Longest subarray with at most two different values - Runtime complexity for a DP solution Consider the problem of finding, for a given input array, the longest subarray with at most two different values. 

For example:

Input: [3,3,3,1,2,1,1,2,3,3,4]
Ans = 5, the longest subarray would be [1,2,1,1,2].

Input: [1,2,3,2,2]
Ans = 4, the longest subarray would be [2,3,2,2].


Below is dynamic programming solution to this problem (in Python, hopefully it's easy to read) using a sliding window that holds a "valid subarray" (the subarray of elements between indices i and j always holds two values at most).

I read on e.g. LeetCode that this solution has a runtime complexity of $O(N)$ where $N$ is the length of the input array, but that's not immediately clear to me since we have two nested loops with $i$ and $j$ and $0\leq i\leq j\leq n$.

Why is the worst-case runtime complexity of this solution $O(N)$ and not $O(N^2)$?

Here's the DP solution in question with those nested loops holding a subarray between $i$ and $j$:

    def longest_subarray_holding_two_diff_values (input_array):
        ans = i = 0
        count = collections.Counter()

        for j, x in enumerate(input_array):
            count[x] += 1

            while len(count) &gt;= 3:
                count[input_array[i]] -= 1
                if count[input_array[i]] == 0:
                    del count[input_array[i]]
                i += 1
            ans = max(ans, j - i + 1)
        return ans
Here is how the sliding window algorithm works (unfortunately, I don't understand your code, so can't say whether this is the same algorithm).

We keep track of two pointers $i,j$, with the following properties: the subarray $A[j],\ldots,A[i]$ contains exactly two values, and it is maximal with respect to $j$ (that is, either $j = 0$ or $A[j-1],\ldots,A[i]$ contains three values). We also keep track of the two values in question $a,b$, and of their last appearance $k_a,k_b$. Finally, we keep track of the longest valid subarray seen so far.

In the initialization phase, we scan the array until we see two different values; if the array is constant, then the answer is the length of the array.

At steady state, we take a peek at $A[i+1]$. If $A[i+1] \in \{a,b\}$, we update $k_a$ or $k_b$, and simply increase $i$. If $A[i+1] \notin \{a,b\}$, then we do two things. First, we update the value of the longest valid subarray seen so far (comparing it to $j-i+1$). Second, suppose that $A[i] = a$; then we set $j = k_b+1$, set $b = A[i+1]$, set $k_b = i+1$, and increment $i$.

Finally, when reaching $i = n$, we update the value of the longest valid subarray (comparing it to $j-i+1$), and output the result.

As you can see, this algorithm performs $O(1)$ operations per iteration, so runs in $O(n)$ time.