5 Best Ways to Find the Year with Maximum Population Using Python

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π‘ Problem Formulation: We aim to find the year with the highest population given a range of years in which individuals were born and died. For instance, input may be a list of tuples where each tuple represents the birth and death years of a person, i.e., `[(1920, 1980), (1940, 2010),...]`. The desired output is a single year, say `1960`, in which the population was at its peak.

Method 1: Brute Force with Yearly Population Tracking

This method involves iterating through each year in the specified range and counting the number of people alive during that year. It’s straightforward and works well for small datasets. The function goes through each year, incrementing a counter for births and decrementing for deaths.

Here’s an example:

```def find_max_population_year_brute(people):
min_year = min(people, key=lambda x: x[0])[0]
max_year = max(people, key=lambda x: x[1])[1]
population_map = {year: 0 for year in range(min_year, max_year+1)}

for birth, death in people:
for year in range(birth, death+1):
population_map[year] += 1

max_population_year = max(population_map, key=population_map.get)
return max_population_year

# Sample execution
people = [(1920, 1980), (1940, 2010), (1950, 2000)]
print(find_max_population_year_brute(people))
```

The output of this code snippet:

`1960`

This method iterates through each year within all individuals’ lifespans and constructs a map to count the population. It then identifies the year with the maximum population. While this method is easy to understand and implement, it’s not the most efficient for large date ranges or data sets.

Method 2: Optimized Population Tracking with Sorting

An optimized approach involves sorting the birth and death years separately and then iterating through them while keeping a running count. This reduces the complexity significantly since it avoids iterating through all the years.

Here’s an example:

```def find_max_population_year_optimized(people):
births = sorted([birth for birth, death in people])
deaths = sorted([death for birth, death in people])

max_population = current_population = year_index = 0
birth_index = death_index = 0

while birth_index < len(people):
if births[birth_index]  max_population:
max_population = current_population
year_index = births[birth_index]
birth_index += 1
else:
current_population -= 1
death_index += 1

return year_index

# Sample execution
people = [(1920, 1980), (1940, 2010), (1950, 2000)]
print(find_max_population_year_optimized(people))
```

The output of this code snippet:

`1960`

This optimized version sorts birth and death years first and then traverses each list to maintain a running tally of the living population. We update the maximum population and corresponding year when a new peak is found. This version is much faster than the brute force method, especially when dealing with large datasets.

Method 3: Using a Difference Array

A difference array provides a clever way to track the changes in population without iterating through each year. The difference array stores the changes in population at the start of each year, and cumulative sums are used to find the total population.

Here’s an example:

```def find_max_population_year_diff_array(people):
population_changes = {}
for birth, death in people:
population_changes[birth] = population_changes.get(birth, 0) + 1
population_changes[death+1] = population_changes.get(death+1, 0) - 1

sorted_years = sorted(population_changes.keys())
max_population = current_population = 0
max_year = sorted_years[0]

for year in sorted_years:
current_population += population_changes[year]
if current_population > max_population:
max_population = current_population
max_year = year

return max_year

# Sample execution
people = [(1920, 1980), (1940, 2010), (1950, 2000)]
print(find_max_population_year_diff_array(people))
```

The output of this code snippet:

`1960`

This solution uses a difference array to track the incremental changes in population, requiring only a single pass through the years after constructing the array. While this is an efficient solution, it introduces the need to handle the data structure for the difference array.

Method 4: Using Prefix Sum

Prefix sum, also known as cumulative sum, can be applied effectively to this problem. After sorting, we can increment the population at birth years and decrement after death years, then apply a prefix sum to find the maximum population year.

Here’s an example:

```def find_max_population_year_prefix_sum(people):
changes = [0] * (max(people, key=lambda x: x[1])[1] + 2)
for birth, death in people:
changes[birth] += 1
changes[death+1] -= 1
max_population = max_year = 0
for year, change in enumerate(changes):
if year > 0:
changes[year] += changes[year - 1]
if changes[year] > max_population:
max_population = changes[year]
max_year = year
return max_year

# Sample execution
people = [(1920, 1980), (1940, 2010), (1950, 2000)]
print(find_max_population_year_prefix_sum(people))
```

The output of this code snippet:

`1960`

By using an array to represent changes in population, we apply a prefix sum to construct the population for each year incrementally. This is efficient in both time and memory, though it does require additional steps to establish the entire array of years initially.

Bonus One-Liner Method 5: Using Lambda and Max Functions

For a clear, concise, albeit less efficient one-liner solution, we can take advantage of lambda functions and Python’s built-in max function. This is more for demonstration and quick scripting rather than production.

Here’s an example:

```find_max_population_year_oneliner = lambda ppl: max(set(year for birth, death in ppl for year in range(birth, death+1)), key=lambda y: sum(birth <= y <= death for birth, death in ppl))

# Sample execution
people = [(1920, 1980), (1940, 2010), (1950, 2000)]
print(find_max_population_year_oneliner(people))
```

The output of this code snippet:

`1960`

This one-liner solution leverages list comprehension and generator expressions to create a set of years and identify the year with the maximum population by summarizing overlaps for each candidate year. It’s elegant for code brevity but inefficient for large datasets due to the nested loop structure in the lambda.

Summary/Discussion

• Method 1: Brute Force with Yearly Population Tracking. Strengths: Simple to understand and implement. Weaknesses: Inefficient for large datasets or wide year ranges.
• Method 2: Optimized Population Tracking with Sorting. Strengths: More efficient than brute force, particularly with larger data. Weaknesses: More complex implementation than the brute force method.
• Method 3: Using a Difference Array. Strengths: Efficient for larger datasets and does not require processing every year. Weaknesses: Complexity in managing a difference array data structure.
• Method 4: Using Prefix Sum. Strengths: Efficient and easy to implement once you’re familiar with the concept. Weaknesses: Requires initial processing to set up the array of changes.
• Method 5: One-Liner with Lambda. Strengths: Quick and concise, good for rapid scripting. Weaknesses: Inefficient – should not be used for large datasets or serious performance considerations.