feat(algorithms, intervals): car pooling

DIRECTORY.md

@@ -106,6 +106,8 @@
     * [Decoding](https://github.com/BrianLusina/PythonSnips/blob/master/algorithms/huffman/decoding.py)
     * [Encoding](https://github.com/BrianLusina/PythonSnips/blob/master/algorithms/huffman/encoding.py)
   * Intervals
+    * Car Pooling
+      * [Test Car Pooling](https://github.com/BrianLusina/PythonSnips/blob/master/algorithms/intervals/car_pooling/test_car_pooling.py)
     * Count Days
       * [Test Count Days Without Meetings](https://github.com/BrianLusina/PythonSnips/blob/master/algorithms/intervals/count_days/test_count_days_without_meetings.py)
     * Full Bloom Flowers

algorithms/intervals/car_pooling/README.md

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+# Car Pooling
+
+You are given a car with a fixed number of seats, denoted by an integer capacity. The car only travels in one direction 
+— eastward — and does not make any U-turns.
+
+You are also provided with an array, trips, where each element trips[i]= [numPassengersi, fromi, toi] represents a group of 
+numPassengersi that must be picked up at location fromi and dropped off at location toi . All locations are measured in 
+kilometers east of the starting point.
+
+Your task is to determine whether it is possible to complete all the trips without exceeding the car’s capacity at any
+point in time.
+
+Return TRUE if all trips can be completed successfully, or FALSE otherwise.
+
+## Constraints
+
+- 1 <= `trips.length` <= 1000
+- `trips[i].length` == 3
+- 1 <= `numPassengers` <= 1000
+- 0 <= fromi < toi <= 1000
+- 1 <= `capacity` <= 10^5
+
+## Examples
+
+![Example 1](./images/examples/car_pooling_example_1.png)
+![Example 2](./images/examples/car_pooling_example_2.png)
+![Example 3](./images/examples/car_pooling_example_3.png)
+
+
+## Solution
+
+The core intuition behind this solution is to simulate the journey of the car using a timeline, tracking when passengers
+get in and get out. This approach aligns with the merge intervals pattern, where we focus on events happening at specific
+points in time, rather than handling each trip separately in isolation. The idea is to accumulate and monitor the number
+of passengers in the car at any moment using a difference array (also known as a prefix sum technique).
+
+In the context of this problem, each trip defines a passenger change over a specific interval — passengers are picked up
+at from and dropped off at to. Instead of iterating over the entire interval (which is inefficient), we can track just
+the start and end of the interval using a fixed-size timeline array.
+
+This strategy is built on two key observations:
+
+1. Each trip contributes a net passenger change at exactly two locations:
+   - Increase at the pickup location (from)
+   - Decrease at the drop-off location (to)
+
+2. The car’s capacity must never be exceeded at any point on the journey, so we monitor the cumulative passenger count
+   as we move forward in time.
+
+This strategy leverages the difference array pattern, which is a version of merge intervals. Rather than merging 
+overlapping intervals explicitly, it simulates all trips together via a timeline of passenger changes.
+
+Let’s break down the steps:
+
+1. A list timestamp of size 1001 (based on constraints) is created to track changes in passenger counts at each kilometer
+   point.
+2. For each trip `[numPassengersi ,fromi, toi]` in trips:
+   - Add numPassengers at `timestamp[fromi]`
+   - Subtract numPassengers at `timestamp[toi]`
+
+   This marks the start and end of each passenger interval without iterating over the full range.
+
+3. To simulate the trip and track capacity, initialize a variable used_capacity = 0. 
+4. Iterate through the timestamp list, adding the passenger changes at each point. If at any point used_capacity > capacity,
+   return FALSE immediately— the car is overloaded. 
+5. If the full timeline is processed without exceeding capacity, return TRUE, indicating that all trips can be accommodated.
+
+Let’s look at the following illustration to get a better understanding of the solution:
+
+![Solution 1](./images/solutions/car_pooling_solution_1.png)
+![Solution 2](./images/solutions/car_pooling_solution_2.png)
+![Solution 3](./images/solutions/car_pooling_solution_3.png)
+![Solution 4](./images/solutions/car_pooling_solution_4.png)
+![Solution 5](./images/solutions/car_pooling_solution_5.png)
+![Solution 6](./images/solutions/car_pooling_solution_6.png)
+![Solution 7](./images/solutions/car_pooling_solution_7.png)
+![Solution 8](./images/solutions/car_pooling_solution_8.png)
+![Solution 9](./images/solutions/car_pooling_solution_9.png)
+![Solution 10](./images/solutions/car_pooling_solution_10.png)
+![Solution 11](./images/solutions/car_pooling_solution_11.png)
+![Solution 12](./images/solutions/car_pooling_solution_12.png)
+
+### Time Complexity
+
+The overall time complexity of the solution is O(n) because:
+
+- The trips list is scanned once to record passenger changes at pickup and drop-off points.
+- Each trip results in two constant-time operations: one increment and one decrement in the timeline array. 
+- The timestamp array (of fixed length 1001) is traversed once to simulate the journey and check capacity constraints.
+
+As the length of timestamp is constant and independent of input size, it contributes O(1) time. So, the total operations 
+scale linearly with the number of trips.
+
+### Space Complexity
+
+The space complexity of the solution is O(1) because a fixed-size array timestamp of length 1001 is used regardless of 
+the input size.

