Start by clarifying the problem constraints (e.g., unlimited coins, positive denominations, target can be 0). Then explain that this is a classic dynamic programming problem where you build up the solution for smaller amounts to find the minimum coins for the target. Present the DP recurrence and analyze time/space complexity, and if time permits, discuss optimizations or alternative approaches.
Pro tip: Mention that greedy doesn't always work (e.g., denominations [1,3,4] and target 6) to show you understand edge cases, and briefly discuss how to reconstruct the actual coins if asked.
Ask about constraints: Are coin denominations unlimited? Can the target be 0? Are all denominations positive? What if no combination exists?
Let dp[i] be the minimum coins to make amount i. Initialize dp[0]=0 and dp[i]=infinity for i>0. For each amount i from 1 to target, dp[i] = min(dp[i - coin] + 1) over all coins ≤ i.
Iterate through amounts and coins, updating dp. After filling, if dp[target] is still infinity, return -1 (or indicate impossible).
Time complexity: O(target * number of coins). Space complexity: O(target) for the DP array.
Mention that BFS can be used for unweighted graph interpretation, and that space can be optimized if only the minimum count is needed (though O(target) is already optimal for this approach).
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