05

2021, Day 5: Hydrothermal Venture

You come across a field of hydrothermal vents on the ocean floor! These vents constantly produce large, opaque clouds, so it would be best to avoid them if possible.

Part 1

They tend to form in lines; the submarine helpfully produces a list of nearby lines of vents (your puzzle input) for you to review. For example:

0,9 -> 5,9
8,0 -> 0,8
9,4 -> 3,4
2,2 -> 2,1
7,0 -> 7,4
6,4 -> 2,0
0,9 -> 2,9
3,4 -> 1,4
0,0 -> 8,8
5,5 -> 8,2

Each line of vents is given as a line segment in the format x1,y1 -> x2,y2 where x1,y1 are the coordinates of one end the line segment and x2,y2 are the coordinates of the other end. These line segments include the points at both ends. In other words:

• An entry like 1,1 -> 1,3 covers points 1,1, 1,2, and 1,3.
• An entry like 9,7 -> 7,7 covers points 9,7, 8,7, and 7,7.

For now, only consider horizontal and vertical lines: lines where either x1 = x2 or y1 = y2.

So, the horizontal and vertical lines from the above list would produce the following diagram:

.......1..
..1....1..
..1....1..
.......1..
.112111211
..........
..........
..........
..........
222111....

In this diagram, the top left corner is 0,0 and the bottom right corner is 9,9. Each position is shown as the number of lines which cover that point or . if no line covers that point. The top-left pair of 1s, for example, comes from 2,2 -> 2,1; the very bottom row is formed by the overlapping lines 0,9 -> 5,9 and 0,9 -> 2,9.

To avoid the most dangerous areas, you need to determine the number of points where at least two lines overlap. In the above example, this is anywhere in the diagram with a 2 or larger - a total of 5 points.

Consider only horizontal and vertical lines. At how many points do at least two lines overlap?

Your puzzle answer was 7438.

Part 2

Unfortunately, considering only horizontal and vertical lines doesn't give you the full picture; you need to also consider diagonal lines.

Because of the limits of the hydrothermal vent mapping system, the lines in your list will only ever be horizontal, vertical, or a diagonal line at exactly 45 degrees. In other words:

• An entry like 1,1 -> 3,3 covers points 1,1, 2,2, and 3,3.
• An entry like 9,7 -> 7,9 covers points 9,7, 8,8, and 7,9.

Considering all lines from the above example would now produce the following diagram:

1.1....11.
.111...2..
..2.1.111.
...1.2.2..
.112313211
...1.2....
..1...1...
.1.....1..
1.......1.
222111....

You still need to determine the number of points where at least two lines overlap. In the above example, this is still anywhere in the diagram with a 2 or larger - now a total of 12 points.

Consider all of the lines. At how many points do at least two lines overlap?

Your puzzle answer was 21406.

Solution Notes

My initial approach for part 1 was to construct a list of locations that are touched by every line and adding those to a dictionary. (Complex numbers are of no help in that case.) Some care has to be taken to avoid off-by-one errors (missing initial/final points) and be able to work with lines in either direction (up and down, left and right), but overall it's a fine and efficient approach.

For part 2, however, this doesn't scale so well: Constructing the point lists for diagonals isn't as easy as it was for the horizontal and vertical lines. I tried a much dumber approach instead: Since all coordinates in the input are guaranteed to be valid hor/vert/diag lines, we can just start at the first index and iterate with increasing/decreasing coordinates depending on where the line's end position is located, stopping after(!) the end point has been reached. This is far slower, but still totally acceptable in terms of runtime. In fact, this approach works so well that code size is significantly lower than for part 1! Using the iterative approach for part 1 too makes the difference shrink a bit, but still it's one of those puzzles where part 2 is actually the easier thing to do, if done properly.

• Part 1, Python (coordinate list): 295 bytes, <100 ms
• Part 1, Python (using part 2 approach): 275 bytes, ~150 ms
• Part 2, Python: 253 bytes, ~250 ms