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
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