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# Lower Envelopes (Cont.) - PowerPoint PPT Presentation

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.

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### Lower Envelopes (Cont.)

Yuval Suede

• 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.

• 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

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

• 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.

• 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

• 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

• 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.

• 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 ..

• 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 !!

• Proof (Cont.)

• We have at most 2n Uj chains, because each sybol is at most once first, and at most once last.

• Proof (Cont.)

• We have at most 2n Uj chains, because each sybol is at most once first, and at most once last.

• 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:

• 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

• 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

• 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):

• 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

• 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.

• 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

• Summing all together:

• 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

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

• We choose and

• Estimate using the previous bound.

• This gives:

• If then

• We chose so

• Recall that

• And since

• We get that

Tight tight Upper Bound

• It is possible to show that :

• But not today …