Recursion and Dynamic Programming

Coin Change (Bottom-Up)

Build a one-dimensional table where each amount stores the fewest coins needed to make it.

Algorithm

@steps

  1. Initialize dp[0] = 0 and all other amounts to an unreachable sentinel.
  2. Scan amounts from 1 through 6.
  3. For each coin, read the earlier cell dp[amount - coin] when it exists.
  4. Write the smallest candidate into the current amount.
  5. Print both the final answer and the full DP array. @end @complexity
  • Time: O(target * coin_count)
  • Space: O(target) @end
bottom-up dynamic programming `dp[a]` is solved from already-computed smaller amounts, so every table cell has a visible dependency.

Dart DSA Implementation

basic.dart 
String listString(List<int> values) => "[${values.join(", ")}]";

void main() {
  final coins = [1, 3, 4];
  const target = 6;
  const inf = target + 1;
  final dp = List<int>.filled(target + 1, inf);
  dp[0] = 0;
  for (var amount = 1; amount <= target; amount++) {
    for (final coin in coins) {
      if (amount >= coin) {
        final candidate = dp[amount - coin] + 1;
        if (candidate < dp[amount]) dp[amount] = candidate;
      }
    }
  }
  print(dp[target]);
  print(listString(dp));
}

@end @output 2 [0, 1, 2, 1, 1, 2, 2] @end