Optimal Bleed Order
Dr. Karlo is researching the impact of various substance infusion orders on patient recovery times in severe bleeding cases. He has developed a procedure called bleed control where substances (e.g., anticoagulants, platelet-rich plasma) need to be administered in a specific sequence to optimize recovery. The effectiveness of each sequence can theoretically be represented using a unique integer score. You are given a list of substances and their pairwise compatibility scores in a matrix form. Your task is to determine the optimal sequence of administering these substances to maximize the total compatibility score, which ultimately correlates with faster and safer recovery. The sequence must use each substance exactly once.
[ [ 0, 2, 3 ], [ 2, 0, 1 ], [ 3, 1, 0 ] ]
Explanation. An optimal sequence is 1 -> 0 -> 2 giving scores of 2 (from 1 to 0) + 3 (from 0 to 2) = 5 or 2 -> 1 -> 0 giving scores of 1 (from 2 to 1) + 2 (from 1 to 0) = 3 or 2 -> 0 -> 1 which gives 3 (from 2 to 0) + 2 (from 0 to 1) = 5. Here two orders give maximum score of 5.
Follow-up: Can this problem be solved more efficiently using non-recursive approaches? Discuss alternative solutions and their potential time-space complexities.
1. All input integers (substances and scores) are zero or positive.\n2. The number of substances N will be at least 1 and no more than 10, due to the complexity and practical limits of administering multiple substances.\n3. The matrix will be square (N x N), where each element `[i][j]` represents the compatibility score between substance i and j when i is administered immediately before j.
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