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Find the minimum number of subarrays needed to validly split each prefix of the input array a.
Denote dp[i] as the minimum number of subarrays needed to validly split [a[0], a[1], … , a[i - 1]], where dp[0] = 0.
Think about the dynamic programming transitions.
If we split the first i elements of the array, the last subarray in this splitting will end with a[i - 1] and start with some a[j], where gcd(a[j], a[i - 1]) ≠ 1. Then, we need to validly split the first j elements of the array, or [a[0]…a[j - 1]].
Iterate over all possible j < i such that gcd(a[j], a[i - 1]) ≠ 1 and let dp[i] = min(dp[i], dp[j] + 1).
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Problems combining dynamic programming with number theory concepts like prime factors and GCD are common in top tech interviews. While the exact question may vary, similar patterns frequently appear in FAANG-level coding interviews.
Number theory is used because the validity condition depends on the greatest common divisor (GCD) between the first and last elements of a subarray. Prime factorization helps efficiently determine whether two numbers share a common factor greater than 1.
The optimal approach uses dynamic programming combined with prime factorization. By tracking prime factors of each number and maintaining the best previous DP states for those factors, we can quickly determine valid split points and minimize the number of subarrays.
A hash map or dictionary is commonly used to map prime factors to the best dynamic programming state seen so far. This allows efficient updates and lookups when processing each element in the array.
