Python - Maximize the Cut Segments

Python – Maximize the Cut Segments

This problem involves cutting a given length into segments of specified sizes such that the total number of segments is maximized. Using dynamic programming, we can determine the optimal way to cut the rod (or ribbon) into segments of lengths a, b, and c so that their sum equals the given length n.

Problem Statement

Given a rod (or ribbon) of length n and three possible cut lengths a, b, and c, determine the maximum number of segments you can obtain by cutting the rod such that the sum of the segment lengths is exactly n. If it is not possible to cut the rod into pieces that sum to n, return 0.

Sample Input and Output

Example 1:

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# Input:
n = 5, a = 5, b = 3, c = 2

# Explanation:
# - One way is to cut one segment of length 5 (total segments = 1).
# - Alternatively, cutting a segment of length 3 and another of length 2 gives 2 segments.
# The maximum number of segments is 2.

# Output:
2

Explanation: Although cutting a single segment of length 5 is possible, the optimal solution is to cut the rod into segments of lengths 3 and 2, yielding a maximum of 2 segments.

Example 2:

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# Input:
n = 9, a = 2, b = 3, c = 4

# Explanation:
# - One optimal way is to cut the rod into segments: 2 + 2 + 2 + 3 = 9, yielding 4 segments.
# Other combinations yield fewer segments.
# The maximum number of segments is 4.

# Output:
4

Solution Approach

We use dynamic programming to solve this problem by breaking it down into subproblems. The idea is to define dp[i] as the maximum number of segments that can be obtained from a rod of length i. The recurrence relation is:

dp[i] = max(dp[i – a], dp[i – b], dp[i – c]) + 1

We initialize the base case with dp[0] = 0 (zero segments for a rod of length 0) and set other values to a very small number (e.g., negative infinity) to represent an unreachable state. We then iterate from 1 to n, updating dp[i] for each possible cut. The final answer will be stored in dp[n]. If dp[n] remains negative, it means no valid cuts can form the rod, and we return 0.

Python Program

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def maximize_cut_segments(n, a, b, c):
    """
    Compute the maximum number of segments the rod of length n can be cut into,
    given the segment lengths a, b, and c.
    
    Parameters:
    n (int): Total length of the rod.
    a, b, c (int): Possible segment lengths.
    
    Returns:
    int: Maximum number of segments that sum up to n.
         Returns 0 if it's not possible to cut the rod accordingly.
    """
    # Initialize dp array where dp[i] represents the maximum segments for length i.
    # We use a very small number to denote an impossible state.
    dp = [-float('inf')] * (n + 1)
    dp[0] = 0  # Base case: 0 segments for a rod of length 0
    
    # Iterate over all lengths from 1 to n
    for i in range(1, n + 1):
        for cut in (a, b, c):
            if i - cut >= 0:
                dp[i] = max(dp[i], dp[i - cut] + 1)
    
    # If dp[n] is still negative, it means no solution exists, so return 0.
    return dp[n] if dp[n] >= 0 else 0

# Test examples

print("Example 1:")
print("Input: n = 5, a = 5, b = 3, c = 2")
print("Output:", maximize_cut_segments(5, 5, 3, 2))
# Expected output: 2

print("\nExample 2:")
print("Input: n = 9, a = 2, b = 3, c = 4")
print("Output:", maximize_cut_segments(9, 2, 3, 4))
# Expected output: 4

Explanation: The dynamic programming approach builds a dp table where each entry represents the maximum number of segments that can form a particular length. For each length i, the algorithm checks whether making a cut of size a, b, or c can lead to a better solution by adding one segment to the optimal solution for i - cut. Finally, the value at dp[n] gives the maximum number of segments for the rod of length n.

Conclusion

This tutorial demonstrated how to solve the “Maximize the Cut Segments” problem using dynamic programming in Python. We discussed the problem statement, walked through sample examples with detailed explanations, and implemented a solution that builds a DP table to efficiently compute the answer. This approach is highly effective for optimization problems involving choices and can be adapted to various similar problems in dynamic programming.

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Python – Maximize the Cut Segments