Zuko's Redemption Mission

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Unit 12 Session 1 Standard (Click for link to problem statements)

Problem Highlights

• 💡 Difficulty: Medium
• ⏰ Time to complete: 30 mins
• 🛠️ Topics: Dynamic Programming

1: U-nderstand

Understand what the interviewer is asking for by using test cases and questions about the problem.

• Established a set (2-3) of test cases to verify their own solution later.
• Established a set (1-2) of edge cases to verify their solution handles complexities.
• Have fully understood the problem and have no clarifying questions.
• Have you verified any Time/Space Constraints for this problem?

• What is the goal of the problem?
  • The goal is to find the fewest number of tokens Zuko needs to gather exactly amount units of supplies.
• What if the amount cannot be reached with the available tokens?
  • If it is impossible to gather the exact amount, return -1.
• What happens if the amount is zero?
  • If the amount is 0, Zuko doesn't need any tokens and the result should be 0.

HAPPY CASE
Input: 
    tokens = [1, 2, 5], amount = 11
Output: 
    3
Explanation:
    Zuko can gather 11 units of supplies with 5 + 5 + 1 tokens.

EDGE CASE
Input: 
    tokens = [2], amount = 3
Output: 
    -1
Explanation:
    It's impossible for Zuko to gather exactly 3 units of supplies with only 2-unit tokens.

EDGE CASE
Input: 
    tokens = [1], amount = 0
Output: 
    0
Explanation:
    Zuko doesn't need any tokens to gather 0 units of supplies.

2: M-atch

Match what this problem looks like to known categories of problems, e.g. Linked List or Dynamic Programming, and strategies or patterns in those categories.

For minimum token problems, we want to consider the following approaches:

• Dynamic Programming (DP): This problem is similar to the coin change problem. We can use a DP array to store the fewest number of tokens required to gather i units of supplies.

3: P-lan

Plan the solution with appropriate visualizations and pseudocode.

General Idea: Use dynamic programming to calculate the minimum number of tokens needed to gather the exact amount. We will build up a solution using a DP array, where dp[i] stores the minimum number of tokens required to gather exactly i units of supplies.

Steps:

1. Base Case:

   • Create a DP array dp initialized to float('inf'), except dp[0] = 0 since 0 units can be gathered with 0 tokens.

2. Token Processing:

   • For each token in tokens, iterate over all possible supply amounts from token to amount.
   • For each amount i, calculate dp[i] as the minimum of the current value dp[i] and dp[i - token] + 1.

3. Return Result:

   • If dp[amount] is still float('inf'), return -1, indicating it's impossible to gather the exact amount.
   • Otherwise, return dp[amount].

4: I-mplement

Implement the code to solve the algorithm.

def zuko_supply_mission(tokens, amount):
    # Initialize DP array with infinity, except dp[0] = 0
    dp = [float('inf')] * (amount + 1)
    dp[0] = 0
    
    # Update dp array for each token
    for token in tokens:
        for i in range(token, amount + 1):
            dp[i] = min(dp[i], dp[i - token] + 1)
    
    # If dp[amount] is still infinity, it means the amount cannot be achieved
    return dp[amount] if dp[amount] != float('inf') else -1

5: R-eview

Review the code by running specific example(s) and recording values (watchlist) of your code's variables along the way.

Example 1:

• Input: tokens = [1, 2, 5], amount = 11
• Expected Output: 3

Example 2:

• Input: tokens = [2], amount = 3
• Expected Output: -1

Example 3:

• Input: tokens = [1], amount = 0
• Expected Output: 0

6: E-valuate

Evaluate the performance of your algorithm and state any strong/weak or future potential work.

Assume n is the amount and m is the number of tokens.

• Time Complexity: O(n * m) because we iterate over each amount from 1 to n for each token in tokens.
• Space Complexity: O(n) to store the DP array.