Minimum Swaps To Make Sequences Increasing

Minimum Swaps To Make Sequences Increasing

What is this problem about?

The Minimum Swaps To Make Sequences Increasing problem gives you two equal-length integer arrays. In each operation, you can swap elements at the same index between the two arrays (arr1[i] with arr2[i]). The goal is to make both arrays strictly increasing simultaneously, using the minimum number of such swaps. This hard Minimum Swaps To Make Sequences Increasing coding problem is a classic two-state dynamic programming challenge.

Why is this asked in interviews?

Microsoft, Meta, Amazon, and Google ask this because it requires tracking two states simultaneously at each position — whether you swapped at the current index or not — and making the optimal decision based on both the current and previous choices. The array and dynamic programming interview pattern is demonstrated in a particularly elegant way here, and the state-transition logic tests structured DP thinking.

Algorithmic pattern used

Two-state DP. At each position i, maintain:

• keep[i] = minimum swaps to make both sequences valid up to index i, if we don't swap at i.
• swap[i] = minimum swaps to make both sequences valid up to index i, if we do swap at i.

Transitions depend on conditions at index i vs i-1:

• If arr1[i] > arr1[i-1] and arr2[i] > arr2[i-1]: no-swap can follow no-swap; swap can follow swap.
• If arr1[i] > arr2[i-1] and arr2[i] > arr1[i-1]: swap can follow no-swap; no-swap can follow swap.

Example explanation

arr1 = [1, 3, 5], arr2 = [1, 2, 4].

• i=1: arr1[1]=3>arr1[0]=1 ✓, arr2[1]=2>arr2[0]=1 ✓ → keep[1]=0, swap[1]=1. Also arr1[1]=3>arr2[0]=1 ✓, arr2[1]=2>arr1[0]=1 ✓ → can cross too.
• i=2: similarly compute keep[2] and swap[2]. Final answer: min(keep[2], swap[2]) = 0 (no swaps needed, already increasing).

Common mistakes candidates make

• Using a single state instead of two (keep/swap) at each index.
• Not initializing base cases: keep[0] = 0, swap[0] = 1.
• Missing the condition that both cross-conditions must be checked together.
• Overwriting state values before using them in the transition.

Interview preparation tip

Two-state DP problems require careful state definition. Before coding, write out the two states explicitly and enumerate all possible transitions with their conditions. Simulate the DP table on a small example (3-4 elements) by hand. Once the transitions are correct on paper, coding becomes straightforward. Practice similar "swap or not swap" DP problems — they appear in stock trading, assignment problems, and sequence alignment scenarios.