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DPChange (M, c, d): bestNumCoins [0] - 0 for m - 1 to M bestNumcoins (m] = inf for i - 1 to d if m >- c[i] if bestNumcoins (m-c[i]] + 1 < bestNumcoins (m] bestNumCoins (m] = bestNumCoins (m-c[i]] + 1 return bestNumCoins (M) a) Convert this pseudocode to Python. b) What is the runtime of this algorithm?

DPChange (M, c, d): bestNumCoins [0] - 0 for m - 1 to M bestNumcoins (m] = inf for i - 1 to d if m >- c[i] if bestNumcoins (m-c[i]] + 1 < bestNumcoins (m] bestNumCoins (m] = bestNumCoins (m-c[i]] + 1 return bestNumCoins (M) a) Convert this pseudocode to Python. b) What is the runtime of this algorithm?

Computer Networking: A Top-Down Approach (7th Edition)
Computer Networking: A Top-Down Approach (7th Edition)
7th Edition
ISBN:9780133594140
Author:James Kurose, Keith Ross
Publisher:James Kurose, Keith Ross
Chapter1: Computer Networks And The Internet
Section: Chapter Questions
Problem R1RQ: What is the difference between a host and an end system? List several different types of end...
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### DPChange(M, c, d) Pseudocode This pseudocode is used to find the minimum number of coins needed to make change for a given amount \( M \) using a list of coin denominations. #### Pseudocode Explanation: 1. **Initialization:** - `bestNumCoins[0] = 0`: Set the base case for making change for 0 amount. 2. **Outer Loop (1 to M):** - For each amount \( m \) from 1 to \( M \), do the following: - Initialize `bestNumCoins[m] = inf` (infinity) to represent initially an impossible large number of coins. 3. **Inner Loop (1 to d):** - Iterate over each coin denomination. - **Check Condition:** - If the current amount \( m \) is greater than or equal to the coin denomination \( c[i] \): - Compare and update the best number of coins: ``` if bestNumCoins[m-c[i]] + 1 < bestNumCoins[m]: bestNumCoins[m] = bestNumCoins[m-c[i]] + 1 ``` - This checks if using the current coin \( c[i] \) results in fewer coins than previously computed for amount \( m \). 4. **Return Result:** - Return `bestNumCoins[M]` which contains the minimum number of coins needed for amount \( M \). ### Questions: a) **Convert this pseudocode to Python.** ```python def DPChange(M, c): bestNumCoins = [float('inf')] * (M + 1) bestNumCoins[0] = 0 for m in range(1, M + 1): for coin in c: if m >= coin: if bestNumCoins[m - coin] + 1 < bestNumCoins[m]: bestNumCoins[m] = bestNumCoins[m - coin] + 1 return bestNumCoins[M] ``` b) **What is the runtime of this algorithm?** The runtime of this algorithm is \( O(M \cdot d) \) where \( M \) is the amount for which we are making change and \( d \) is the number of different coin denominations. This is because for each amount up to \( M \), we iterate through each denomination.
Transcribed Image Text:### DPChange(M, c, d) Pseudocode This pseudocode is used to find the minimum number of coins needed to make change for a given amount \( M \) using a list of coin denominations. #### Pseudocode Explanation: 1. **Initialization:** - `bestNumCoins[0] = 0`: Set the base case for making change for 0 amount. 2. **Outer Loop (1 to M):** - For each amount \( m \) from 1 to \( M \), do the following: - Initialize `bestNumCoins[m] = inf` (infinity) to represent initially an impossible large number of coins. 3. **Inner Loop (1 to d):** - Iterate over each coin denomination. - **Check Condition:** - If the current amount \( m \) is greater than or equal to the coin denomination \( c[i] \): - Compare and update the best number of coins: ``` if bestNumCoins[m-c[i]] + 1 < bestNumCoins[m]: bestNumCoins[m] = bestNumCoins[m-c[i]] + 1 ``` - This checks if using the current coin \( c[i] \) results in fewer coins than previously computed for amount \( m \). 4. **Return Result:** - Return `bestNumCoins[M]` which contains the minimum number of coins needed for amount \( M \). ### Questions: a) **Convert this pseudocode to Python.** ```python def DPChange(M, c): bestNumCoins = [float('inf')] * (M + 1) bestNumCoins[0] = 0 for m in range(1, M + 1): for coin in c: if m >= coin: if bestNumCoins[m - coin] + 1 < bestNumCoins[m]: bestNumCoins[m] = bestNumCoins[m - coin] + 1 return bestNumCoins[M] ``` b) **What is the runtime of this algorithm?** The runtime of this algorithm is \( O(M \cdot d) \) where \( M \) is the amount for which we are making change and \( d \) is the number of different coin denominations. This is because for each amount up to \( M \), we iterate through each denomination.
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