An asymptotic lower bound for the maximal-number-of-runs function

1
An asymptotic lower bound for the
maximal-number-of-runs function
F. Franek Q. Yang
Dept. of Computing SoftwareMcMaster
UniversityHamilton, Ontario
PSC06, August 2006
Slide 1
2
We show that
e
A
gt0
E
N0
A
N N0 ?(N) (a-e)N
where a
0.927
and where ?() is the maxrun function
Slide 2
3

1. runs in strings and the maxrun function.
2. Creating strings rich in runs.
3. Obtaining asymptotic lower bounds for maxrun
   from the method described in 2.
4. Conclusion and further research.

Slide 3
4
What is a run? An efficient encoding of
repetitions Main 1989 (start,period,power,t
ail) (4,3,3,2)
1
1
1
1
1
1
1
1
1
1
2
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
a
b
b
a
b
a
a
b
a
a
b
a
a
b
b
a
a
b
a
b
s4 p3 w3 t2
Slide 4
5
Though there can be O(n log n) repetetions in a
string of length n Crochemore 1981, runs in
any string can be listed in a linear time and
space Kolpakov Kucherov 2000
Slide 5
6
The maxrun function ?( ) is defined
as ?(N)maxr(u) u string uN wherer(u)
number of runs in string u
Slide 6
7

• Simple observations about ?( )
• ?(N2) ?(N)1
• ?(N1) ?(N)
• ?(N1) ?(N) for some N
• ?(N1) ?(N)2 for some N

Slide 6
8

• B. Smyth et al 2005, 2006 conjectured that
• ?(N) lt N
• ?(N1) ?(N)2
• ?(N) ?2(N) (cube-free)

?(N) 6.3N Rytter 2006
Slide 7
9
Franek, Simpson, Smyth in 2003 presented a
construction of a sequence xn of binary strings
that are rich in runs
Slide 8
10
xy if ? ? ?
x? º ?y
x?y if ? ?
010010 º 101101 0100101101010010 º 010010
01001010010
010010 if x 0
101101 if x 1
g(x1..n)
g(x1) º º g(xn)
Slide 9
11
r(g(0) º g(0)) 2r(g(0))1r(g(1) º g(1))
2r(g(1))1r(g(0) º g(1)) r(g(0))r(g(1))1r(g(
1) º g(0)) r(g(1))r(g(0))1 (computer search
up to length 20 did not find any better pair of
strings.)
Slide 10
12
Let ?(x) is the number of pairs 00 or 11 in x,
then g(x) 6x-?(x)-2(x-?(x)-1)
4x?(x)2?(g(x)) xr(g(x))
r(x)2x(x-1) r(x)3x-1
Slide 11
13
x0 0xn1 g(xn)xn1 4xnxn-12r(xn1
) r(xn)3xn-1
Slide 12
14
Why this result cannot be considered as giving a
lower bound?
ax
6.3x
x
?(x)
xn
xn1
Slide 13
15
How to assure dense coverage to get a lower
bound? Through multiplication based on xn (see
slide 12). xn1 4xnxn-12 r(xn1)
r(xn)3xn-1
-2-3
Slide 14
16
Thus, if xn?yn, then xn1?yn1 and
a
Slide 16
17
x0
x0
x1
x0
1
2
1
k

lt lt
lt
x1
x1
x2
x1
1
2
1
k

lt lt
lt

xn
xn
xn1
x1
k
1
2
1

lt lt
lt
Slide 17
18
http//www.cas.mcmaster.ca/franek
Slide 18