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Alternating Groups III

Difficulty: Hard

Problem Description

There are some red and blue tiles arranged circularly. You are given an array of integers colors and a 2D integers array queries. The color of tile i is represented by colors[i]: colors[i] == 0 means that tile i is red, and colors[i] == 1 means that tile i is blue. An alternating group is a contiguous subset of tiles in the circle with alternating colors. You have to process queries of two types: determine the count of alternating groups with a specified size or change the color of a specific tile. Return an array containing the results of the queries of the first type in order.

Key Insights

  • The tiles are arranged in a circular manner, meaning the first and last tiles are adjacent.
  • An alternating group consists of contiguous tiles with alternating colors.
  • Efficiently counting alternating groups of a specific size is crucial given the constraints.
  • Updates to the colors may affect the groups, thus requiring a dynamic approach.

Space and Time Complexity

Time Complexity: O(n + q), where n is the number of tiles and q is the number of queries. Each query can be processed in constant time after an initial setup. Space Complexity: O(n), for storing the groups and possibly auxiliary data structures.

Solution

To solve the problem, we can use a combination of a data structure to track the current segments of alternating colors and a method to efficiently count these segments.

  1. Segment Tracking: Create an initial list of segments where each segment contains the start index, end index, and the color of the segment.
  2. Processing Queries:
    • For type 1 queries (counting groups), traverse the segments and check for groups of the requested size.
    • For type 2 queries (updating colors), modify the respective segment and check if it merges with neighboring segments or splits an existing one.

This approach helps maintain the alternating groups efficiently and answers each query in constant time after the initial setup.

Code Solutions

def alternatingGroups(colors, queries):
    n = len(colors)
    segments = []
    
    # Initial segment creation
    start = 0
    while start < n:
        end = start
        while end < n and colors[end] == colors[start]:
            end += 1
        segments.append((start, end - 1, colors[start]))
        start = end

    # Wrap around for circular array
    if segments[0][2] == segments[-1][2]:
        new_start = segments.pop()
        segments[0] = (new_start[0], segments[0][1], segments[0][2])
    
    answer = []
    
    for query in queries:
        if query[0] == 1:  # Count groups
            size = query[1]
            count = 0
            for i in range(len(segments)):
                length = segments[i][1] - segments[i][0] + 1
                if length >= size:
                    count += length - size + 1
            answer.append(count)
        elif query[0] == 2:  # Update color
            index, new_color = query[1], query[2]
            if colors[index] != new_color:
                colors[index] = new_color  # Update the color
                # Logic to adjust segments accordingly
                # This part would need to be implemented
                # to handle merging or splitting segments.
    
    return answer
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