The Super Ugly Number interview question is an advanced variation of the classic "Ugly Number" problem. A super ugly number is a positive integer whose prime factors are all contained in a given list of primes. For example, if the given primes are [2, 7, 13, 19], then [1, 2, 4, 7, 8, 13, 14, 16, 19, 26, 28, 32] are the first few super ugly numbers. Your task is to find the nth super ugly number given the list of primes and the value of n.
This problem is frequently asked by companies like Microsoft and Google to test a candidate's understanding of dynamic programming and priority queues (heaps). It requires you to efficiently generate numbers in increasing order without skipping any or repeating them. It evaluates your ability to manage multiple pointers or indices and your capacity to optimize a solution from a naive approach to one that handles large inputs within the time limit.
The core algorithmic pattern is Dynamic Programming or the Min-Heap (Priority Queue) approach. The DP approach involves maintaining an array dp to store the super ugly numbers found so far and an array of pointers where each pointer corresponds to a prime in the input list. For each new super ugly number, you calculate the next potential values by multiplying each prime with the super ugly number at its respective pointer. The minimum of these values becomes the next super ugly number, and the corresponding pointers are incremented.
Let primes = [2, 5] and n = 4.
dp[0] = 1. Pointers: p2=0, p5=0.2 * dp[p2] = 2, 5 * dp[p5] = 5.
dp[1] = 2, increment p2 to 1.2 * dp[p2] = 2 * 2 = 4, 5 * dp[p5] = 5 * 1 = 5.
dp[2] = 4, increment p2 to 2.2 * dp[p2] = 2 * 4 = 8, 5 * dp[p5] = 5 * 1 = 5.
dp[3] = 5, increment p5 to 1.
The 4th super ugly number is 5.A common mistake is failing to handle duplicate candidates. If multiple primes lead to the same minimum value, all their pointers must be incremented to avoid repeating the same number in the dp array. Another mistake is using a simple heap without considering that the number of primes can be large, which might lead to performance issues if not careful with the heap operations.
When preparing for the Super Ugly Number coding problem, practice both the pointer-based DP approach and the heap-based approach. The pointer approach is usually O(n * k) where k is the number of primes, while the heap approach is O(n * log k). Knowing the trade-offs between these two is a great talking point during an interview. Also, ensure you are comfortable with the Dynamic Programming interview pattern for generating sequences.
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