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Pretty straightforward setup question, basically just grouping and subtracting.
First, clarify the input format: whether usage records are given as a list of (day, cluster, usage) tuples or a 2D array. Then, aggregate usage per cluster per day using a hash map or by sorting, and compute remaining capacity as fixed capacity minus total usage for each distinct day, outputting rows in ascending day order.
Pro tip: Mention that you would handle edge cases like usage exceeding capacity (remaining could be negative) and ensure the output includes all clusters for every distinct day, even if a cluster had zero usage that day.
Ask whether usage is given as a list of records or a matrix, and confirm that output should be a 2D array with rows for each distinct day in ascending order and columns for each cluster.
Collect all unique days from the input and sort them in ascending order to determine the row order of the output.
Use a hash map keyed by day and cluster to sum usage, or sort records by day and cluster and accumulate. This handles multiple records for the same day and cluster.
For each distinct day and each cluster, subtract the aggregated usage from the fixed daily capacity. If no usage exists for a cluster on a day, the remaining capacity is the full capacity.
Construct the 2D array by iterating over sorted days and clusters, placing remaining capacities in the correct order. Discuss time and space complexity.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
No constraint on switching so you just take the max remaining capacity across clusters for each day and sum them up.
Model the problem as a maximum flow or bipartite matching where each day's cluster capacity is a source of GPUs and the workload can draw from any cluster each day. Since there is no switching limit, the total maximum GPUs is simply the sum of the maximum daily capacities, but verify if the workload has a total demand or per-day constraints. If the workload can freely pick any cluster each day, the answer is the sum of the maximum capacity per day across all clusters.
Pro tip: Clarify whether the workload can use multiple clusters on the same day and whether there is a total GPU requirement; if not, the problem reduces to summing daily maxima, but mentioning flow-based generalization shows depth.
Confirm that the workload can choose any cluster each day without switching penalties and that there is no limit on the number of clusters used per day. Identify if there is a total demand or if we are maximizing total GPUs over the period.
If the goal is to maximize total GPUs accessed over all days, and the workload can use all available capacity each day, then the maximum is the sum over days of the maximum capacity among clusters for that day.
For each day, find the cluster with the highest remaining capacity. Sum these daily maxima to get the total maximum GPUs.
If the workload has a fixed total demand or per-day demand, or if there are constraints like using only one cluster per day, then model as a flow problem. But given 'freely pick any cluster each day with no switching limit', the sum of daily maxima is correct.
Test with small examples to ensure the sum of daily maxima matches the expected maximum. If there are dependencies between days (e.g., capacity changes due to usage), adjust accordingly.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Model the problem as a dynamic programming over days and clusters, where the state includes the number of switches used so far. For each day, decide to stay on the current cluster or switch to another, updating the maximum GPUs accordingly. Optimize by precomputing prefix sums of GPUs per cluster to quickly compute daily gains.
Pro tip: Clarify whether K is the maximum number of switches allowed or exactly K, and whether switching on the same day is allowed. Also, discuss time and space complexity trade-offs, and consider if K is small enough for a DP with state (day, cluster, switches) or if a more efficient approach like divide and conquer DP optimization is needed.
Let dp[i][j][k] represent the maximum GPUs obtainable up to day i, ending on cluster j, having used exactly k switches. Initialize for day 0 with 0 switches.
For each day i and cluster j, either stay on j (no switch) or switch from another cluster l (switch count increases by 1). Take the maximum over all possibilities.
Precompute prefix sums of GPUs per cluster to compute the total GPUs gained from day a to day b on a cluster in O(1) time, enabling efficient transition calculations.
Initialize dp for day 0, iterate through days, and finally take the maximum over all clusters and switch counts up to K. Return the maximum GPUs.
The naive DP is O(D * C^2 * K). If C or K is large, consider optimizations like maintaining best previous states or using divide and conquer DP if the cost function is monotonic.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.