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Maximum Population Year

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Arrays

You are given a 2D integer array logs where each logs[i] = [birth_i, death_i] 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 year x's population if x is in the inclusive range [birth_i, death_i - 1]. Note that the person is not counted in the year that they die.

Return the earliest year with the maximum population.

Example 1:

Input: logs = [[1993,1999],[2000,2010]]
Output: 1993
Explanation: The maximum population is 1, and 1993 is the earliest year with this population.

Example 2:

Input: logs = [[1950,1961],[1960,1971],[1970,1981]]
Output: 1960
Explanation: 
The maximum population is 2, and it had happened in years 1960 and 1970.
The earlier year between them is 1960.

Constraints:

1 <= logs.length <= 100
1950 <= birth_i < death_i <= 2050

Try coding it in LeetCode

Maximum Population Year

Easy

Asked by:

61 views

Topics:

Arrays

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

Return the earliest year with the maximum population.

Example 1:

Input: logs = [[1993,1999],[2000,2010]]
Output: 1993
Explanation: The maximum population is 1, and 1993 is the earliest year with this population.

Example 2:

Input: logs = [[1950,1961],[1960,1971],[1970,1981]]
Output: 1960
Explanation: 
The maximum population is 2, and it had happened in years 1960 and 1970.
The earlier year between them is 1960.

Constraints:

1 <= logs.length <= 100
1950 <= birth_i < death_i <= 2050

Try coding it in LeetCode

Solution

Clarifying Questions

When you get asked this question in a real-life environment, it will often be ambiguous (especially at FAANG). Make sure to ask these questions in that case:

What are the constraints on the birth_i and death_i values? Specifically, what's the range of years we should consider?
Can the birth year and death year be the same? If so, should that year be considered part of the person's life?
If there are multiple years with the same maximum population, which year should I return (the earliest or the latest)?
Is the input array guaranteed to be non-empty, or should I handle the case where the input is empty or null?
Are the birth years always strictly less than the death years, or can a birth year be greater than or equal to a death year?

Brute Force Solution

Approach

To find the year with the most people alive, we can just check every single year within the given range. For each year, we'll count how many people were alive during that year based on their birth and death years.

Here's how the algorithm would work step-by-step:

Consider all the years between the earliest birth year and the latest death year from the given data.
For each of these years, go through the list of people and check if they were alive during that specific year.
A person is considered alive in a year if the year is between their birth year and death year (inclusive).
Count how many people were alive for each year.
Keep track of the year with the highest number of people alive.
If multiple years have the same highest number of people alive, choose the earliest one.

Code Implementation

def maximum_population_year_brute_force(people):    earliest_birth_year = min(birth for birth, death in people)
    latest_death_year = max(death for birth, death in people)

    maximum_population = 0
    maximum_population_year = earliest_birth_year

    for year in range(earliest_birth_year, latest_death_year + 1):
        current_population = 0

        # Need to check each year to find most populus
        for birth_year, death_year in people:

            if birth_year <= year <= death_year:

                current_population += 1

        # Need to keep track of the highest year value
        if current_population > maximum_population:

            maximum_population = current_population
            maximum_population_year = year

        # Check that earlier year is returned in tie
        elif current_population == maximum_population and year < maximum_population_year:

            maximum_population_year = year

    return maximum_population_year

Big(O) Analysis

Time Complexity

O(n*m) – Let n be the number of years between the earliest birth year and the latest death year, and m be the number of people in the input. The algorithm iterates through each of the n years. For each year, it iterates through all m people to check if they were alive during that year. Therefore, the time complexity is O(n*m), representing the nested loop structure where we iterate over the years and then iterate over the people for each year.

Space Complexity

O(1) – The algorithm iterates through a range of years, but it only stores a few scalar variables: the current year being checked, the number of people alive in that year, the maximum population count seen so far, and the corresponding year with the maximum population. The number of years to check is determined by the earliest birth year and latest death year, but the space used does not depend on the number of people, N. Since the space used by these variables is constant regardless of the input size, the auxiliary space complexity is O(1).

Optimal Solution

Approach

The goal is to find the year when the most people were alive, given a list of birth and death years. Instead of checking each year individually, we'll track how the population changes each year to find the peak efficiently.

