 # Equality(ISBIAS) Editorial

Author: sahi1422
Tester enchom
Editorialist: Sabre

Prerequisites: STL

Difficulty: Easy

Effective Question: Check if the number of maximal inc/dec contiguous subsequences in a given range(L,R) are equal or not, for Q queries.

L=1,R=13
MIS:3
MDS:4
Verdict: NO

Explanation:

We fix flags(store positions) 0 and 1 to denote the behaviour of an arbitrary subsequence: Going from 0 to 1 implies the subsequence is increasing and vice versa. A single loop running over the length is sufficient for this task.

A few observations follow:

1. This sequence will always be alternating in nature: 010101… or 1010101… and hence any subsequence of these flags will be so too.
2. Not counting the last flag, # of 0’s will give # of MIS and # of 1’s will give the # of MDS in range L=1 and R=N. This is true for any arbitrary subsequence too(with any L or R).

Now we take Q queries each consisting of a “L” and a “R”.

Now each L and R will be between some positions of our preset flags. They may even coincide onto these flags(4 cases). For instance this is one possible case: Now we find the nearest left flag wrt L(is 0 for above) and nearest right flag wrt R(is 0 for above). There maybe 4 possible cases:(00,01,10,11) We treat the cases (00,11) as similiar and (01,10) as similiar.

1. Case: 00/11
Only odd lengths will be allowed.
Eg: 010/101 length=3;
Eg: 01010/10101 length=5;

and so on. So here we note that the #0’s=#1’s if we exclude the last flag. From observation 2 the #MIS=#MDS always for this case.

1. Case: 01/10
Only even lengths will be allowed.
01,0101,010101…,10,1010,101010,…

and so on. So here we note that the #0’s!=#1’s if we exclude the last flag. From observation 2 the #MIS and #MDS will never be equal for this case.

So it all comes down to finding the nearest left flag wrt L and nearest right flag wrt R !

There can be many efficient ways to do this but one way that comes to mind is to use a multiset because of its very efficient insert,find,erase operations.

It will be a simple 5 step process(Assuming L and R are not present initially in the set):

Complexity:
O(N) for fixing the flags. O(logN+logN+logN) for each query, total Q queries so: O(3QlogN).
Overall complexity: O(N+3QlogN) which is O(N+Q).

Sabre

Enjoyed writing my first unofficial editorial Thought of writing taster instead of tester(if you know, you know xD) 3 Likes

Nice approach. My approach was to store all the increasing and decreasing subsequences as ranges and find out the number of intersections with [L, R]
O(N) to make ranges and O(QLgN) for queries = O(N+QLgN)
My Code

one of the interesting submissions in this problem was this by @avi_gp

1 Like

Thanks, precise code and a good approach too.

1 Like

I did queries in O (1)
Check out my solution
https://www.codechef.com/viewsolution/28757897

3 Likes

I used an auxiliary array initialized it with zeroes than update an index with 1 if it’s greater than previous value in the actual array else remain 0 signifies that whether sequence increased or decreased at that index.

I observe that if the value at L+1 is not equal to the value at R then answer is yes
Else No

2 Likes

https://www.codechef.com/viewsolution/28967994
I got TLE in this solution , please tell how to avoid it?

U r getting tle bcoz u iterating for each given ranges making complexity of your code of an order of O (n^2).
Just check above solutions for an idea.

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Because its O((R-L)Q) and (R-L) in worst case can be N so O(NQ) which is of order 10^10, you need to reduce this order to 10^5-10^6 ish. Read the editorial and check solutions posted above.

Quite commendable nice… 1 Like

Thank you

Please tell me where i am wrong in my approach!

https://www.codechef.com/viewsolution/28901776

that was amazing!

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How did you got this approach…is it just by observing the pattern in auxiliary array or there is some logic behind it ??

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Thank you!!

i used segment tree for this however got tle .complexity was just O(Qlog(N)+N).also got segmentation for 1st case.can you take a look at the solution.
https://www.codechef.com/viewsolution/28976764

It is an observation but a true fact
If the values at L+1 and R is same then the no. Of increase and decrease in between will not
Will always diff by at least 1 you can check it out. But if the values are different then for sure the increase and decrease is equal.

I am getting TLE in some of the test cases…can somebody please explain where i am going wrong??
here is my submission :- https://www.codechef.com/viewsolution/28973671

I am not getting the approaches used in above solutions as I am new to programming .
Can you explain how to reduce time complexity of a written code , for example - reducing complexity of : https://www.codechef.com/viewsolution/28967994
Is there any way to reduce complexity?

There is no need to count the number of increasing/decreasing sub sequences. Just observe the nature of sequence(inc/dec) formed by first two and the last two elements of the series between the given L, R bounds ( both inclusive). If the nature of both sequences is same then verdict is NO, otherwise YES. It’s because after the end of every increasing sequence, there has to be a decreasing sequence before the next increasing sequence begins. Therefore in any given L-R span, the difference of counts of both of these sequences cannot be more than 1. Think about it. Here’s my solution : ISBIAS solution

7 Likes

Try increasing the size of segment tree.