## \-\-\- Day 11: Cosmic Expansion --- You continue following signs for "Hot Springs" and eventually come across an [observatory](https://en.wikipedia.org/wiki/Observatory). The Elf within turns out to be a researcher studying cosmic expansion using the giant telescope here. He doesn't know anything about the missing machine parts; he's only visiting for this research project. However, he confirms that the hot springs are the next-closest area likely to have people; he'll even take you straight there once he's done with today's observation analysis. Maybe you can help him with the analysis to speed things up? The researcher has collected a bunch of data and compiled the data into a single giant _image_ (your puzzle input). The image includes _empty space_ ( `.`) and _galaxies_ ( `#`). For example: ``` ...#...... .......#.. #......... .......... ......#... .#........ .........# .......... .......#.. #...#..... ``` The researcher is trying to figure out the sum of the lengths of the _shortest path between every pair of galaxies_. However, there's a catch: the universe expanded in the time it took the light from those galaxies to reach the observatory. Due to something involving gravitational effects, _only some space expands_. In fact, the result is that _any rows or columns that contain no galaxies_ should all actually be twice as big. In the above example, three columns and two rows contain no galaxies: ``` v v v ...#...... .......#.. #......... >..........< ......#... .#........ .........# >..........< .......#.. #...#..... ^ ^ ^ ``` These rows and columns need to be _twice as big_; the result of cosmic expansion therefore looks like this: ``` ....#........ .........#... #............ ............. ............. ........#.... .#........... ............# ............. ............. .........#... #....#....... ``` Equipped with this expanded universe, the shortest path between every pair of galaxies can be found. It can help to assign every galaxy a unique number: ``` ....1........ .........2... 3............ ............. ............. ........4.... .5........... ............6 ............. ............. .........7... 8....9....... ``` In these 9 galaxies, there are _36 pairs_. Only count each pair once; order within the pair doesn't matter. For each pair, find any shortest path between the two galaxies using only steps that move up, down, left, or right exactly one `.` or `#` at a time. (The shortest path between two galaxies is allowed to pass through another galaxy.) For example, here is one of the shortest paths between galaxies `5` and `9`: ``` ....1........ .........2... 3............ ............. ............. ........4.... .5........... .##.........6 ..##......... ...##........ ....##...7... 8....9....... ``` This path has length `9` because it takes a minimum of _nine steps_ to get from galaxy `5` to galaxy `9` (the eight locations marked `#` plus the step onto galaxy `9` itself). Here are some other example shortest path lengths: - Between galaxy `1` and galaxy `7`: 15 - Between galaxy `3` and galaxy `6`: 17 - Between galaxy `8` and galaxy `9`: 5 In this example, after expanding the universe, the sum of the shortest path between all 36 pairs of galaxies is `374`. Expand the universe, then find the length of the shortest path between every pair of galaxies. _What is the sum of these lengths?_