use strict;
my $str = 'Here’s an extended version of the previous explanation, now including **problem definitions**, **approaches**, and **examples** for each interval problem:
---
### 1. **Basic Interval Problems**
#### a) **Finding Overlaps Between Two Intervals**
- **Problem Definition**: Given two intervals, determine if they overlap. Two intervals overlap if they share at least one point in common.
- **Approach**:
- Check if the start of one interval is before the end of the other and vice versa.
- **Example**:
- Interval A = [1, 5], Interval B = [4, 9].
- Output: True (they overlap).
```python
def do_intervals_overlap(interval1, interval2):
return interval1[0] <= interval2[1] and interval2[0] <= interval1[1]
# Example
interval1 = [1, 5]
interval2 = [4, 9]
print(do_intervals_overlap(interval1, interval2)) # True
```
#### b) **Union of Two Intervals**
- **Problem Definition**: Given two intervals, return the union of the intervals if they overlap or are adjacent. If they do not overlap, return both intervals.
- **Approach**:
- If the intervals overlap, return the minimum start and maximum end of the two intervals.
- **Example**:
- Interval A = [1, 5], Interval B = [4, 9].
- Output: [1, 9].
```python
def union_intervals(interval1, interval2):
if do_intervals_overlap(interval1, interval2):
return [min(interval1[0], interval2[0]), max(interval1[1], interval2[1])]
return [interval1, interval2]
# Example
print(union_intervals([1, 5], [4, 9])) # [1, 9]
```
#### c) **Intersection of Two Intervals**
- **Problem Definition**: Find the intersection of two intervals. The intersection is the range that is covered by both intervals.
- **Approach**:
- If the intervals overlap, the intersection is the maximum of the start points and the minimum of the end points.
- **Example**:
- Interval A = [1, 5], Interval B = [3, 9].
- Output: [3, 5].
```python
def intersection_intervals(interval1, interval2):
if do_intervals_overlap(interval1, interval2):
return [max(interval1[0], interval2[0]), min(interval1[1], interval2[1])]
return None
# Example
print(intersection_intervals([1, 5], [3, 9])) # [3, 5]
```
---
### 2. **Multiple Intervals Problems**
#### a) **Merging Overlapping Intervals**
- **Problem Definition**: Given a set of intervals, merge all the overlapping ones.
- **Approach**:
- Sort the intervals by their start time, then iterate through them and merge overlapping intervals.
- **Example**:
- Intervals = [[1, 5], [2, 6], [8, 10]].
- Output: [[1, 6], [8, 10]].
```python
def merge_intervals(intervals):
intervals.sort(key=lambda x: x[0])
merged = [intervals[0]]
for i in range(1, len(intervals)):
if merged[-1][1] >= intervals[i][0]:
merged[-1][1] = max(merged[-1][1], intervals[i][1])
else:
merged.append(intervals[i])
return merged
# Example
print(merge_intervals([[1, 5], [2, 6], [8, 10]])) # [[1, 6], [8, 10]]
```
#### b) **Finding Gaps Between Intervals**
- **Problem Definition**: Given a set of intervals, find the gaps where no interval exists.
- **Approach**:
- Sort intervals by start time. Identify gaps by checking the end of one interval and the start of the next.
- **Example**:
- Intervals = [[1, 5], [7, 10]].
- Output: [5, 7].
```python
def find_gaps(intervals):
intervals.sort(key=lambda x: x[0])
gaps = []
for i in range(1, len(intervals)):
if intervals[i][0] > intervals[i-1][1]:
gaps.append([intervals[i-1][1], intervals[i][0]])
return gaps
# Example
print(find_gaps([[1, 5], [7, 10]])) # [[5, 7]]
```
---
### 3. **Complex Interval Operations**
#### a) **Interval Difference**
- **Problem Definition**: Given two intervals, find the difference between them, i.e., the part of the first interval that does not overlap with the second.
- **Approach**:
- If the intervals overlap, return the non-overlapping portions of the first interval.
- **Example**:
- Interval A = [1, 10], Interval B = [5, 7].
- Output: [[1, 5], [7, 10]].
```python
def interval_difference(A, B):
if not do_intervals_overlap(A, B):
return [A] # No overlap
result = []
if A[0] < B[0]:
result.append([A[0], B[0]])
if A[1] > B[1]:
result.append([B[1], A[1]])
return result
# Example
print(interval_difference([1, 10], [5, 7])) # [[1, 5], [7, 10]]
```
#### b) **Interval Scheduling**
- **Problem Definition**: Given a set of intervals, find the maximum number of non-overlapping intervals that can be selected.
