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Try to define a recursive approach. For the ith candies, there will be one of the two following cases:
If the i - 1 previous candies are already distributed into k bags for the ith candy, you can have k * dp[n - 1][k] ways to distribute the ith candy. We need then to solve the state of (n - 1, k).
If the i - 1 previous candies are already distributed into k - 1 bags for the ith candy, you can have dp[n - 1][k - 1] ways to distribute the ith candy. We need then to solve the state of (n - 1, k - 1).
This approach will be too slow and will traverse some states more than once. We should use memoization to make the algorithm efficient.
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Yes, variations of distribution and combinatorial DP problems are common in FAANG-style interviews. This problem tests understanding of dynamic programming, recurrence relations, and combinatorial reasoning.
A 2D dynamic programming table (or a 1D optimized array) is typically used. It stores intermediate counts of distributions for different numbers of candies and children, allowing the solution to be built incrementally.
The optimal approach uses dynamic programming. Define a state dp[i][j] representing the number of ways to distribute i candies among j children. Using a recurrence that accounts for adding the candy to an existing child or starting a new group, the solution can be built in O(n × k) time.
Yes. Since each DP state depends only on the previous row, the 2D table can be compressed into a 1D array. This reduces the space complexity from O(n × k) to O(k) while keeping the same time complexity.