Here's how the algorithm would work step-by-step:

Imagine a timeline of years. For each person, note down their birth year as a population increase and their death year as a population decrease.
Sort all these 'birth' and 'death' events by year.
Go through the sorted events year by year. Whenever you encounter a birth, increase the current population. When you encounter a death, decrease the current population.
Keep track of the current population for each year and also track the maximum population seen so far and the year it occurred.
As you go through all the years, the year with the highest recorded population is your answer.

Code Implementation

def maximum_population_year(people):
    events = []
    for birth, death in people:
        events.append((birth, 1))
        events.append((death, -1))

    # Sort events by year. Crucial for timeline processing.
    events.sort()

    current_population = 0
    max_population = 0
    max_population_year = 0

    for year, change in events:
        current_population += change

        # Update max population if needed.
        if current_population > max_population:
            max_population = current_population
            max_population_year = year

    return max_population_year

Big(O) Analysis

Time Complexity

O(n log n) – The primary cost comes from sorting the birth and death year events. If there are n people, there will be 2n events (birth and death). Sorting these 2n events using an efficient sorting algorithm like merge sort or quicksort will take O(n log n) time. The subsequent iteration through the sorted events takes O(n) time to track population changes, which is dominated by the sorting time. Therefore, the overall time complexity is O(n log n).

Space Complexity

O(N) – The algorithm creates a timeline of birth and death events. In the worst case, where each person has a unique birth and death year, the number of events will be 2N, where N is the number of people (birth/death year pairs). These events are stored for sorting. Therefore, the auxiliary space required to store these events is directly proportional to the number of people, N, resulting in O(N) space complexity.

Edge Cases

Case	How to Handle
Empty logs array	Return a default value, such as 1900, as no population information is available, or throw an exception if no default is acceptable.
logs array with a single entry	The maximum population year is the birth year of that single entry.
All birth and death years are the same	The birth year will be the year with the maximum population, as everyone is born and dies in the same year.
Overlapping lifespans leading to high population in certain years	The algorithm should correctly count the overlapping years when determining maximum population.
Non-overlapping lifespans (everyone lives in distinct time periods)	The algorithm should identify the earliest birth year among these distinct periods and return that year as the year with maximum population during that distinct period.
Large input data (large logs array and/or large year ranges)	Use an efficient algorithm like prefix sum or difference array to avoid time complexity issues from iterating through all possible years.
Years outside of the specified range (e.g., before 1900 or after 2100)	Handle the years according to the problem specifications, perhaps by truncating them or handling them as invalid input.
Birth year is equal to the death year	Treat this as the person being alive for that single year; the population should increment by one for that year.

Empty logs array

How to Handle:

Return a default value, such as 1900, as no population information is available, or throw an exception if no default is acceptable.

logs array with a single entry

How to Handle:

The maximum population year is the birth year of that single entry.

All birth and death years are the same

How to Handle:

The birth year will be the year with the maximum population, as everyone is born and dies in the same year.

Overlapping lifespans leading to high population in certain years

How to Handle:

The algorithm should correctly count the overlapping years when determining maximum population.

Non-overlapping lifespans (everyone lives in distinct time periods)

How to Handle:

The algorithm should identify the earliest birth year among these distinct periods and return that year as the year with maximum population during that distinct period.

Large input data (large logs array and/or large year ranges)

How to Handle:

Use an efficient algorithm like prefix sum or difference array to avoid time complexity issues from iterating through all possible years.

Years outside of the specified range (e.g., before 1900 or after 2100)

How to Handle:

Handle the years according to the problem specifications, perhaps by truncating them or handling them as invalid input.

Birth year is equal to the death year

How to Handle:

Treat this as the person being alive for that single year; the population should increment by one for that year.

Solution

Clarifying Questions

When you get asked this question in a real-life environment, it will often be ambiguous (especially at FAANG). Make sure to ask these questions in that case:

What are the constraints on the birth_i and death_i values? Specifically, what's the range of years we should consider?
Can the birth year and death year be the same? If so, should that year be considered part of the person's life?
If there are multiple years with the same maximum population, which year should I return (the earliest or the latest)?
Is the input array guaranteed to be non-empty, or should I handle the case where the input is empty or null?
Are the birth years always strictly less than the death years, or can a birth year be greater than or equal to a death year?

