Lower Envelopes (Cont.)

1. Lower Envelopes (Cont.) Yuval Suede

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

3. Reminder • 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

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

5. Towards Tight Upper Bound • Let W = a1a2 .. al be a sequence • A non-repetitive chain in W is contiguous subsequence U = aiai+1 .. ai+k consisting of k distinct symbols. • A sequence W is m-decomposable if it can be partitioned to at most mnon-repetitive chains.

6. Towards Tight Upper Bound • 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

7. Towards Tight Upper Bound • Proof: • Let w be a sequence. We define a linear ordering on 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

8. Towards Tight Upper Bound • 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.

9. Towards Tight Upper Bound • Proof (Cont.) • Let Uj and Uj+1be two consecutive chains: U1 .. UjUj+1 .. • Let a be the last symbol of Uj and and(i) its indexand let b be the first symbol of Uj+1 and (i+1) its index :a b U1 .. UjUj+1 ..

10. Towards Tight Upper Bound • Claim: • The i-th position is the last of aor 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 Uj). • So we get the forbidden sequence ababa !!

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

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

13. Towards Tight Upper Bound • Proposition (7.4.2) : Let m,n ≥ 1 and p ≤ m be integers, and let m = m1 + m2 + .. mpbe a partition of m into p addends, then there is partition n = n1 + n2 + .. + np + n* such that:

14. Towards Tight Upper Bound • Proof: • Let w = DS(3,n) attaining • Let u1u2 .. umbe a partition of w into non-repetitive chains where : w1 = u1u2 .. Um1 w2 = um1+1um1+2 .. Um2 … wp

15. Towards Tight Upper Bound • We divide the symbols of w into 2 classes: • A symbol a is local if it occurs in at most one of the parts wk • 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 • Letnk be the number of local symbols in wk

16. Local Symbols • By deleting all non-local symbols from wk we get mk-decomposable sequence over nk symbols (no ababa) • This can contains consecutive repetitions, but at most mk-1 (only at the boundaries of uj) • We remain with DS sequence with length at most (the contribution of local-symbols):

17. Non-local Symbols • A non-local symbol is middle symbol in a part of WK if it appears before and after Wk • Otherwise it is non-middlesymbol in Wk

18. Non local -Contribution of middle • For each Wk: • Delete all local symbols. • Deleteall non-middlesymbols. • 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. Non local -Contribution of middle • Each sequence Wk cannot contain b .. a .. b there is a before and a after • Remaining sequence of Wk is non-repetitive chain. • Total contribution of middle symbols in W is at most m +

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

21. Non local -Contribution of non-middle • The remaining starting symbols contain no abab because there is a following Wk • 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

22. Towards Tight Upper Bound • Summing all together:

23. Towards Tight Upper Bound • 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

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

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

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