Asked in: DOCUSIGN
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All Testcases Passed ✔
Solution
def getMinVulnerablity(vulnerablity, k):
def helper(middle_value):
result = 0
current = 0
// ... rest of solution available after purchaseExplanation
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To solve the problem of selecting a subsequence of exactly k servers from an array of vulnerabilities, such that no two chosen servers are adjacent, and minimizing the maximum vulnerability among the chosen servers, we need a strategy that balances combinatorial selection with optimization constraints. Here’s a detailed thought process and approach framework:
1. **Understanding the Problem:**
We are given:
- An array of vulnerabilities representing servers.
- We want to select exactly k servers from this array.
- The chosen servers must not be adjacent (i.e., if you pick server i, you cannot pick i-1 or i+1).
- Among all valid subsequences that satisfy the above, we want to minimize the maximum vulnerability of the chosen servers.
This is an optimization problem with a combinatorial constraint (no adjacent picks) and an objective function (minimize the maximum vulnerability in the subsequence).
2. **Key Constraints and Observations:**
- Since we must pick exactly k servers, and no two are adjacent, the maximum k allowed is at most (n + 1)/2, because you can't pick more servers than half the array without adjacency.
- Minimizing the maximum vulnerability means you want to keep the chosen servers' vulnerabilities as low as possible, but still satisfy the adjacency and count constraints.
- The adjacency condition heavily restricts which indices can be chosen together, creating a sparse subsequence.
- We want to find the minimum possible "maximum vulnerability" in any valid subsequence of length k.
3. **Potential Approaches to the Problem:**
**A. Brute Force Approach (Impractical):**
- Enumerate all subsequences of length k with no two adjacent servers.
- For each subsequence, compute the max vulnerability.
- Return the minimal maximum.
- This approach is exponentially expensive and infeasible for n up to 10^5.
**B. Greedy or Dynamic Programming:**
- Because of the adjacency restriction, dynamic programming could be used to find the maximum or minimum sum of selected elements.
- But here, the problem is to minimize the maximum element chosen, not sum or count.
- DP solutions would be complex since the "max" function isn't additive, but we could explore if DP can be adapted with binary search or pruning.
**C. Binary Search on the Answer:**
- This is a common technique for problems where you minimize or maximize a value under constraints.
- The idea is to guess a maximum vulnerability threshold M and check if it's possible to pick k servers, non-adjacent, all with vulnerabilities ≤ M.
- If yes, try a smaller M; if no, try a larger M.
- This reduces the problem to a decision problem for a fixed M.
4. **Implementing the Decision Problem (Feasibility Check):**
For a fixed M:
- Construct a boolean array where each position is True if vulnerability[i] ≤ M, False otherwise.
- Now the problem reduces to picking k servers (indices) from the True positions, with no two adjacent indices selected.
- We need to check if it is possible to select k True positions such that no two are adjacent.
5. **Checking Feasibility Efficiently:**
To check if you can pick k servers non-adjacent from positions marked True:
- Iterate through the array.
- Whenever you encounter a True position, you can choose it (increment count), then skip the next position (due to adjacency constraint).
- Continue this greedy selection until you either pick k servers or finish the array.
- If you manage to pick k servers, it means for the current M, the subsequence exists.
6. **Binary Search Range:**
- The minimal possible vulnerability is at least the minimum value in the array.
- The maximum possible vulnerability is the maximum value in the array.
- Perform binary search in this range to find the minimum M for which the feasibility test passes.
7. **Algorithm Summary:**
- Set low = min(vulnerability), high = max(vulnerability).
- While low < high:
- mid = (low + high) // 2
- Check feasibility with threshold mid.
- If feasible, update high = mid.
- Else, update low = mid + 1.
- After the loop, low will hold the minimal maximum vulnerability possible.
8. **Time Complexity and Optimization:**
- Each feasibility check is O(n) since it scans through the array once.
- Binary search performs O(log(max_val - min_val)) iterations.
- Overall complexity is O(n log(max_val - min_val)), which is efficient for the constraints.
9. **Additional Points:**
- Pay attention to edge cases like when k=1 (just pick the smallest vulnerability).
- When all vulnerabilities are the same, answer is that vulnerability.
- Ensure careful handling of the adjacency condition during feasibility.
10. **Why This Approach Works:**
- Binary search on the answer space transforms a complex combinatorial optimization into a series of decision problems.
- The decision problem itself is simple and solvable greedily.
- Combining these techniques results in an efficient and elegant solution.
By thinking along these lines, you move from an intractable brute force idea to a feasible and scalable approach leveraging binary search and greedy feasibility checks to find the minimum possible maximum vulnerability under the constraints given.
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