Brute Force Solution

Approach

Here's how the algorithm would work step-by-step:

Consider all the years between the earliest birth year and the latest death year from the given data.
For each of these years, go through the list of people and check if they were alive during that specific year.
A person is considered alive in a year if the year is between their birth year and death year (inclusive).
Count how many people were alive for each year.
Keep track of the year with the highest number of people alive.
If multiple years have the same highest number of people alive, choose the earliest one.

Code Implementation

def maximum_population_year_brute_force(people):    earliest_birth_year = min(birth for birth, death in people)
    latest_death_year = max(death for birth, death in people)

    maximum_population = 0
    maximum_population_year = earliest_birth_year

    for year in range(earliest_birth_year, latest_death_year + 1):
        current_population = 0

        # Need to check each year to find most populus
        for birth_year, death_year in people:

            if birth_year <= year <= death_year:

                current_population += 1

        # Need to keep track of the highest year value
        if current_population > maximum_population:

            maximum_population = current_population
            maximum_population_year = year

        # Check that earlier year is returned in tie
        elif current_population == maximum_population and year < maximum_population_year:

            maximum_population_year = year

    return maximum_population_year

Big(O) Analysis

Time Complexity

Space Complexity

Optimal Solution

Approach

Here's how the algorithm would work step-by-step:

Imagine a timeline of years. For each person, note down their birth year as a population increase and their death year as a population decrease.
Sort all these 'birth' and 'death' events by year.
Go through the sorted events year by year. Whenever you encounter a birth, increase the current population. When you encounter a death, decrease the current population.
Keep track of the current population for each year and also track the maximum population seen so far and the year it occurred.
As you go through all the years, the year with the highest recorded population is your answer.

Code Implementation

def maximum_population_year(people):
    events = []
    for birth, death in people:
        events.append((birth, 1))
        events.append((death, -1))

    # Sort events by year. Crucial for timeline processing.
    events.sort()

    current_population = 0
    max_population = 0
    max_population_year = 0

    for year, change in events:
        current_population += change

        # Update max population if needed.
        if current_population > max_population:
            max_population = current_population
            max_population_year = year

    return max_population_year

Big(O) Analysis

Time Complexity

Space Complexity

Edge Cases

Case	How to Handle
Empty logs array	Return a default value, such as 1900, as no population information is available, or throw an exception if no default is acceptable.
logs array with a single entry	The maximum population year is the birth year of that single entry.
All birth and death years are the same	The birth year will be the year with the maximum population, as everyone is born and dies in the same year.
Overlapping lifespans leading to high population in certain years	The algorithm should correctly count the overlapping years when determining maximum population.
Non-overlapping lifespans (everyone lives in distinct time periods)	The algorithm should identify the earliest birth year among these distinct periods and return that year as the year with maximum population during that distinct period.
Large input data (large logs array and/or large year ranges)	Use an efficient algorithm like prefix sum or difference array to avoid time complexity issues from iterating through all possible years.
Years outside of the specified range (e.g., before 1900 or after 2100)	Handle the years according to the problem specifications, perhaps by truncating them or handling them as invalid input.
Birth year is equal to the death year	Treat this as the person being alive for that single year; the population should increment by one for that year.

Empty logs array

How to Handle:

Return a default value, such as 1900, as no population information is available, or throw an exception if no default is acceptable.

logs array with a single entry

How to Handle:

The maximum population year is the birth year of that single entry.

All birth and death years are the same

How to Handle:

The birth year will be the year with the maximum population, as everyone is born and dies in the same year.

Overlapping lifespans leading to high population in certain years

How to Handle:

The algorithm should correctly count the overlapping years when determining maximum population.

Non-overlapping lifespans (everyone lives in distinct time periods)

How to Handle:

The algorithm should identify the earliest birth year among these distinct periods and return that year as the year with maximum population during that distinct period.

Large input data (large logs array and/or large year ranges)

How to Handle:

Use an efficient algorithm like prefix sum or difference array to avoid time complexity issues from iterating through all possible years.

Years outside of the specified range (e.g., before 1900 or after 2100)

How to Handle:

Handle the years according to the problem specifications, perhaps by truncating them or handling them as invalid input.

Birth year is equal to the death year

How to Handle:

Treat this as the person being alive for that single year; the population should increment by one for that year.