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.

TypeScript DSA Implementation

basic.ts 
function listString(values: number[]): string { return `[${values.join(", ")}]`; }
const coins = [1, 3, 4];
const target = 6;
const inf = target + 1;
const dp = Array(target + 1).fill(inf);
dp[0] = 0;
for (let amount = 1; amount <= target; amount++) {
  for (const coin of coins) {
    if (amount >= coin) {
      const candidate = dp[amount - coin] + 1;
      if (candidate < dp[amount]) dp[amount] = candidate;
    }
  }
}
console.log(dp[target]);
console.log(listString(dp));

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