• August 20th 2012, 06:25 AM
baxy77bax
Hi,

well i need some help with the following problem. I have posted fractions of this problem before and got very helpful comments. Now when the Proof is nearly done i need for someone to check it and tell me how bad the proof is. So let me recapitulate. This is a toy problem connected to CS class. What i have is :

Defn_1:

$x_{1} = a$
$x_{p+1} = x_{p} + A[x_{p}]$

where
1. 0<A[a]<=n
2. A[a]-1 <= A[a+1] for 0<=a<=A[l]
3. A[a]-1 = A[a+1] for n-A[l]< a <= n

A[a] is an array value on position a. A[l] is the last array value such that A[l-1] < A[l] and A[l] > A[l+1] , su A[l] is the last local maximum

Defn_2:

$y_{1} \leq A[a]$
$y_{p+1} \leq A[a+\sum^{p}_{n=1} y_{n}]$

Problem:

If y>=1 then $k\leq \sum^{k}_{p=1} y_{p} \leq \sum^{k}_{p=1}A[x_{p}]$.

Proof:

Suppose $y \geq 1$. Suppose $\sum^{k}_{p=1} y_{p} < k$. By plugging in $y_{p+1} = y_{p} = 1$ we get $\sum^{k}_{p=1} y_{p} = k < k$ which is a contradiction. Thus for $y \geq 1$, $\sum^{k}_{p=1} y_{p} \geq k$. Furthermore, by definition 2 we have $y_{1} \leq A[a]$ and $y_{p+1} \leq A[a+\sum^{p}_{n=1} y_{n}]$
thus:

$\sum^{k}_{p=1} y_{p} = y_{1}+y_{2}+ .. y_{k}$
$\sum^{k}_{p=1} y_{p} \leq A[a] + A[a+\sum^{p}_{f=1} y_{f}] + ... A[a+\sum^{k-1}_{f=1} y_{f}]$ by defn 2
$\sum^{k}_{p=1} y_{p} \leq A[x_{1}] + A[x_{2}] + ... A[x_{k}]$ by defn 1
$\sum^{k}_{p=1} y_{p} \leq \sum^{k}_{p=1} A[x_{p}]$

Therefore if y>=1 then $k\leq \sum^{k}_{p=1} y_{p} \leq \sum^{k}_{p=1}A[x_{p}]$.

_________________

So this is how far i came. However i feal that i am missing something since my domain (array length) is limited. Or should i say, i feal that the above would hold if array was infinite in size. Also to depict the problem i am going to give an example:

A is my array

. . 0 1 2 3 4 5 6 7 8 9 0 1 2 3
A: 7 6 5 4 6 5 4 3 2 5 4 3 2 1

the claim that i should prove is, if i make k jumps where k im my example let say is 1, the the distance if i start in 0 -> A[0]=7 equals 10 since |A[0]| + |A[7]| = 10. But this will not hold if my k =4 because then i am out of range. and this is what is bugging me. all examples that i have seen up until now are working with sets that are infinite in size and i have not seen an example that is limited by a size....

so please help by giving an advice on what should i read or what should i do to make my proof and problem definition correct. And please check what nonsense have i written in the section above. Remember that i am still learning how to prove claims.

Thank you

baxy
• September 11th 2012, 11:33 PM
chiro