Kth Smallest Amount With Single Denomination Combination - Solution & Explanation

This approach uses a min-heap to efficiently find the k-th smallest amount. We can generate multiples of each coin denomination and insert them into a min-heap, keeping track of the smallest values across all sequences.

The solution initializes a min-heap with each coin denomination. The tuple (coin, coin) is inserted into the heap, where every element is a pair of current_value and the base coin denomination. We repeatedly pop the smallest value from the heap, and after k iterations, the k-th smallest element is returned. After popping, we push the next multiple of that coin into the heap.

Another approach is to count numbers that are multiples of any given denomination using mathematics and number theory concepts to find the k-th smallest efficiently. This avoids insertion into a heap by logically deducing the k-th smallest multiple.

This approach uses binary search to find the k-th smallest value by searching over a range of potential answers. For each midpoint in the range, we calculate how many numbers are divisible by each coin denomination and adjust the search region based on whether the count is less than or greater than k.