KSUBSUM - Editorial

editorial
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#1

PROBLEM LINKS

Practice
Contest

DIFFICULTY

EASY

EXPLANATION

Given an array A of N integers, we are asked to find the Kth maximum sum of a contiguous subarray of elements. Finding the Maximum sum of a subarray is a well known problem. If the array elements are all non-negative, we can use binary search to find the answer in O(n log S) time, where S is the maximum sum of a subarray. In this problem, the values can be negative. As with the other problems in this set, look at the constraints carefully, N ≤ 10,000 and K ≤ 2012. We go through the array from left to right and at each index i, we find all K maximum sums of subarrays, ending at index i. If S is the prefix sum array, where S[i] = A1 + A2 + … + A[i], then all subarray sums ending at index i can be computed using S[i] - S[j], for j = 0 to i-1 and considering S[0] = 0. But we only need top K of them, so we can subtract S[j] s in non-decreasing order and only K of them. This requires us to maintain the array S in sorted order and this can be done similar to insertion sort in O(K) time per insertion.

Now that we have the top-K subarray sums ending at index i, we can compare them with the current top-K best answers so far and pick some of them and drop others. Note that at each step you only need to maintain K minimum prefix sums and K maximum subarray sums so far. Given the best K sums so far and the current K sums, we can merge the two sorted arrays and get the updated best K sums. This can also be done in O(K) time. The overall time complexity is O(NK). Maintaining a set ( or heap ) in which each insertion is additional O(log K) only increases the running time by more than 10x and may not fit in the given time limit.

SETTER’S SOLUTION

Can be found here.

TESTER’S SOLUTION

Can be found here.

5 Likes
#2

i am getting runtime error…please help…
http://www.codechef.com/viewsolution/1761272

#3

@shubham2892 : The number of sub arrays is N(N-1)/2 which is O(N^2) . k1 , k2 , k3 are bounded in this problem . But the total number of sub arrays are not and they can be a huge number like 50000000 for N = 10000 . You will have count variable going up to that . And you are making array access with count variable . That’s the cause of runtime error .

2 Likes
#4

why am i getting a runtime error here

#5

@admin: Can you explain how would you do this-

If the array elements are all non-negative, we can use binary search to find the answer in O(n log S) time, where S is the maximum sum of a subarray.

#6

Using priority queue :

6 Likes
#7

I solved this question using Multiset.But, still there is some doubt.
AC solution: https://www.codechef.com/viewsolution/10907889
TLE solution: https://www.codechef.com/viewsolution/10907825

Both of them is having complexity O(n^2logn).Why one is getting TLE with slight change in approach?

#8

Lets consider the 3rd test case

Given array is 20 -15 10 -15

so cumulativeSum array becomes 20 5 15 0

now till each position of array you will have certain possible sums

1 -> 20

2 -> 5,-15

3 -> 15,-5,0

4 -> 0,-20,-5,-15

Now in your first approach you are putting these elements in the multiset row wise
like 20 then 5 and -15 and so on.

And in your second approach you are putting these elements in the multiset col wise
like 20,5,15,0 and then -15,-5,-29 and so on.

This making your complexity same but as the traversal is different so the number of
times the multiset will be updated is variable.

I see that as the only reason for one getting accepted and other getting tle.

So O(n^2 logn) won’t pass always as stated in the editorial too.

1 Like
#9

Still, i guess the test cases suited my AC approach, because by reversing inner loop we can no way guarantee optimization.Its just sheer luck that my solution passed in O(n^2 logn). :slight_smile:

#10

i think O(n^2logn) should not work. According to given constraints unless it is useless

#11

anybody please tell, what’s wrong with my solution.
https://www.codechef.com/viewsolution/22155789

#12

Look at this you will understand.

#13

Getting a TLE for this solution. I have used a priority queue of size K to insert into for each comparison in the prefix array. Time complexity is O(N^2 logK).

https://www.codechef.com/viewsolution/27951352

#14

don’t erase all sub array sum in set which is less than current subarray sum , take a pointer and store starting element of set and erase it,and insert new subarray sum. check this solution.
https://www.codechef.com/viewsolution/38939885