# 5 Best Methods to Find the Number of Tasks Worked on Within Specific Intervals in Python

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π‘ Problem Formulation: The challenge is to calculate how many tasks are being worked on during different periods of time. Imagine you are tracking task progress throughout the day. You have start and end times for each task, and you want to know the number of tasks ongoing during specific intervals. For example, given tasks with intervals `[(1, 4), (2, 3), (3, 5)]` and query intervals like `[(1, 2), (2, 3)]`, you want to determine how many tasks are active during each query interval: here, 1 task in the first interval, and 2 tasks in the second.

## Method 1: Brute Force Comparison

This method involves iterating over each task interval and query interval to count the tasks that are active within each query interval’s bounds. It’s simple and straightforward, suitable for a small number of tasks, but becomes less efficient as the number of tasks grows due to its O(n*m) time complexity, where n is the number of tasks and m is the number of query intervals.

Here’s an example:

```def count_tasks_in_intervals(tasks, queries):
return [sum(start <= q_start < end for start, end in tasks) for q_start, q_end in queries]

tasks = [(1, 4), (2, 3), (3, 5)]
# Query intervals
queries = [(1, 2), (2, 3)]

```

Output:

`[1, 2]`

This code snippet defines a function `count_tasks_in_intervals` that takes the list of tasks and queries and returns a list with the count of tasks active for each query interval. It uses list comprehension to iterate and count the tasks active in each interval, which while comprehensible and simplistic in small cases, may be inefficient for larger datasets.

## Method 2: Sorting and Pointers

Method 2 optimizes the search by first sorting the intervals and then using pointers to efficiently traverse and count task overlaps. Sorting the task intervals helps in reducing the number of comparisons necessary. This improves the time complexity slightly but introduces an initial cost of sorting.

Here’s an example:

```def count_tasks_in_intervals(tasks, queries):
results = []
for q_start, q_end in queries:
count = 0
if start < q_end:
if end > q_start:
count += 1
else:
break
results.append(count)
return results

tasks = [(1, 4), (2, 3), (3, 5)]
queries = [(1, 2), (2, 3)]

```

Output:

`[1, 2]`

In this method, after sorting the tasks based on their start times, the nested loops are now more efficient because we can break out of the inner loop earlier due to the sorted order of task intervals. It results in slightly better performance than brute force, especially when the intervals have a large disparity in their time ranges.

## Method 3: Using Interval Trees

An interval tree is a data structure that allows efficient searching of all intervals that overlap with a given interval or point. This method offers significant performance improvements for larger sets of data, particularly when many overlaps are expected, as the interval tree is built in O(n log n) time complexity and querying is done in O(log n + m) time, where m is the number of overlapping intervals.

Here’s an example:

```from intervaltree import IntervalTree

tree = IntervalTree()
tree[start:end] = True
return [len(tree[q_start:q_end]) for q_start, q_end in queries]

tasks = [(1, 4), (2, 3), (3, 5)]
queries = [(1, 2), (2, 3)]

```

Output:

`[1, 2]`

The code snippet demonstrates how to use the `intervaltree` Python library to construct an interval tree from the given task intervals and then query this tree to count the number of tasks in each query interval. This approach is more sophisticated and scales well with the size of the input data.

## Method 4: Segment Trees

Segment trees are another type of data structure suitable for scenarios involving interval queries. They offer a balance between tree depth and query/update efficiency, performing both operations in O(log n) time after an O(n log n) build time. They are especially useful for dynamic datasets where intervals can be frequently added or removed.

Here’s an example:

```# Segment tree implementation is omitted for brevity
# Assuming the existence of a SegmentTree class
# Segment tree-based implementation would go here
pass

# This code is solely conceptual as a full implementation of a segment tree is complex and lengthy.
# Refer to an actual Segment Tree library or implementation for functional code.
```

The segment tree implementation would require significantly more code and is not shown here in full, but it would involve building a tree from the task intervals and then querying it to get the count for each interval. This method requires some understanding of advanced data structures and algorithms.

## Bonus One-Liner Method 5: Using List Comprehensions and Python’s `sum` Function

A one-liner approach using list comprehensions and Python’s built-in `sum` function can be a concise (although not necessarily the most efficient) way to achieve our goal.

Here’s an example:

```tasks = [(1, 4), (2, 3), (3, 5)]
queries = [(1, 2), (2, 3)]

print([[sum(start <= q_start < end for start, end in tasks) for q_start, q_end in queries] for q_start, q_end in queries])
```

Output:

`[[1, 2], [1, 2]]`

This code snippet uses a nested list comprehension to create a list of lists that contains the counts of active tasks for each interval. While extremely concise, this approach is essentially a condensed version of the brute force method and suffers from the same inefficiency in time complexity.

## Summary/Discussion

• Method 1: Brute Force Comparison. Strengths: Easy to understand and implement. Good for small datasets. Weaknesses: Poor scalability with large datasets.
• Method 2: Sorting and Pointers. Strengths: Improved efficiency over brute force for datasets with large disparities in intervals. Weaknesses: Initial sort adds overhead, and it’s not the most optimal method for dynamic datasets.
• Method 3: Using Interval Trees. Strengths: Significant performance improvements, especially with many intervals. Good for large, static datasets. Weaknesses: Requires additional library and understanding of interval trees.
• Method 4: Segment Trees. Strengths: Balanced performance for both querying and updating intervals. Good for dynamic datasets. Weaknesses: Complex to understand and implement.
• Method 5: One-Liner List Comprehension. Strengths: Extremely concise and elegant. Weaknesses: Inefficient and not practical for large datasets.