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Minimum Number of Coins to be Added

Difficulty: Medium

Problem Description

You are given a 0-indexed integer array coins, representing the values of the coins available, and an integer target. An integer x is obtainable if there exists a subsequence of coins that sums to x. Return the minimum number of coins of any value that need to be added to the array so that every integer in the range [1, target] is obtainable.

Key Insights

  • The problem requires ensuring that all integers from 1 to the target can be formed using the available coins.
  • By sorting the coins and iterating through the range up to the target, we can determine gaps where additional coins are needed.
  • A greedy approach can be used to fill these gaps by incrementally adding the smallest missing values.

Space and Time Complexity

Time Complexity: O(n log n) due to sorting the coins, where n is the length of the coins array. The subsequent processing is O(n). Space Complexity: O(1) for the main variables used, as we are modifying the coins in place and not using extra space proportional to the input size.

Solution

The solution uses a greedy algorithm approach:

  1. Sort the coins to ensure we can evaluate them in increasing order.
  2. Initialize a variable to keep track of the smallest number that we cannot form (current_target).
  3. Iterate through the sorted coin array and check if the current coin can help form the current_target. If not, we need to add coins until we can form all integers up to the target.
  4. Each time we can't form a number, we add coins with values equal to the current_target until we reach the target.

Code Solutions

def min_coins_to_add(coins, target):
    coins.sort()  # Sort the coins
    current_target = 1  # Start by trying to create the number 1
    coins_needed = 0  # Count of additional coins needed

    for coin in coins:
        while current_target < coin:  # While current_target cannot be formed
            coins_needed += 1  # We need to add a coin
            current_target += current_target  # Double the current_target

        current_target += coin  # We can now form current_target with this coin

    while current_target <= target:  # Fill in any remaining gaps until target
        coins_needed += 1
        current_target += current_target  # Again double the current_target

    return coins_needed
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