## Prompt

## Answer

# Process Scheduling Algorithm: Minimum Cores Required

## Problem Analysis

The problem at hand involves determining the minimum number of CPU cores necessary to execute a set of processes that have defined start and end times. The challenge arises from the overlap of the execution times of these processes.

## Solution Approach

To address this, we can utilize a **sweep line algorithm** where we treat both the start and end times as events and track the number of active processes at any given time. The core idea involves:

**Event Creation**: Create events for each process' start and end time.**Event Sorting**: Sort these events, prioritizing end events before start events if they occur at the same time.**Core Counting**: Use a counter to track the current number of cores needed as we process each event sequentially.

### Key Considerations:

- Start events will increase the core requirement.
- End events will decrease the core requirement.
- The maximum count during this process will yield the minimum required cores.

## Implementation

Here’s how to implement the aforementioned strategy in Python:

```
def getMinCores(start, end):
# Create events for start (+1) and end (-1) times.
events = []
for i in range(len(start)):
events.append((start[i], 1)) # Process start
events.append((end[i] + 1, -1)) # Process end (exclusive end)
# Sort events; end events should come before start in case of a tie.
events.sort(key=lambda x: (x[0], x[1]))
max_cores = 0
current_cores = 0
# Sweep line algorithm to count the cores required.
for event in events:
current_cores += event[1] # Update current cores
max_cores = max(max_cores, current_cores) # Keep track of maximum cores needed
return max_cores
# Example usage:
if __name__ == '__main__':
start = [1, 3, 4]
end = [3, 5, 6]
print(getMinCores(start, end)) # Output: 2
```

### Code Explanation:

**Event Creation**: For each process, we create and append two tuples: one for the start time and one for the end time incremented by one (to denote that the process is not running after`end`

).**Sorting**: Sorting is done to ensure that in case of simultaneous events, the end event takes precedence.**Counting Cores**: A loop iterates over the sorted events and updates the count of current cores based on whether it’s a start (+1) or end (-1) event, simultaneously tracking the maximum core usage.

### Complexity:

**Time Complexity**: O(n log n) due to sorting.**Space Complexity**: O(n) for the events array.

## Conclusion

This approach efficiently calculates the minimum number of CPU cores required to run overlapping processes while adhering to constraints and ensuring performance considerations. The provided code is ready for testing and integration into larger systems.

## Description

This program uses a sweep line algorithm to determine the minimum number of CPU cores needed to execute overlapping processes defined by their start and end times, effectively managing core utilization in processor scheduling.