algorithms/intervals/car_pooling/__init__.py

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+from typing import List, Tuple
+
+
+def car_pooling(trips: List[List[int]], capacity: int) -> bool:
+    """
+    Calculates and checks whether it is possible to pick and drop of passengers on the given trips given the cars
+    capacity
+    Args:
+        trips(list): The trips that the car makes while collecting passengers on the route
+        capacity(int): capacity of the car
+    Returns:
+        bool: whether it is possible to collect and drop all passengers along the route
+    """
+    # For every trip, create two entries, the start with the passenger count and end with the passenger count
+    events: List[Tuple[int, int]] = []
+
+    # Create the events
+    for trip in trips:
+        num, start, end = trip
+        # Mark the increase and decrease in passengers at pickup and drop-off points
+        events.append((start, num))  # pick up
+        events.append((end, -num))  # drop off
+
+    # sort by the location
+    # If locations are equal, the negative value (drop-off)
+    # will naturally come before the positive value (pick-up)
+    events.sort()
+
+    # This keeps track of the current vehicle's capacity, which will be used along the trip to check how many passengers
+    # have been carried so far
+    current_occupancy = 0
+
+    # process events chronologically
+    for event in events:
+        location, change = event
+        current_occupancy += change
+        if current_occupancy > capacity:
+            return False
+
+    return True
+
+
+def car_pooling_bucket(trips: List[List[int]], capacity: int) -> bool:
+    # Initialize a timeline to track changes in passenger count at each km mark (0 to 1000)
+    timestamp = [0] * 1001
+
+    # Mark the increase and decrease in passengers at pickup and drop-off points
+    for trip in trips:
+        num_passengers, start, end = trip
+        timestamp[start] += num_passengers  # Pick up passengers
+        timestamp[end] -= num_passengers  # Drop off passengers
+
+    # Simulate the car's journey by applying the passenger changes
+    used_capacity = 0
+    for passenger_change in timestamp:
+        used_capacity += passenger_change  # Update current passenger count
+        if used_capacity > capacity:  # Check if capacity is exceeded
+            return False  # Trip configuration is invalid
+
+    return True  # All trips are valid within capacity

algorithms/intervals/car_pooling/test_car_pooling.py

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+import unittest
+from typing import List
+from parameterized import parameterized
+from algorithms.intervals.car_pooling import car_pooling, car_pooling_bucket
+
+CAR_POOLING_TEST_CASES = [
+    ([[3, 2, 6], [1, 4, 7], [2, 5, 8]], 5, False),
+    ([[2, 0, 4], [3, 2, 6], [1, 5, 8]], 4, False),
+    ([[1, 0, 3], [2, 2, 5], [3, 4, 6]], 6, True),
+    ([[2, 1, 5], [3, 3, 7]], 4, False),
+    ([[2, 1, 5], [3, 3, 7]], 5, True),
+    ([[3, 2, 6], [1, 4, 7], [2, 5, 8]], 5, False),
+    ([[1, 0, 4], [2, 2, 6], [3, 5, 8]], 6, True),
+    ([[50, 0, 50], [51, 1, 51]], 100, False),
+    ([[25, 0, 4], [25, 2, 6], [25, 5, 8]], 50, True),
+]
+
+
+class CarPoolingTestCases(unittest.TestCase):
+    @parameterized.expand(CAR_POOLING_TEST_CASES)
+    def test_car_pooling(self, trips: List[List[int]], capacity: int, expected: bool):
+        actual = car_pooling(trips, capacity)
+        self.assertEqual(expected, actual)
+
+    @parameterized.expand(CAR_POOLING_TEST_CASES)
+    def test_car_pooling_bucket_sort(
+        self, trips: List[List[int]], capacity: int, expected: bool
+    ):
+        actual = car_pooling_bucket(trips, capacity)
+        self.assertEqual(expected, actual)
+
+
+if __name__ == "__main__":
+    unittest.main()