Problem Description

You have a long painting represented as a number line where multiple colored segments overlap. Each segment is defined by [start, end, color] representing a half-closed interval [start, end) painted with a unique color.

When segments overlap, their colors mix together. Mixed colors are represented as sets - for example, if colors 2, 4, and 6 overlap at a position, the mixed color is {2,4,6}. For output simplicity, you need to return the sum of colors in each set rather than the set itself (so {2,4,6} becomes 12).

Your task is to describe the final painting using the minimum number of non-overlapping segments, where each segment shows the mixed color sum for that interval.

Key points:

• Input: segments[i] = [start_i, end_i, color_i]
• Output: painting[j] = [left_j, right_j, mix_j] where mix_j is the sum of all colors in that interval
• Half-closed intervals [a, b) include point a but exclude point b
• Return only painted portions (exclude unpainted parts)
• Segments can be returned in any order

Example: Given segments = [[1,4,5],[1,7,7]]:

• Interval [1,4) has both colors 5 and 7, so the mixed sum is 12
• Interval [4,7) has only color 7
• Output: [[1,4,12],[4,7,7]]

The solution uses a difference array technique with a sweep line approach. It marks color changes at segment boundaries (adding color at start, subtracting at end), sorts these events, computes running sums to get the color mix at each interval, and constructs the final non-overlapping segments from consecutive events where the color sum is non-zero.

Intuition

Think about what happens when we paint overlapping segments on a number line. At any point, we need to track which colors are "active" and sum them up. The key insight is that colors only change at segment boundaries - when a segment starts or ends.

Imagine walking along the number line from left to right. As we move, the only positions where the color mix changes are:

• When a new segment starts (a color gets added to the mix)
• When a segment ends (a color gets removed from the mix)

This suggests we don't need to check every single point on the line. We only need to examine "event points" where segments begin or end.

Consider the example [[1,4,5],[1,7,7]]:

• At position 1: both colors 5 and 7 start (total: +5, +7)
• At position 4: color 5 ends (total: -5)
• At position 7: color 7 ends (total: -7)

If we track these changes using a difference array concept, we can mark +color when a segment starts and -color when it ends. Then by maintaining a running sum as we sweep through these events, we automatically get the correct color mix for each interval.

The approach becomes:

1. Mark all color changes at boundary points in a map/dictionary
2. Sort these boundary points
3. Calculate the running sum between consecutive boundaries
4. Each pair of consecutive boundaries with a non-zero sum forms a segment in our answer

This efficiently handles all overlaps without explicitly checking every possible position, reducing the problem to processing only the critical points where changes occur.

Learn more about Prefix Sum and Sorting patterns.

Solution Approach

The solution implements a sweep line algorithm with a difference array technique:

Step 1: Build the difference array

d = defaultdict(int)
for l, r, c in segments:
    d[l] += c
    d[r] -= c

We use a dictionary d to track color changes at boundary points. For each segment [l, r, c]:

• Add color c at the start position l (color becomes active)
• Subtract color c at the end position r (color becomes inactive)

This leverages the half-closed interval property - color is active at l but not at r.

Step 2: Sort the boundary points

s = sorted([[k, v] for k, v in d.items()])

Convert the dictionary to a sorted list of [position, color_change] pairs. Sorting ensures we process events from left to right along the number line.

Step 3: Calculate running sums

n = len(s)
for i in range(1, n):
    s[i][1] += s[i - 1][1]

Transform each color change into the actual color sum at that position using prefix sums. After this step, s[i][1] contains the total color sum for the interval starting at position s[i][0].

Step 4: Build the result segments

return [[s[i][0], s[i + 1][0], s[i][1]] for i in range(n - 1) if s[i][1]]

Create segments from consecutive boundary points:

• Left boundary: s[i][0]
• Right boundary: s[i + 1][0]
• Color sum: s[i][1]

The condition if s[i][1] filters out intervals with zero color sum (unpainted regions).

Time Complexity: O(n log n) where n is the number of segments, dominated by sorting. Space Complexity: O(n) for storing the boundary points and their color changes.

