11240 - Antimonotonicity

[jan_holmes]

Got TLE with DP (Top Down). How did you guys solve this problem without getting TLE ?

[asif_rahman0]

Hints:
There are only two direction. One is '<' & another is '>'.
Then make a table with memo[N][2]. That's all.

[DanS]

Hi, can I have some extra test cases?

[Hackson]

There can be at most 30000 numbers. Surely simple DP does not work, and you need further optimizations.

[sclo]

...

[sunny]

I did greedy.

My solution is O(n) without any extra memory(not even an array to store all numbers).

[WingletE]

Could somebody tell me how to use DP in this problem?

[little joey]

Could somebody tell me how to use DP in this problem?

You can look at the sample solution at the Waterloo site (google for it). It is a quadratic time complexity solution. They missed a crucial observation to make it linear, but it is not very hard to find.

To sunny: I wouldn't call it 'greedy', I think technically it is still DP, but with a 'zero-dimensional DP-array'. Unless you have a different solution, of course.

[sunny]

To sunny: I wouldn't call it 'greedy', I think technically it is still DP, but with a 'zero-dimensional DP-array'. Unless you have a different solution, of course.

Actually, I was little confused to call it greedy too. But as in my solution I always try to keep the bigger(if the sequence is towards to up)/smaller(if towards to down) values as last element , so I considered this as 'greedy'.

[little joey]

Never mind the technicalities, we're talking about the same thing

[StatujaLeha]

Could somebody tell me how to use DP in this problem?

To sunny: I wouldn't call it 'greedy', I think technically it is still DP, but with a 'zero-dimensional DP-array'. Unless you have a different solution, of course.

My solution isn't based on DP. It works in O(n) and also can be modified to avoid storing of sequence in the array(but i use array for convenience).

[Ivan]

I am not going to comment on a linear solution (I believe that no correct greedy
solution exists).

My advice to people that are trying to look at this as a typical DP problem:

1. The n^2 solution is obvious.

Let L be the length of the longest alternating (Mary) sequence ending
at element i of the given (Fred or let call it A) sequence such that the
last element is a "low" element.


Let H be the length of longest alternating (Mary) sequence ending at
element i of the given A sequence such that the last element is a "high" element.

L = max(H[j] + 1) for all j < i such that A[j] > A.
H = max(L[j] + 1) for all j < i such that A[j] < A.

In code terms this looks something this:

for every i from 1 to n
for every j from 1 to i - 1 { do the above }

2. To reduce the complexity we use a common technique to reduce the inner loop.
Instead of iterating through every j in order to find the above mentioned
maximum we can do a range maximum query (RMQ) - which can be easily done in
logarithmic time leading to overall complexity of n * log(n).

In the above context the query is done over a range of VALUES (not indexes)
of elements of A looking for such a VALUE whose index i corresponds
to the largest H (or L). The mentioned query can be easily answered
using an appropriate data structure such as segment tree, binary indexed
tree or whichever you prefer.

So we can replace the inner loop with something like this:

L = get_max_H(A + 1, n) + 1; insert_new_L();
H[i] = get_max_L(1, A[i] - 1) + 1; insert_new_H();


The idea of this technique is worth remembering since it can be quite useful
in some DP problems.

[Parleen]

Please tell me what's wrong in my code? I couldnt find bug in my problem
Or send some critical input/output here :

Code: Select all

#include<stdio.h>
#include<math.h>

long a,b,c,i,j,k;
void main()
{
long par,inp,test,max,count,t;

scanf("%ld",&test);

for(t=1;t<=test;t++)
{
scanf("%ld",&inp);
max=0;count=0;

if(inp>0)
{
 count=1;max=1;
 par=0;
 scanf("%ld",&a);
 b=a;
 for(i=2;i<=inp;i++)
 {
 scanf("%ld",&a);

	if(par%2==0 && b>a)
	{count++;par++;b=a;}

	else if(par%2!=0 && b<a)
	{par++;count++;b=a;}

	else
	{
	if(count>max)
	   max=count;

	   count=1;
	   par=0;
	   b=a;
	}
 }
}

if(count>max)
   max=count;

printf("%ld\n",max);

}
}  

[KirllButin]

I'm trying to solve it in O(n), but don't understand what's wrong with this code
Please help me

Code: Select all

//deleted

subsequence can be not cosequtive

[Robert Gerbicz]

I am not going to comment on a linear solution (I believe that no correct greedy
solution exists).

There is a dp solution using O(n) time and O(1) memory, and it is trivial to prove that the algorithm is correct. It's a little surprising.