#1854

Easy

LeetCode Problem Workspace

Maximum Population Year

Find the earliest year with the highest population using an array plus counting approach.

Problem Statement

You are given a list of people's birth and death years. For each person, their birth year and death year are given as intervals. A person's death year is exclusive, meaning they are not alive in the year they die. Your task is to find the earliest year with the highest population during that period.

The population of a year is determined by counting the number of people whose birth year is less than or equal to that year and whose death year is greater than that year. You need to return the earliest year with the maximum population count.

Examples

Example 1

Input: logs = [[1993,1999],[2000,2010]]

Output: 1993

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

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 <= birthi < deathi <= 2050

Solution Approach

Array plus Counting

You can approach this problem by iterating over each year and counting the number of people alive in that year using an array. The array represents changes in population at each year, where you increment at birth years and decrement at death years.

Prefix Sum

A more efficient approach is to use a prefix sum. By creating an array to track population changes, a prefix sum can be computed to find the population for each year. This will allow you to track the population over time without needing to recount for every individual year.

Tracking Maximum Population

As you calculate the population for each year, keep track of the year with the highest population. If multiple years have the same population, return the earliest one.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity of the array plus counting approach is O(n), where n is the number of years being considered. The space complexity can be O(k), where k is the range of years, depending on how you store population changes.

What Interviewers Usually Probe

Check if the candidate optimizes the approach by using a prefix sum technique.
Look for the candidate's ability to handle ties in population values.
Test if the candidate can efficiently count population using a simple array.

Common Pitfalls or Variants

Common pitfalls

Not correctly handling the exclusive nature of death years.
Forgetting to return the earliest year in case of ties in population counts.
Using inefficient methods that do not optimize population tracking over a range of years.

Follow-up variants

Use a different method to track population changes, such as a hash map.
Consider a problem where some birth years are earlier than others, requiring a more complex population calculation.
Limit the input size and check how the candidate handles edge cases like maximum years in a small range.

FAQ

What is the time complexity of the maximum population year problem?

The time complexity can vary depending on the approach, but an optimal solution using prefix sums generally has a time complexity of O(k), where k is the range of years.

How does prefix sum help in solving this problem?

Prefix sum helps by efficiently calculating the population at each year without needing to count people repeatedly, leading to faster computations over large ranges of years.

What should I do if multiple years have the same maximum population?

If multiple years have the same population, return the earliest one. This can be handled by keeping track of both the population and the year in each iteration.

What is the role of counting in the maximum population year problem?

Counting helps determine how many people are alive during each year by tracking population changes at birth and death years, allowing for an efficient solution.

How does the maximum population year problem relate to other problems in array manipulation?

The problem uses techniques like array manipulation and counting, which are also commonly used in problems dealing with ranges, intervals, and aggregate calculations.

terminal

Solution

Solution 1: Difference Array

We notice that the range of years is $[1950,..2050]$. Therefore, we can map these years to an array $d$ of length $101$, where the index of the array represents the value of the year minus $1950$.

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class Solution:
    def maximumPopulation(self, logs: List[List[int]]) -> int:
        d = [0] * 101
        offset = 1950
        for a, b in logs:
            a, b = a - offset, b - offset
            d[a] += 1
            d[b] -= 1
        s = mx = j = 0
        for i, x in enumerate(d):
            s += x
            if mx < s:
                mx, j = s, i
        return j + offset

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