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This one took me a while to even understand what was being asked.
First, clarify the problem statement and constraints, as the phrasing is ambiguous. Then, model the effect of flipping a subarray on the number of operational servers, and determine how to maximize the number of distinct counts achievable. Consider the range of possible counts and whether all values in that range can be achieved.
Pro tip: Don't rush to code; demonstrate strong communication by restating the problem in your own words and confirming assumptions with the interviewer. This shows you prioritize understanding over speed.
Ask questions to resolve ambiguities: What exactly is 'redundancy'? Is it the number of distinct counts of 1s across all possible configurations after any number of flips? Are we allowed to flip any subarray any number of times?
Let the initial array have k ones. Flipping a subarray of length L changes the number of ones by L - 2*(number of ones in the subarray). The new count can be any integer between 0 and n, but not all may be reachable.
Analyze which counts of 1s are achievable. Since you can flip any subarray, you can effectively change the count by any even number? Actually, flipping a single element changes count by ±1, so any count between 0 and n is achievable if you can flip individual elements. But flipping a subarray flips multiple elements at once; however, by flipping overlapping subarrays, you can achieve any count.
If any count from 0 to n is achievable, then the maximum redundancy is n+1. But check if there are constraints: e.g., if you can only flip once, then the reachable counts are limited. The problem says 'you can pick any consecutive subarray and flip all values in it' — it doesn't specify a limit on the number of flips. So likely multiple flips are allowed, making all counts reachable.
Verify with small n: n=1, initial [0] -> flip [0] gives [1], counts {0,1} -> redundancy 2 = n+1. n=2, initial [0,0] -> flip [0,0] gives [1,1] count 2; flip [0] gives [1,0] count 1; so counts {0,1,2} -> redundancy 3. So answer is n+1.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Graph problem with a constraint on edges you can actually traverse.
Model the network as a graph where edges connect devices with frequency difference ≤1. The problem reduces to finding the diameter of each connected component, then taking the maximum across all components. Use BFS from any node to find the farthest node, then BFS from that node to find the diameter.
Pro tip: Clarify whether the graph is a tree or general graph; if general, the double-BFS method only works for trees, so you may need to run BFS from every node or use more advanced algorithms. Also, consider edge cases like disconnected components and single-node components.
Represent devices as nodes and add edges between devices whose frequency difference is at most 1. This creates a graph where communication corresponds to connectivity.
Use BFS or DFS to find all connected components. The maximum distance must be within a single component.
For each component, compute its diameter (longest shortest path). If the component is a tree, use double BFS; otherwise, use BFS from each node or Floyd-Warshall for small graphs.
Take the maximum diameter across all components as the final answer. Handle edge cases like empty graph or isolated nodes.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
This is the kind of problem where brute force is obvious but clearly won't scale.
Sort the array and use a contribution technique: for each element as the maximum, compute its contribution to the sum over all subsets where it is the maximum. The contribution of the i-th smallest element is arr[i] * (sum_{k=1}^{i+1} k * C(i, k-1)) modulo 1e9+7. Precompute binomial coefficients and powers of two to compute the sum efficiently.
Pro tip: Derive a closed-form for the inner sum: sum_{k=1}^{m} k * C(m-1, k-1) = 2^{m-2} * (m+1) for m >= 1, which simplifies the computation to O(n log n) after sorting. Mention that this avoids O(n^2) and shows strong mathematical insight.
Confirm the problem details: subsets are non-empty, order doesn't matter, and modulo is 1e9+7. Sort the array to easily identify maximums.
For each element, consider it as the maximum of a subset. Count how many subsets have this element as maximum and compute the sum of sizes of those subsets.
If the element is at index i (0-based) in sorted order, there are i elements smaller. For a subset of size k (1 ≤ k ≤ i+1), choose k-1 from the i smaller elements. The sum of sizes is sum_{k=1}^{i+1} k * C(i, k-1).
Use the identity sum_{k=1}^{m} k * C(m-1, k-1) = 2^{m-2} * (m+1) for m ≥ 1, where m = i+1. This reduces the per-element computation to O(1) after precomputing powers of two.
Iterate through the sorted array, compute each element's contribution as arr[i] * (2^{i-1} * (i+2)) mod MOD, sum them up, and return the result modulo 1e9+7.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.