Lower Envelopes Cont Yuval Suede Reminder Lower Envelope

Lower Envelopes (Cont. ) Yuval Suede

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.

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

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

Towards Tight Upper Bound • Let W = a 1 a 2. . 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 m non-repetitive chains.

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 2 n-decomposable and so

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

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

Towards Tight Upper Bound • 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. . 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 !!

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

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

Towards Tight Upper Bound • Proposition (7. 4. 2) : Let m, n ≥ 1 and p ≤ m be integers, and let m = m 1 + m 2 +. . mp be a partition of m into p addends, then there is partition n = n 1 + n 2 +. . + np + n* such that:

Towards Tight Upper Bound • Proof: – Let w = DS(3, n) attaining – Let u 1 u 2. . um be a partition of w into non-repetitive chains where : w 1 = u 1 u 2. . Um 1 w 2 = um 1+1 um 1+2. . Um 2 … wp

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 • Let nk be the number of local symbols in wk

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):

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-middle symbol in Wk

Non local -Contribution of middle • For each Wk: – 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 pdecomposable.

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 +

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.

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

Towards Tight Upper Bound • Summing all together:

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

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

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