- **Approach**:
- Sort intervals by end time. Greedily select intervals that do not overlap with the previously selected one.
- **Example**:
- Intervals = [[1, 3], [2, 4], [3, 5]].
- Output: [[1, 3], [3, 5]].
```python
def interval_scheduling(intervals):
intervals.sort(key=lambda x: x[1]) # Sort by end times
result = []
last_end = float(\'-inf\')
for interval in intervals:
if interval[0] >= last_end:
result.append(interval)
last_end = interval[1]
return result
# Example
print(interval_scheduling([[1, 3], [2, 4], [3, 5]])) # [[1, 3], [3, 5]]
```
---
### 4. **Advanced Interval Challenges**
#### a) **Finding the Smallest Range Covering All Points**
- **Problem Definition**: Given several lists of intervals, find the smallest range that includes at least one interval from each list.
- **Approach**:
- Use a sliding window and min-heap to track the smallest range covering all lists.
- **Example**:
- Lists = [[1, 5], [6, 9], [7, 10]].
- Output: (6, 7).
```python
import heapq
def smallest_range(lists):
min_heap = []
max_val = float(\'-inf\')
# Add the first element of each list to the heap
for i, l in enumerate(lists):
heapq.heappush(min_heap, (l[0], i, 0))
max_val = max(max_val, l[0])
min_range = float(\'inf\'), None, None
while min_heap:
min_val, list_idx, element_idx = heapq.heappop(min_heap)
if max_val - min_val < min_range[0]:
min_range = (max_val - min_val, min_val, max_val)
if element_idx + 1 == len(lists[list_idx]):
break # We\'ve reached the end of one list
next_val = lists[list_idx][element_idx + 1]
max_val = max(max_val, next_val)
heapq.heappush(min_heap, (next_val, list_idx, element_idx + 1))
return min_range[1], min_range[2]
# Example
lists = [[1, 5], [6, 9], [7, 10]]
print(smallest_range(lists)) # (6, 7)
```
#### b) **K-Interval Coverage**
- **Problem Definition**: Given a set of intervals and an integer `k`, find the maximum number of intervals that cover any point or range.
- **Approach**:
- Sort events (start and end of intervals), track the number of overlapping intervals at each event, and find the maximum count.
- **Example**:
- Intervals = [[1, 4], [2, 6], [3, 8]], k = 2.
- Output: 2.
```python
def max_k_interval_coverage(intervals, k):
points = []
for interval in intervals:
points.append((
interval[0], \'start\'))
points.append((interval[1], \'end\'))
points.sort()
coverage = 0
max_coverage = 0
for point, kind in points:
if kind == \'start\':
coverage += 1
if coverage == k:
max_coverage = max(max_coverage, coverage)
else:
coverage -= 1
return max_coverage
# Example
print(max_k_interval_coverage([[1, 4], [2, 6], [3, 8]], 2)) # 2
```
---
### 5. **Geometric Interval Problems**
#### a) **Rectangle Intersection via Intervals**
- **Problem Definition**: Given two rectangles, where each rectangle is defined by two intervals (x-axis and y-axis), find the intersection of these rectangles.
- **Approach**:
- Find the intersection of intervals on both x and y axes. The intersection of two rectangles is the combination of the intersecting intervals on both axes.
- **Example**:
- Rectangle A = ([1, 5], [2, 6]), Rectangle B = ([3, 7], [4, 9]).
- Output: [[3, 5], [4, 6]].
```python
def rectangle_intersection(A, B):
x_intersect = intersection_intervals([A[0][0], A[0][1]], [B[0][0], B[0][1]])
y_intersect = intersection_intervals([A[1][0], A[1][1]], [B[1][0], B[1][1]])
if x_intersect and y_intersect:
return [x_intersect, y_intersect]
return None
# Example
A = ([1, 5], [2, 6])
B = ([3, 7], [4, 9])
print(rectangle_intersection(A, B)) # [[3, 5], [4, 6]]
```
---
This version includes **problem definitions**, **approaches**, and **examples** for all the interval problems, along with code implementations. Each section is structured to make it easier to understand and apply to similar problems.';
my $regex = qr/\#(.*?)\*\*(.*?)\*\*/mp;
my $subst = '\\#\\1\\2';
my $result = $str =~ s/$regex/$subst/rg;
print "The result of the substitution is' $result\n";
Please keep in mind that these code samples are automatically generated and are not guaranteed to work. If you find any syntax errors, feel free to submit a bug report. For a full regex reference for Perl, please visit: http://perldoc.perl.org/perlre.html