Kth Smallest Amount With Single Denomination Combination

Kth Smallest Amount With Single Denomination Combination - Problem

Array Math Binary Search Bit Manipulation Combinatorics Number Theory Hard

You are given an integer array coins representing coins of different denominations and an integer k.

You have an infinite number of coins of each denomination. However, you are not allowed to combine coins of different denominations.

Return the k^th smallest amount that can be made using these coins.

Input & Output

Example 1 — Basic Case

$ Input: coins = [3,5], k = 4

› Output: 9

💡 Note: The amounts we can make are: 3, 5, 6, 9, 10, 12, 15, ... The 4th smallest is 9.

Example 2 — Single Coin

$ Input: coins = [5], k = 2

› Output: 10

💡 Note: With only coin 5, we can make: 5, 10, 15, 20, ... The 2nd smallest is 10.

Example 3 — Common Multiples

$ Input: coins = [2,4], k = 3

› Output: 6

💡 Note: Coin 2 makes: 2, 4, 6, 8, ... Coin 4 makes: 4, 8, 12, ... Combined unique amounts: 2, 4, 6, 8, ... The 3rd smallest is 6.

Constraints

1 ≤ coins.length ≤ 15
1 ≤ coins[i] ≤ 25
1 ≤ k ≤ 2 × 10⁹

Visualization

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Asked in

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The key insight is using binary search on the answer space combined with efficient counting. We binary search for the kth smallest amount and use mathematical formulas to count valid amounts ≤ any given value. The optimal approach uses inclusion-exclusion principle to handle overlapping multiples. Time: O(log(k×max) × 2^n), Space: O(1).

Common Approaches

✓ Backtracking

⏱️ Time: N/A Space: N/A

Binary Search on Answer

⏱️ Time: O(log(k×max(coins)) × n) Space: O(1)

Binary search on possible answer values. For each candidate answer, count how many valid amounts are ≤ that value. The kth smallest is the smallest value with count ≥ k.

Brute Force Generation

⏱️ Time: O(L log L) Space: O(L)

Generate amounts by multiplying each coin by 1, 2, 3... up to some limit. Collect all unique amounts, sort them, and return the kth element.

Mathematical Optimization with Inclusion-Exclusion

⏱️ Time: O(log(k×max) × 2^n) Space: O(1)

Binary search combined with inclusion-exclusion principle to avoid overcounting when coins have common multiples. Count amounts that are multiples of any coin using GCD and LCM calculations.

Algorithm Steps — Algorithm Steps

Code -

solution.c — C

Time & Space Complexity

Time Complexity

⏱️

⚡ Linearithmic

Space Complexity

✓ Linear Space

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Smart Actions

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