x. σ (n) enotes the maximum number of primitively roote istinct squares over all strings of length n containing exactly istinct symbols. A singleton refers to a symbol in a string that occurs exactly once, a pair occurs exactly twice, a triple occurs exactly three times, an in general an k-tuple (k times).

7 Outline 1 Introuction 2 (, n ) Table 3 Conjecture Reformulations 4 Relatively Short Square-Maximal Strings Structure 5 Conclusions

8 (,n-) Table Basic Properties n (, n-) Table: σ (n) with 1 10 an 1 n 10 For all n 2: 1 σ (n) σ (n+1) 2 σ (n) σ +1 (n+1) 3 σ (n) < σ +1 (n+2) 4 σ (n) = σ +1 (n+1) for n 2 5 σ (n) n for n 2 6 σ (2) σ 1 (2 1) 1

9 Outline 1 Introuction 2 (, n ) Table 3 Conjecture Reformulations 4 Relatively Short Square-Maximal Strings Structure 5 Conclusions

10 Theorem 1 Theorem 1 For all n 2, σ (n) n σ (2) = for all 2 n - Proof. n < 2, constant uner the iagonal. n > 2, smaller or equal than the iagonal value.

11 Theorem 2 Theorem 2 For all n 2, σ (n) n σ (2+1) σ (2) 1 for all 2-1 n - -1 = 1 Proof. is the least s.t. σ (2) >. Remove the singleton, σ 1 (2 1) = σ (2). σ (2) σ 1 (2 2) 1, an σ 1 (2 2) = 1. Thus σ (2).

12 Theorem 3 Theorem 3 For all 2, if σ (2+1), then 1 σ (n) n for all n 2 2 σ (n) n 1 for all n > n - +1 Proof. σ (2) = σ (2+1) =. n > 2, smaller than the iagonal value.

13 Theorem 4 Theorem 4 For all 2, if σ (2) = σ (2+1), then 1 σ (n) n for all n 2 2 σ (n) n 1 for all n > n - -1 = 1 Proof. To show σ (2) = σ (2+1) =. is the least s.t. σ (2) >. σ (2) σ 1 (2 1) 1, an σ 1 (2 1) = 1. Thus σ (2).

14 Outline 1 Introuction 2 (, n ) Table 3 Conjecture Reformulations 4 Relatively Short Square-Maximal Strings Structure 5 Conclusions

15 Relatively Short Square-Maximal Strings Structure We investigate the structure of square-maximal strings on the main iagonal. If σ (2) = then at least one of the square maximal string is in the form of aabbcceeff... If σ (2) > then the square maximal string is a counterexample. We investigate its structure an raw conclusions for counterexamples with n 4. n -

16 Pairs Lemma 1 Let is the least s.t. for some x, s(x) = σ (2) >. Then x oes not contain a pair. Proof. n - The pair: x[i 0 ] = x[i 1 ] = C. Occurs in only one square. Replace the first C with a new symbol Ĉ. 1 σ +1 (2) σ (2) 1. Occurs in a non-trivial run uvcwuvcwu. Remove wuv between C s. k σ (2 k) σ (2) k, where k = w + u + v.

17 Triples Lemma 2 Let is the least s.t. for some x, s(x) = σ (2) >. Then x can only contain a triple x[i 0 ] = x[i 1 ] = x[i 2 ] = C that satisfies: 1 x[i 0 ] an x[i 1 ] occur in a run r 1 = u 1 v 1 Cw 1 u 1 v 1 Cw 1 u 1, where u 1 1, 2 x[i 1 ] an x[i 2 ] occur in a run r 2 = u 2 v 2 Cw 2 u 2 v 2 Cw 2 u 2, where u 2 1, an where i 1 i 0 i 2 i 1, 3 either u 1 v 1 is a proper suffix of u 2 v 2, or w 2 u 2 is a proper prefix of w 1 u 1.

18 Triples (cont.) Proof. r 1 : u 1 v 1 Cw 1 u 1 v 1 Cw 1 u 1 r 2 : u 2 v 2 Cw 2 u 2 v 2 Cw 2 u 2 Show it is impossible to have only two symbols occur in a run. Show it is impossible to have three symbols occur in the same run. Show it is impossible to have both ens are long.

19 Singletons Estimation Lemma 3 Let is the least s.t. for some x, s(x) = σ (2) >. Then x has at least 2 3 singletons. Proof. Let u 1v 1 is a proper suffix of u 2v 2, a = u 1[0]. a occurs at least 6 times in the r 1 an r 2. We assign 5 a s to the triple. It can be shown this assignment is mutually isjoint with others. r 1 : u 1v 1Cw 1u 1v 1Cw 1u 1 r 2 : u 2v 2Cw 2u 2v 2Cw 2u 2 m 0: the number of triples, m 1: the number of other multiply occurring symbols (at least 4 times), m 2: the number of singletons. 2 8m 0+4m 1+m 2 (1) 2m 0+m 1+m 2 (2) Thus, m 2 2 3

20 Theorem 5 Theorem 5 For all n 2, σ (n) n σ (4) 3 for all 2 n - Proof. is the least s.t. σ (2) >. Remove 2 3 singletons. σ (4 ) σ (2) > an 3 =. Thus, σ (4 ) > 3.

21 Theorem 5 (cont.) We construct (,n-3) table, the conjecture upper boun is true if all the main iagonal entries satisfies σ (4) n In general term, (,n-k) table may be constructe an the conjecture is equivalent with σ ((k + 1)) k for all 2 on the main iagonal.

22 Outline 1 Introuction 2 (, n ) Table 3 Conjecture Reformulations 4 Relatively Short Square-Maximal Strings Structure 5 Conclusions

23 Conclusions We exhibit the usefulness of investigating the main iagonal of (,n-) table for tackling the conjecture upper boun. To prove the conjecture by showing that the first counterexample has an impossible structure. i.e. it cannot contain an k-tuple, or if it contains an k-tuple, then it must contain another symbol with a frequency > k. To isprove the conjecture by fining a counterexample on the iagonal. The Hirsch conjecture was recently isprove by Santos by exhibiting a violation on the iagonal with = 20. Let s remark the techniques we use for pushing up the main iagonal can be applicable to the verification of the conjecture upper boun.

24 References A. S. Fraenkel an J. Simpson, How Many Squares Can a String Contain?, Journal of Combinatorial Theory Series A, 82, 1 (1998), L. Ilie, A simple proof that a wor of length n has at most 2n istinct squares, Journal of Combinatorial Theory Series A, 112, 1 (2005) L. Ilie, A note on the number of squares in a wor, Theoretical Computer Science, 380, 3 (2007), F. Santos, A counterexample to the Hirsch conjecture, arxiv: v1 (2010). A. Deza, F. Franek, an M. Jiang, A -step approach for istinct squares in strings, AvOL Technical Report 2011/01, Dept. of Computing an Software, McMaster University, Canaa.

25 T HAN K YOU!