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.

( )n

• 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:

Reminder

( )s n

3( ) ( ) 2 ln( ) 3n n n n n

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 m non-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

Towards Tight Upper Bound

( , )m n

3( ) (2 , )n n n

• 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

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 Bound

• Proof (Cont.)– Let Uj and Uj+1 be two consecutive chains:

U1 .. Uj Uj+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 .. 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 .. 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 Bound

a b

• 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 Bound

3( ) (2 , )n n n

• 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 Bound

3( ) (2 , )n n n

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

Towards Tight Upper Bound

1

( , ) 4 4 * ( , *) ( , )p

k kk

m n m n p n m n

• Proof:– Let w = DS(3,n) attaining

– Let u1u2 .. um be a partition of w into non-repetitive chains where :

w1 = u1u2 .. Um1

w2 = um1+1um1+2 .. Um2

…wp

Towards Tight Upper Bound

( , )m n

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

• 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):

1 1

[ 1 ( , )] ( , )p p

k k k k kk k

m m n m m n

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 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 middle

( , *)p n

• 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

1

* *p

kk

n n

*kn

• 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

Non local -Contribution of non-middle

* *

1

( 1 2 1) 2p

k ki

m n m n

*kn

*kn

• Summing all together:

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 Bound

1

( , ) 4 4 * ( , *) ( , )p

k kk

m n m n p n m n

2m

1 2( , ) 4 4 * (2, *) ( , ) ( , )2 2m mm n m n n n n

(2, ) 2n n

24 log 6m m n

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

• We choose and • Estimate using the previous

bound.• This gives:

Towards Tight Upper Bound

3( )m A

2logmpm

2 3log ( 1)kmm m Ap

( , *)p n

* *2

1

( , ) 4 4 4 log 6 ( , )p

k kk

m n m n p p n m n

* * *4 4 4 6 8( 1) 10( )m n m n m n n

8 10m n

• If then • We chose so • Recall that • And since • We get that

Towards Tight Upper Bound

( )km A ( )k m

3( )m A *log m

3( ) (2 1, )n n n

(2 1) ( ) 1n n *

3( ) ( log )n O n n

Tight tight Upper Bound

• It is possible to show that :

• But not today …

3( ) 4 ( ) ( ( ))n n n O n n