# 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) >= 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

• Not everybody can read python. Is it possible to rewrite your algorithm for people who are not python experts? Apr 16 '20 at 8:22

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.
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.