# Smallest Subarray with a given Sum

## Problem

Given an array of positive numbers and a positive number ‘S,’ find the length of the **smallest contiguous subarray whose sum is greater than or equal to ‘S’**. Return 0 if no such subarray exists.

{% hint style="info" %}
For example:

```
Input: [2, 1, 5, 2, 3, 2], S=7 
Output: 2
Explanation: The smallest subarray with a sum
great than or equal to '7' is [5, 2].
```

```
Input: [2, 1, 5, 2, 8], S=7 
Output: 1
Explanation: The smallest subarray with a sum
greater than or equal to '7' is [8].
```

```
Input: [3, 4, 1, 1, 6], S=8 
Output: 3
Explanation: Smallest subarrays with a sum 
greater than or equal to '8' are [3, 4, 1] 
or [1, 1, 6].
```

{% endhint %}

### Thought Process

* We will use a dynamic window (adjust size as we go depending on condition)

* The condition that will determine our window size is if the total sum is greater than the *s*, if this is true then we will continually shrink our window till the total sum is less than *s*.

## Solution

```
def smallest_subarray_with_given_sum(s, nums):
        minWindow = float("inf")
        totalSum = 0
        start = 0
        
        
        for i in range(len(nums)):
            totalSum+=nums[i]
            while totalSum >= s:
                minWindow = min(minWindow, i+1-start)
                totalSum-=nums[start]
                start+=1
        if minWindow == float("inf"):
            return 0
        return minWindow


```

## Key Points

* Using a dynamic window

* When the total sum of a contigous subarray is greater than *s* is when we start to shrink our window

## Time Complexity

* **Time:** $$O(N)$$ because the outer `for` loop runs for all elements, and the inner `while` loop processes each element only once; therefore, the time complexity of the algorithm will be $$O(N+N)$$ , which is asymptotically equivalent to $$O(N)$$.

* **Space:** O(1)


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