The Elves resume water filtering operations! Clean water starts flowing over the edge of Island Island.
They offer to help you go over the edge of Island Island, too! Just hold on tight to one end of this impossibly long rope and they'll lower you down a safe distance from the massive waterfall you just created.
As you finally reach Snow Island, you see that the water isn't really reaching the ground: it's being absorbed by the air itself. It looks like you'll finally have a little downtime while the moisture builds up to snow-producing levels. Snow Island is pretty scenic, even without any snow; why not take a walk?
There's a map of nearby hiking trails (your puzzle input) that indicates paths (.), forest (#), and steep slopes (^, >, v, and <).
For example:
#.#####################
#.......#########...###
#######.#########.#.###
###.....#.>.>.###.#.###
###v#####.#v#.###.#.###
###.>...#.#.#.....#...#
###v###.#.#.#########.#
###...#.#.#.......#...#
#####.#.#.#######.#.###
#.....#.#.#.......#...#
#.#####.#.#.#########v#
#.#...#...#...###...>.#
#.#.#v#######v###.###v#
#...#.>.#...>.>.#.###.#
#####v#.#.###v#.#.###.#
#.....#...#...#.#.#...#
#.#########.###.#.#.###
#...###...#...#...#.###
###.###.#.###v#####v###
#...#...#.#.>.>.#.>.###
#.###.###.#.###.#.#v###
#.....###...###...#...#
#####################.#
You're currently on the single path tile in the top row; your goal is to reach the single path tile in the bottom row. Because of all the mist from the waterfall, the slopes are probably quite icy; if you step onto a slope tile, your next step must be downhill (in the direction the arrow is pointing). To make sure you have the most scenic hike possible, never step onto the same tile twice. What is the longest hike you can take?
In the example above, the longest hike you can take is marked with O, and your starting position is marked S:
#S#####################
#OOOOOOO#########...###
#######O#########.#.###
###OOOOO#OOO>.###.#.###
###O#####O#O#.###.#.###
###OOOOO#O#O#.....#...#
###v###O#O#O#########.#
###...#O#O#OOOOOOO#...#
#####.#O#O#######O#.###
#.....#O#O#OOOOOOO#...#
#.#####O#O#O#########v#
#.#...#OOO#OOO###OOOOO#
#.#.#v#######O###O###O#
#...#.>.#...>OOO#O###O#
#####v#.#.###v#O#O###O#
#.....#...#...#O#O#OOO#
#.#########.###O#O#O###
#...###...#...#OOO#O###
###.###.#.###v#####O###
#...#...#.#.>.>.#.>O###
#.###.###.#.###.#.#O###
#.....###...###...#OOO#
#####################O#
This hike contains 94 steps. (The other possible hikes you could have taken were 90, 86, 82, 82, and 74 steps long.)
Find the longest hike you can take through the hiking trails listed on your map. How many steps long is the longest hike?
Your puzzle answer was 2050.
As you reach the trailhead, you realize that the ground isn't as slippery as you expected; you'll have no problem climbing up the steep slopes.
Now, treat all slopes as if they were normal paths (.). You still want to make sure you have the most scenic hike possible, so continue to ensure that you never step onto the same tile twice. What is the longest hike you can take?
In the example above, this increases the longest hike to 154 steps:
#S#####################
#OOOOOOO#########OOO###
#######O#########O#O###
###OOOOO#.>OOO###O#O###
###O#####.#O#O###O#O###
###O>...#.#O#OOOOO#OOO#
###O###.#.#O#########O#
###OOO#.#.#OOOOOOO#OOO#
#####O#.#.#######O#O###
#OOOOO#.#.#OOOOOOO#OOO#
#O#####.#.#O#########O#
#O#OOO#...#OOO###...>O#
#O#O#O#######O###.###O#
#OOO#O>.#...>O>.#.###O#
#####O#.#.###O#.#.###O#
#OOOOO#...#OOO#.#.#OOO#
#O#########O###.#.#O###
#OOO###OOO#OOO#...#O###
###O###O#O###O#####O###
#OOO#OOO#O#OOO>.#.>O###
#O###O###O#O###.#.#O###
#OOOOO###OOO###...#OOO#
#####################O#
Find the longest hike you can take through the surprisingly dry hiking trails listed on your map. How many steps long is the longest hike?
Your puzzle answer was 6262.
A very interesting twist on the standard pathfinding algorithms – but unfortunately, the Longest Path Problem is NP-hard, and it shows.
Due to the direction constraints, part 1 is perfectly manageable with a simple DFS; the only optimization required is that recursion must only happen at actual junctions, otherwise Python's recursion limit will hit.
Running the same DFS directly for part 2 is futile; it will eventually terminate, but at least in Python, it's far too slow to be acceptable. The problem needs to be simplified first, and this can be done by observing that there is only a relatively low number of junctions in the maze, and the distances between each junction can be pre-computed. The start and goal positions count as junctions too, and thus the resulting distance matrix is a weighted undirected graph:
With this simplified representation, the final DFS becomes a little quicker: one minute instead of several hours. This is still too long to be really practical, so I thought of another optimization: Instead of having an actual Python set of visited nodes, a bitfield of visited nodes is also possible. This can be constructed by mapping each junction / graph node to a power of two (which is easy, as there's only 36 nodes). This does indeed make things faster, but only by a factor of two. Python itself seems to be a limitation here, so I escalated one step further and made the Python code generate a small C program that contains the distance matrix in a long switch...case statement. With this, I get down to sub-second runtimes for everything from Python graph generation and C compilation to execution of the compiled program.
- Part 1, Python (direct maze DFS): 417 bytes, ~400 ms
- Part 2, Python (graph DFS using sets): 425 bytes, ~60 s
- Part 2, Python (graph DFS using bitfields): 503 bytes, ~35 s
- Part 2, Python (generating and running C code): 789 bytes, ~700 ms