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You are given a 2D integer array logs where each logs[i] = [birthi, deathi] indicates the birth and death years of the i-th person.

The population of some year x is the number of people alive during that year. The i-th person is counted in the population of year x if x is in the inclusive range [birthi, deathi - 1]. Note that the person is not counted in the death year.

Return the year with the maximum population.

Clarifying Questions

  1. Range of Years: What is the possible range of years we are dealing with?
    • Typically, the constraints might be from, e.g., 1950 to 2050.
  2. Length and Values of Logs: What is the expected range of values for logs and how long can it be?
    • This helps ascertain the upper limits on performance and storage.

Strategy

We will use a strategy involving a difference array to determine the population changes efficiently:

  1. Create an Array for Population Changes:
    • Mark every birth year with a +1 and every death year with a -1. Note that the death year should decrement the next year, because the person is not counted in the death year.
  2. Calculate Prefix Sum:
    • Use a prefix sum approach to find out the number of people alive in each year.
  3. Determine the Maximum Population Year:
    • Traverse the prefix sum array to find the year with the highest population.

Steps in Detail

  1. Initialize an array, population_changes, to track changes in population for each year.
  2. Loop through the logs and apply the changes to this array.
  3. Compute the prefix sum of the population_changes array.
  4. Find the year with the maximum population from the prefix sum array.

Code

def maxPopulationYear(logs):
    # Define the range based on typical constraints (assuming 1950 to 2050)
    start_year = 1950
    end_year = 2051
    num_years = end_year - start_year
    
    # Array to store population changes
    population_changes = [0] * num_years
    
    # Apply population changes at birth and just after death
    for birth, death in logs:
        population_changes[birth - start_year] += 1
        if death < end_year:
            population_changes[death - start_year] -= 1
            
    # Use prefix sum to compute population for each year
    max_population = 0
    current_population = 0
    max_year = start_year
    
    for year in range(num_years):
        current_population += population_changes[year]
        if current_population > max_population:
            max_population = current_population
            max_year = start_year + year
    
    return max_year

# Example usage
logs = [[1993,1999], [2000,2010], [1975,2005], [1970,1982], [1980,2040], [1950,1961]]
print(maxPopulationYear(logs))  # Output should be a year with maximum population

Time Complexity

  1. Initialization and Array Operations: O(Y), where Y is the range of years (e.g., 101 for 1950 to 2050).
  2. Processing Logs: O(N), where N is the number of logs.
  3. Prefix Sum Calculation: O(Y), where Y is again the range of years.

Thus, the overall time complexity is O(N + Y), which is efficient given that typically Y (number of years) is fixed and relatively small.

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