Cut Them All

An automated cutting machine is used to cut rods into segments. The cutting machine can only hold a rod of minLength or more. A rod is marked with the necessary cuts and their lengths are given as an array in the order they are marked. Determine if it is possible to plan the cuts so the last cut is from a rod at least minLength units long.

Example

n = 3

lengths = [4, 3, 2]

minLength = 7

The rod is initially sum(lengths) = 4 + 3 + 2 = 9 units long. First cut off the segment of length 4 + 3 = 7 leaving a rod 9 - 7 = 2. Then check that the length 7 rod can be cut into segments of lengths 4 and 3. Since 7 is greater than or equal to minLength = 7, the final cut can be made. Return “Possible”.

Example

n = 3

lengths = [4, 2, 3]

minLength = 7

The rod is initially sum(lengths) = 4 + 2 + 3 = 9 units long. In this case, the initial cut can be of length 4 or 4 + 2 = 6. Regardless of the length of the first cut, the remaining piece will be shorter than minLength. Because n - 1 = 2 cuts cannot be made, the answer is “Impossible.”

Function Description

Complete the function cutThemAll in the editor below.

cutThemAll has the following parameter(s):

int lengths[n]: the lengths of the segments, in order

int minLength: the minimum length the machine can accept Returns

string: “Possible” if all n-1 cuts can be made. Otherwise, return the string “Impossible” Constraints

2 ≤ n ≤ 105 1 ≤ t ≤ 109 1 ≤ lengths[i] ≤ 109 The sum of the elements of lengths equals the uncut rod length.

Input Format For Custom Testing The first line contains an integer, n, the number of elements in lengths.

Each line i of the n subsequent lines (where 0 ≤ i < n) contains an integer, lengths[i].

The next line contains an integer, minLength, the minimum length accepted by the machine.

Sample Case 0 Sample Input For Custom Testing

STDIN Function

4 → lengths[] size n = 4 3 → lengths[] = [3, 5, 4, 3] 5 4 3 9 → minLength= 9 Sample Output

Possible Explanation

The uncut rod is 3 + 5 + 4 + 3 = 15 units long. Cut the rod into lengths of 3 + 5 + 4 = 12 and 3. Then cut the 12 unit piece into lengths 3 and 5 + 4 = 9. The remaining segment is 5 + 4 = 9 units and that is long enough to make the final cut.

Sample Case 1 Sample Input For Custom Testing

STDIN Function

3 → lengths[] size n = 3 5 → lengths[] = [5, 6, 2] 6 2 12 → minLength= 12 Sample Output

Impossible Explanation

The uncut rod is 5 + 6 + 2 = 13 units long. After making either cut, the rod will be too short to make the second cut.

Solution

To solve this problem, you can follow these steps:

Initial Check: First, sum all elements in the lengths array. If the sum is less than minLength, return “Impossible”.
Iterative Check: Start from the beginning of the lengths array and keep a running sum.
- At each step, check if the remaining length of the rod (totalLength - runningSum) is greater than or equal to minLength.
- If yes, then we should also check if the runningSum itself can be broken down into the lengths in the array leading up to this point.
Final Check: If you find such a configuration, return “Possible”. If you exhaust all possibilities, return “Impossible”.

Here’s a Python function to implement this algorithm:

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def cutThemAll(lengths, minLength):
    totalLength = sum(lengths)
    
    # Step 1: Initial Check
    if totalLength < minLength:
        return "Impossible"
    
    runningSum = 0
    for i in range(len(lengths)):
        runningSum += lengths[i]
        
        # Step 2: Iterative Check
        if totalLength - runningSum >= minLength:
            tempSum = 0
            for j in range(i + 1):
                tempSum += lengths[j]
                if runningSum - tempSum == minLength:
                    # Step 3: Final Check
                    return "Possible"
                    
    return "Impossible"

# Sample Test 1
print(cutThemAll([3, 5, 4, 3], 9))  # Output should be "Possible"

# Sample Test 2
print(cutThemAll([5, 6, 2], 12))  # Output should be "Impossible"

In this function, runningSum keeps track of the length of the rod at the current step, and tempSum checks if the remaining lengths in the array can sum up to the minLength.

This should solve the problem as specified.