1. Lower Envelopes (Cont.) Yuval Suede

2. Reminder Lower Envelope is the graph of the pointwise minimum of the (partially defined) functions. Letbe the maximum number of pieces in the lower envelope.

3. Each Davenport-Schinzel sequence of order s over n symbols corresponds to the lower envelope of a suitable set of n curves with at most s intersections between each pair. DS sequence important property: – There is no subsequence of the form Reminder

4. Let be the maximum possible length of Davenport-Schinzel sequence of order s over n symbols. Upper bound: Reminder

5. Towards Tight Upper Bound Let W = a 1 a 2.. a l be a sequence A non-repetitive chain in W is contiguous subsequence U = a i a i+1.. a i+k consisting of k distinct symbols. A sequence W is m-decomposable if it can be partitioned to at most m non-repetitive chains.

6. Let denote the maximum possible length of m-decomposable DS(3,n). Lemma (7.4.1): Every DS(3,n) is 2n-decomposable and so Towards Tight Upper Bound

7. Proof: – Let w be a sequence. We define a linear orderingon the symbols of w: we set a b if the first occurrence of a in w precedes the first occurrence of b in w. – We partition w into maximal strictly decreasing chains to the ordering – For example: 123242156543 -> 1|2|32|421|5|6543 Towards Tight Upper Bound

8. Proof (Cont.) – Each strictly decreasing chain is non-repetitive. – It is sufficient to show that the number of non- repetitive chains is at most 2n. Towards Tight Upper Bound

9. Proof (Cont.) – Let U j and U j+1 be two consecutive chains: U 1.. U j U j+1.. – Let a be the last symbol of U j and and (i) its index and let b be the first symbol of U j+1 and (i+1) its index : a b U 1.. U j U j+1.. Towards Tight Upper Bound

10. Claim: – The i-th position is the last of a or the first of b – if not, there should be b before a (b.. ab) – And there should be a after the b (b.. ab.. a) – And because of there should be a before the first b (otherwise the (i+1)-th position could be appended to U j ). – So we get the forbidden sequence ababa !! Towards Tight Upper Bound

11. Proof (Cont.) – We have at most 2n U j chains, because each sybol is at most once first, and at most once last. Towards Tight Upper Bound

12. Proof (Cont.) – We have at most 2n U j chains, because each sybol is at most once first, and at most once last. Towards Tight Upper Bound

13. Proposition (7.4.2) : Let m,n ≥ 1 and p ≤ m be integers, and let m = m 1 + m 2 +.. m p be a partition of m into p addends, then there is partition n = n 1 + n 2 +.. + n p + n* such that: Towards Tight Upper Bound

14. Proof: – Let w = DS(3,n) attaining – Let u 1 u 2.. u m be a partition of w into non-repetitive chains where : w 1 = u 1 u 2.. U m1 w 2 = u m1+1 u m1+2.. U m2 … w p Towards Tight Upper Bound

15. We divide the symbols of w into 2 classes: A symbol a is local if it occurs in at most one of the parts w k A symbol a is non-local if it appears in at least two distinct parts. Let n* be the number of distinct non-local symbols Let n k be the number of local symbols in w k

16. By deleting all non-local symbols from w k we get m k -decomposable sequence over n k symbols (no ababa) This can contains consecutive repetitions, but at most m k -1 (only at the boundaries of u j ) We remain with DS sequence with length at most (the contribution of local-symbols): Local Symbols

17. Non-local Symbols A non-local symbol is middle symbol in a part of W K if it appears before and after W k Otherwise it is non-middle symbol in W k

18. Non local -Contribution of middle For each W k : – Delete all local symbols. – Delete all non-middle symbols. – Delete all symbols (but one) of each contiguous repetition (we delete at most m middle symbols) – The resulting sequence is DS(3,n*) Claim: The resulting sequence is p- decomposable.

19. Each sequence W k cannot contain b.. a.. b there is a before and a after Remaining sequence of W k is non-repetitive chain. Total contribution of middle symbols in W is at most m + Non local -Contribution of middle

20. We divide non-middle symbols of W k to starting and ending symbols. Let be the number of distinct starting symbols in W k. A symbol is starting in at most one part, so we have We remove from W k all but starting symbols and all contiguous repetitions in each W k. Non local -Contribution of non-middle

21. The remaining starting symbols contain no abab because there is a following W k What is left of Wk is DS(2, ) that has length at most 2 -1 Total number of starting symbols in all W is at most Non local -Contribution of non-middle

22. Summing all together: Towards Tight Upper Bound

23. The recurrence can be used to prove better and better bound. 1st try: we assume m is a power of 2. We choose p=2, m1=m2= and we get : Using we estimate the last expression by Towards Tight Upper Bound

24. 2 nd try: we assume (the tower function) for an integer We choose and Estimate using the previous bound. This gives: Towards Tight Upper Bound

25. If then We chose so Recall that And since We get that Towards Tight Upper Bound

26. Tight tight Upper Bound It is possible to show that : But not today …