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Lower Envelopes (Cont.). Yuval Suede. 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. . Reminder.

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Lower Envelopes (Cont.)

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Lower envelopes cont

Lower Envelopes (Cont.)

Yuval Suede


Reminder

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.


Reminder1

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


Reminder2

Reminder

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

  • Upper bound:


Towards tight upper bound

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.


Towards tight upper bound1

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


Towards tight upper bound2

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 bound3

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.


Towards tight upper bound4

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 ..


Towards tight upper bound5

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 !!


Towards tight upper bound6

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.


Towards tight upper bound7

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.


Towards tight upper bound8

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:


Towards tight upper bound9

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


Towards tight upper bound10

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


Local symbols

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

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


Non local contribution of middle

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.


Non local contribution of middle1

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

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 middle1

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 bound11

Towards Tight Upper Bound

  • Summing all together:


Towards tight upper bound12

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


Towards tight upper bound13

Towards Tight Upper Bound

  • 2nd try: we assume (the tower function) for an integer

  • We choose and

  • Estimate using the previous bound.

  • This gives:


Towards tight upper bound14

Towards Tight Upper Bound

  • If then

  • We chose so

  • Recall that

  • And since

  • We get that


Tight tight upper bound

Tight tight Upper Bound

  • It is possible to show that :

  • But not today …


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