Counting according to query

Question

Given an array of N positive elements. Lets suppose we list all N × (N+1) / 2 non-empty continuous subarrays of the array A and then replaced all the subarrays with the maxi

Answer 1

I believe I have a solution in O(N + Q*log N) (More about time complexity). The trick is to do a lot of preparation with your array before even the first query arrives.

1. For each number, figure out where is the first number on left / right of this number that is strictly bigger.

Example: for array: 1, 8, 2, 3, 3, 5, 1 both 3's left block would be position of 8, right block would be the position of 5.

This can be determined in linear time. This is how: Keep a stack of previous maximums in a stack. If a new maximum appears, remove maximums from the stack until you get to a element bigger than or equal to the current one. Illustration:

In this example, in the stack is: [15, 13, 11, 10, 7, 3] (you will of course keep the indexes, not the values, I will just use value for better readability).

Now we read 8, 8 >= 3 so we remove 3 from stack and repeat. 8 >= 7, remove 7. 8 < 10, so we stop removing. We set 10 as 8's left block, and add 8 to the maximums stack.

Also, whenever you remove from the stack (3 and 7 in this example), set the right block of removed number to the current number. One problem though: right block would be set to the next number bigger or equal, not strictly bigger. You can fix this with simply checking and relinking right blocks.

2. Compute what number is how many times a maximum of some subsequence.

Since for each number you now know where is the next left / right bigger number, I trust you with finding appropriate math formula for this.

Then, store the results in a hashmap, key would be a value of a number, and value would be how many times is that number a maximum of some subsequence. For example, record [4->12] would mean that number 4 is the maximum in 12 subsequences.

Lastly, extract all key-value pairs from the hashmap into an array, and sort that array by the keys. Finally, create a prefix sum for the values of that sorted array.

3. Handle a request

For request "exactly k", just binary search in your array, for more/less thank``, binary search for key k and then use the prefix array.