# OETW - EDITORIAL

Setter: DOlaBMOon
Tester: Hussain
Editorialist: Taranpreet Singh

Easy

Prefix Sum

### PROBLEM:

Given an array A consisting of only 1 and 2. Find number of unique values between 1 and 2*N which can represented as sum of a sub-array of array A.

Let us call a value X reachable, there exist a sub-array with sum X.

### SUPER QUICK EXPLANATION

•   Find largest value $X$, such that $X$ is reachable while $X+1$ is not reachable, then all values up to $X$ and values $V$ which satisfy $X < V \leq S$ and $X \equiv V \pmod{2}$ are reachable, where $S$ is sum of array.


### EXPLANATION

This problem have very simple implementation, if you get the intuition. So, I’ll be mostly focusing on idea and examples, Implementation can be referred from solutions.

Firstly, Note that to reach an odd value, we need at least one occurrence of digit 1. Easy enough, right? So, If array A doesn’t contain digit 1 at all, we know, all even values are reachable.

Consider following examples

Array 1 2 2 2 2

We can manually see that all values from [1,9] is reachable.

Array 2 2 1 2 2

Values from [1,5] and 7 and 9 is reachable, while Sum 6 and 8 are not reachable.

Now, consider array 2 1 2 2 2

All values from [1,9] except 8 is reachable.

What do we learn from these examples?? We can see, that there is always an upper limit X up to which all values are reachable, while only values above X which have same parity as X are reachable.

Time for a general statement
Formally, Once we consider a sub-array with sum which includes both leftmost as well as rightmost occurrence of digit 1, Parity of this sum, i.e. X cannot change, so, values V > X, having different parity with X, can never be reachable.

Now that we have proved that by the Maximum value, we can calculate the answer easily, we now try to calculate this Maximum value.

For this, consider another example of format 1,2,2,2\dotsc having N elements, prove that all values from [1, 2*N-1] is reachable.

Proof can be found below.

Click to view

For Odd values, we can include the 1 and get remaining sum using 2s. Even values can be obtained by excluding 1. But value 2*N can never be reached as sum of array is 2*N-1

So, we now know that by selecting a sub-array beginning or ending with one, we can reach all values up to sum of sub-array. So, The upper bound is actually the maximum sum of sub-array which either start or end with 1.

Also, The values above X sum having same parity as X can be reached by including remaining 2s.

This is it. Author checks for every position of 1 in array A, the largest sub-array sum either starting or ending at 1 using prefix sum array, while it suffices to check only sub-arrays [L, N] and [1,R] where L and R correspond to leftmost and rightmost position of 1 respectively, take the maximum of both.

That’s all, guys. Just implement it when you get the idea and That’s it.

Need another hint to calculate answer when X is known? see below.

Click to view

Answer is X+(S-X)/2. Still didn’t get how?

Click to view

X values in range [1,X] are reachable and (S-X)/2 values in range [X+1,S] are reachable.

Also, using same observation, answer can also be calculated as S-min(C1, C2), where S is sum of array, C1 is Number of 2s before leftmost 1 and C2 is number of 2s after rightmost 1.

Can you prove it how?? This is an exercise for you.

### AUTHOR’S AND TESTER’S SOLUTIONS:

Feel free to Share your approach, If it differs. Suggestions are always welcomed.

7 Likes

The following sample case was given for this problem:

INPUT

1
3
2 1 2

OUTPUT

4

And I totally understand why it is 4 for this test case. But I found solutions which got AC during the contest even after giving wrong answer for this particular test case. There are two solutions which I found faulty in this respect which got AC during the contest.

20307252 by hroarr in COOK98B (Output for this case: 5)

20314080 by daniyal1998 in COOK98B (Output for this case: 2)

It might be the case that many faulty solutions were submitted who got accepted during the contest in both Division 1 and Division 2. Or, it might be the case with other problems in this contest as well.

I believe that the test cases were very loose and weak for this problem.

I recently emailed this concern to bugs@codechef.com, as I thought it must be reported.

I have a doubt.
Consider the given array
2 2 1 2 2 1 2 2 1 2 2 2 2 2 2
here the prefix sum of the largest subarray with 1st and last number 1 is 11.
but z=18 is reachable which is greater than 11 and even which seems to be an counterexample to above solution.
Could you please clarify if i am getting it right.

Seems like you misunderstood my statement.

I meant, either starting with, or ending with. Both are not necessary. In your example, the largest sub-array starting or ending with 1 has sum 23. Only values 24 and 26 are unreachable.

1 Like

Tester’s solution is incorrect!

1
5
2 2 1 2 1


Returns 7. The correct answer is 8.

Here is the proof for S−min(C1,C2)

We observe that if we get 1 2 2 2 or 1 2 2 1 1 2 or 2 2 1 2 1 i.e. any subarray which is starting or with ending with 1 we can make every number till its sum.

Now, if we see in this 2 2 1 2 2 1 1 2 all possible numbers till prefix sum at 7 is possible. While after that it won’t be possible. Also we can consider that all numbers till suffix sum at 3 is possible. While before that it won’t be possible. Because of each 2 we won’t be able to make a number.

Hence, with other examples it can be shown that we need to know number of 2’s just before first occurrence of 1 either from left or right side whichever is optimal and subtract it from total sum.

2 Likes

okay.Got it.Thanks

Actually while implementing the solution we need first check one thing ,whether the array is starting with 1 or ending with 1 .If the answer is “YES” we need to print the sum of the array straight forward.If the answer is “NO” we need compute the answer using the above method.Just be aware of otherwise you might get wrong answers while solving it.( although the editorialist mentioned this many people tend to forget this point)

1 Like

Then what action is taken with this respect ? @taran_1407 @admin

1 Like

I think we do not need to check this case specially. Though no harm mentioning

Well, i think we’ll have to wait for admin statement.

A short python solution.

I can’t proof this but it got an AC.

if array has only twos:
ans= sum/2

else if array starts and ends with 2:
ans= sum-1

else
ans=sum

Here’s my solution.

update ur solution as-- if array has only twos: ans=sum/2

1 Like

for second line, for test case 2 2 1 2 2 output is 7 because 6 and 8 can not be made… but according to ur second point, ans=sum-1 so it would be 9-1=8 but ans is 7… i am getting confused

exactly…

i think ur approach is wrong…
actual answer will be sum-min(c1,c2) where c1 is no of 2’s before leftmost 1 and c2 is no of 2’s after rightmost 1

I understood what the right solution should be. I think the test cases were not strong enough.

//