Sub-arrays

 1.  Find maximum sum of a subarray

Brute- time: O(n^2)
go through each subarray and thus find the maximum sum

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Optimal- time: O(n)

use Kadane Algorithm
at each element either extend old subarray or start a new one.

Handles negatives as well as positives
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2. Find maximum sum of subarray of size K

only positives-

use sliding window of fix size K,
and move both pointers simultaneously thus simulating fix window traversing array
 
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3. Find maximum sum of distinct subarray of size K

Brute-  time: O(nK)    space: O(n)
Use hashing ,
after each expansion and shrinking of sub array, check if this subarray contains duplicates or not.
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Optimal- time: O(n)   space: O(n)
instead of checking each time if subarray contains duplicates or not, we will use a distinct counter
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4. Find length of longest subarray with sum <= K
(array has only non negatives)

we use sliding window
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5. Find length of longest subarray with sum = K

Case-1: both positives & negatives

we use prefix sum + hashing

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Note-
In questions of printing subarrays or subsequences, we have to have to do with brute force, no other option .
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6. Count subarrays with sum = K

Case-1: only positives

use sliding window pattern-3

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Case-2: non negatives

use below identity:
count(sum = k) = count(sum ≤ k) − count(sum ≤ k−1)
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Case-3: both positives & negatives

use prefix sum + hashmap


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