Example Walkthrough

Let's trace through the algorithm with segments = [[1,4,5],[3,6,7],[5,8,3]]:

Step 1: Build the difference array

For each segment, we mark color changes:

• [1,4,5]: Add 5 at position 1, subtract 5 at position 4
• [3,6,7]: Add 7 at position 3, subtract 7 at position 6
• [5,8,3]: Add 3 at position 5, subtract 3 at position 8

Our difference array d:

d = {1: 5, 4: -5, 3: 7, 6: -7, 5: 3, 8: -3}

Step 2: Sort the boundary points

Convert to sorted list:

s = [[1, 5], [3, 7], [4, -5], [5, 3], [6, -7], [8, -3]]

Step 3: Calculate running sums

Transform each position to show the actual color sum starting from that position:

• Position 1: sum = 5
• Position 3: sum = 5 + 7 = 12
• Position 4: sum = 12 + (-5) = 7
• Position 5: sum = 7 + 3 = 10
• Position 6: sum = 10 + (-7) = 3
• Position 8: sum = 3 + (-3) = 0

After running sums:

s = [[1, 5], [3, 12], [4, 7], [5, 10], [6, 3], [8, 0]]

Step 4: Build result segments

Create segments from consecutive pairs where color sum > 0:

• [1, 3) with color sum 5
• [3, 4) with color sum 12
• [4, 5) with color sum 7
• [5, 6) with color sum 10
• [6, 8) with color sum 3
• Skip [8, ?) since color sum is 0

Final result: [[1,3,5], [3,4,12], [4,5,7], [5,6,10], [6,8,3]]

This gives us the minimum number of non-overlapping segments describing the painted regions with their mixed color sums.

Solution Implementation

Time and Space Complexity

Time Complexity: O(n log n) where n is the number of segments.

• Creating the dictionary d and iterating through all segments takes O(n) time, where each segment contributes at most 2 entries (start and end points).
• Converting the dictionary to a list of key-value pairs takes O(k) time where k is the number of unique points, and k ≤ 2n.
• Sorting the list s takes O(k log k) time, which is O(n log n) in the worst case since k ≤ 2n.
• The prefix sum calculation loop takes O(k) time, which is O(n).
• Building the final result list takes O(k) time, which is O(n).
• Overall: O(n) + O(n log n) + O(n) + O(n) = O(n log n).

Space Complexity: O(n)

• The dictionary d stores at most 2n entries (start and end points for each segment), requiring O(n) space.
• The sorted list s contains at most 2n elements, requiring O(n) space.
• The result list contains at most 2n - 1 intervals in the worst case, requiring O(n) space.
• Overall space complexity is O(n).

Learn more about how to find time and space complexity quickly.

Common Pitfalls

Pitfall 1: Integer Overflow with Large Color Values

Problem: When multiple segments with large color values overlap, their sum might exceed the maximum integer limit. For instance, if you have many segments each with color value near 10^9 overlapping at the same position, the sum could overflow.

Solution:

• In Python, integers have arbitrary precision so this isn't an issue
• In languages like Java or C++, use long or long long data types:

// C++ example
map<long long, long long> color_changes;
vector<vector<long long>> result;

Pitfall 2: Forgetting to Handle Empty Segments (Zero Color Sum)

Problem: After calculating prefix sums, some intervals might have a color sum of 0 (where all colors have ended). Including these would violate the requirement to "exclude unpainted parts."

Incorrect approach:

# Wrong: includes segments with 0 color
return [[sorted_positions[i][0], sorted_positions[i + 1][0], sorted_positions[i][1]] 
        for i in range(num_positions - 1)]

Solution: Always check if the color sum is non-zero before adding to result:

# Correct: filters out zero-color segments
if sorted_positions[i][1]:  # Only non-zero color sums
    result.append([...])

Pitfall 3: Misunderstanding Half-Closed Intervals

Problem: Treating intervals as closed [start, end] instead of half-closed [start, end) leads to incorrect color mixing at boundary points.

Example mistake:

# Wrong: treating end as inclusive
for start, end, color in segments:
    color_changes[start] += color
    color_changes[end + 1] -= color  # Incorrect!

Solution: Remember that the end point is exclusive:

# Correct: end is already exclusive in [start, end)
for start, end, color in segments:
    color_changes[start] += color
    color_changes[end] -= color  # Color stops at end, not end+1

Pitfall 4: Not Handling Adjacent Segments with Same Color Sum

Problem: The algorithm naturally merges adjacent segments with the same color sum, which is correct. However, some might incorrectly try to keep them separate or add extra logic to merge them.

Solution: The algorithm already handles this correctly - consecutive boundary points naturally create minimal segments. No additional merging logic is needed.