Coin Change (Bottom-Up)

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

Algorithm

@steps

Initialize dp[0] = 0 and all other amounts to an unreachable sentinel.
Scan amounts from 1 through 6.
For each coin, read the earlier cell dp[amount - coin] when it exists.
Write the smallest candidate into the current amount.
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.

Rust DSA Implementation

basic.rs

fn list_string(values: &[i32]) -> String {
    format!("[{}]", values.iter().map(|v| v.to_string()).collect::<Vec<_>>().join(", "))
}

fn main() {
    let coins = [1, 3, 4];
    let target = 6usize;
    let inf = (target + 1) as i32;
    let mut dp = vec![inf; target + 1];
    dp[0] = 0;
    for amount in 1..=target {
        for coin in coins {
            let coin = coin as usize;
            if amount >= coin {
                let candidate = dp[amount - coin] + 1;
                if candidate < dp[amount] {
                    dp[amount] = candidate;
                }
            }
        }
    }
    println!("{}", dp[target]);
    println!("{}", list_string(&dp));
}

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