[Advent of Code 2021] Day 7 | The Treachery of Whales

Part 1

A giant whale has decided your submarine is its next meal, and it's much faster than you are. There's nowhere to run!

Suddenly, a swarm of crabs (each in its own tiny submarine - it's too deep for them otherwise) zooms in to rescue you! They seem to be preparing to blast a hole in the ocean floor; sensors indicate a massive underground cave system just beyond where they're aiming!

The crab submarines all need to be aligned before they'll have enough power to blast a large enough hole for your submarine to get through. However, it doesn't look like they'll be aligned before the whale catches you! Maybe you can help?

There's one major catch - crab submarines can only move horizontally.

You quickly make a list of the horizontal position of each crab (your puzzle input). Crab submarines have limited fuel, so you need to find a way to make all of their horizontal positions match while requiring them to spend as little fuel as possible.

For example, consider the following horizontal positions:

16,1,2,0,4,2,7,1,2,14

This means there's a crab with horizontal position 16, a crab with horizontal position 1, and so on.

Each change of 1 step in horizontal position of a single crab costs 1 fuel. You could choose any horizontal position to align them all on, but the one that costs the least fuel is horizontal position 2:

• Move from 16 to 2: 14 fuel
• Move from 1 to 2: 1 fuel
• Move from 2 to 2: 0 fuel
• Move from 0 to 2: 2 fuel
• Move from 4 to 2: 2 fuel
• Move from 2 to 2: 0 fuel
• Move from 7 to 2: 5 fuel
• Move from 1 to 2: 1 fuel
• Move from 2 to 2: 0 fuel
• Move from 14 to 2: 12 fuel

This costs a total of _37_ fuel. This is the cheapest possible outcome; more expensive outcomes include aligning at position 1 (41 fuel), position 3 (39 fuel), or position 10 (71 fuel).

Determine the horizontal position that the crabs can align to using the least fuel possible. How much fuel must they spend to align to that position?

Proposed solution: use the median as the meetup point for the crabs – this will always result in the shortest total path for all crabs

Time complexity: O(n log(n))

Space complexity: O(n)

#!/usr/bin/env python3
import sys

if len(sys.argv) != 2:
    print("Usage: {} <input file>".format(sys.argv[0]))
    sys.exit(1)

file_input = open(sys.argv[1], "r").read().strip().split("\n")
positions = [int(x) for x in file_input[0].split(",")]
positions.sort()
meetup = positions[len(positions) // 2]

total = 0
for position in positions:
    total += abs(meetup - position)
print("Total Fuel Cost: " + str(total))

Python's sorting function uses TimSort which has O(n log(n)) time complexity and worst case O(n) space complexity. This is the bottleneck operation, then just select the median as the meetup spot and sum the distances from every other element in the list.

Why the median? By definition, it will produce the shortest total path for all elements. Even if the dataset is skewed one way or the other, choosing any element before or after the median produces a larger total path for all elements.

❯ python3 solution7.py input7
Total Fuel Cost: 341534

Part 2

The crabs don't seem interested in your proposed solution. Perhaps you misunderstand crab engineering?

As it turns out, crab submarine engines don't burn fuel at a constant rate. Instead, each change of 1 step in horizontal position costs 1 more unit of fuel than the last: the first step costs 1, the second step costs 2, the third step costs 3, and so on.

As each crab moves, moving further becomes more expensive. This changes the best horizontal position to align them all on; in the example above, this becomes 5:

• Move from 16 to 5: 66 fuel
• Move from 1 to 5: 10 fuel
• Move from 2 to 5: 6 fuel
• Move from 0 to 5: 15 fuel
• Move from 4 to 5: 1 fuel
• Move from 2 to 5: 6 fuel
• Move from 7 to 5: 3 fuel
• Move from 1 to 5: 10 fuel
• Move from 2 to 5: 6 fuel
• Move from 14 to 5: 45 fuel

This costs a total of _168_ fuel. This is the new cheapest possible outcome; the old alignment position (2) now costs 206 fuel instead.

Determine the horizontal position that the crabs can align to using the least fuel possible so they can make you an escape route! How much fuel must they spend to align to that position?

Proposed solution: since longer paths now follow a cummulative fuel cost formula, the median as a meetup spot will no longer work – but the mean will

Time complexity: O(n log(n))

Space complexity: O(n)

#!/usr/bin/env python3
import sys

if len(sys.argv) != 2:
    print("Usage: {} <input file>".format(sys.argv[0]))
    sys.exit(1)

file_input = open(sys.argv[1], "r").read().strip().split("\n")
positions = [int(x) for x in file_input[0].split(",")]
positions.sort()
meetup = sum(positions) // len(positions)

total = 0
for position in positions:
    total += sum([i for i in range(1, abs(meetup - position)+1)])
print("Total Fuel Cost: " + str(total))

The same concept except we have to do a series summation since fuel cost increases at every step. If the input range is really sparse then we could see this impact time/memory performance but all inputs are in the range 0 ≤ x ≤ 2000.

Also, it seems like the meetup point could be off by one no matter which way you round; I was lazy and just tried both manually.

❯ python3 solution7.py input7
Total Fuel Cost: 93397632