hauleth.dev
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ft: add day 15.2022
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2022/day15.livemd
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2022/day15.livemd
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<!-- livebook:{"persist_outputs":true} -->
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# Day 15
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```elixir
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Mix.install([
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{:kino_aoc, git: "https://github.com/ljgago/kino_aoc"}
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])
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```
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<!-- livebook:{"output":true} -->
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```
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:ok
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```
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## Section
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<!-- livebook:{"attrs":{"day":"15","session_secret":"ADVENT_OF_CODE_SESSION","variable":"puzzle_input","year":"2022"},"kind":"Elixir.KinoAOC.HelperCell","livebook_object":"smart_cell"} -->
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```elixir
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{:ok, puzzle_input} =
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KinoAOC.download_puzzle("2022", "15", System.fetch_env!("LB_ADVENT_OF_CODE_SESSION"))
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```
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<!-- livebook:{"output":true} -->
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```
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{:ok,
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"Sensor at x=2288642, y=2282562: closest beacon is at x=1581951, y=2271709\nSensor at x=2215505, y=2975419: closest beacon is at x=2229474, y=3709584\nSensor at x=275497, y=3166843: closest beacon is at x=-626874, y=3143870\nSensor at x=1189444, y=2115305: closest beacon is at x=1581951, y=2271709\nSensor at x=172215, y=2327851: closest beacon is at x=-101830, y=2000000\nSensor at x=3953907, y=1957660: closest beacon is at x=2882446, y=1934422\nSensor at x=685737, y=2465261: closest beacon is at x=1581951, y=2271709\nSensor at x=1458348, y=2739442: closest beacon is at x=1581951, y=2271709\nSensor at x=3742876, y=2811554: closest beacon is at x=3133845, y=3162635\nSensor at x=437819, y=638526: closest beacon is at x=-101830, y=2000000\nSensor at x=2537979, y=1762726: closest beacon is at x=2882446, y=1934422\nSensor at x=1368739, y=2222863: closest beacon is at x=1581951, y=2271709\nSensor at x=2743572, y=3976937: closest beacon is at x=2229474, y=3709584\nSensor at x=2180640, y=105414: closest beacon is at x=3011118, y=-101788\nSensor at x=3845753, y=474814: closest beacon is at x=3011118, y=-101788\nSensor at x=2493694, y=3828087: closest beacon is at x=2229474, y=3709584\nSensor at x=2786014, y=3388077: closest beacon is at x=3133845, y=3162635\nSensor at x=3593418, y=3761871: closest beacon is at x=3133845, y=3162635\nSensor at x=856288, y=3880566: closest beacon is at x=2229474, y=3709584\nSensor at x=1757086, y=2518373: closest beacon is at x=1581951, y=2271709\nSensor at x=2853518, y=2939097: closest beacon is at x=3133845, y=3162635\nSensor at x=1682023, y=1449902: closest beacon is at x=1581951, y=2271709\nSensor at x=3360575, y=1739100: closest beacon is at x=2882446, y=1934422\nSensor at x=2904259, y=1465606: closest beacon is at x=2882446, y=1934422\nSensor at x=3078500, y=3564862: closest beacon is at x=3133845, y=3162635\nSensor at x=2835288, y=1011055: closest beacon is at x=2882446, y=1934422\nSensor at x=2998762, y=2414323: closest beacon is at x=2882446, y=1934422\n"}
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```
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```elixir
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points =
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puzzle_input
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|> String.split("\n", trim: true)
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|> Enum.map(fn line ->
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%{"sx" => sx, "sy" => sy, "bx" => bx, "by" => by} =
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Regex.named_captures(
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~r/x=(?<sx>-?\d+), y=(?<sy>-?\d+):.*x=(?<bx>-?\d+), y=(?<by>-?\d+)/,
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line
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)
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%{
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sensor: {String.to_integer(sx), String.to_integer(sy)},
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beacon: {String.to_integer(bx), String.to_integer(by)}
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}
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end)
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circles =
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points
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|> Enum.map(fn %{beacon: {bx, by}, sensor: {sx, sy}} ->
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radius = abs(bx - sx) + abs(by - sy)
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{{sx, sy}, radius}
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end)
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```
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<!-- livebook:{"output":true} -->
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```
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[
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{{2288642, 2282562}, 717544},
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{{2215505, 2975419}, 748134},
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{{275497, 3166843}, 925344},
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{{1189444, 2115305}, 548911},
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{{172215, 2327851}, 601896},
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{{3953907, 1957660}, 1094699},
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{{685737, 2465261}, 1089766},
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{{1458348, 2739442}, 591336},
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{{3742876, 2811554}, 960112},
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{{437819, 638526}, 1901123},
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{{2537979, 1762726}, 516163},
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{{1368739, 2222863}, 262058},
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{{2743572, 3976937}, 781451},
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{{2180640, 105414}, 1037680},
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{{3845753, 474814}, 1411237},
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{{2493694, 3828087}, 382723},
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{{2786014, 3388077}, 573273},
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{{3593418, 3761871}, 1058809},
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{{856288, 3880566}, 1544168},
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{{1757086, 2518373}, 421799},
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{{2853518, 2939097}, 503865},
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{{1682023, 1449902}, 921879},
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{{3360575, 1739100}, 673451},
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{{2904259, 1465606}, 490629},
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{{3078500, 3564862}, 457572},
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{{2835288, 1011055}, 970525},
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{{2998762, 2414323}, 596217}
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]
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```
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```elixir
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defmodule RangeSet do
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defstruct ranges: []
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def new(), do: %__MODULE__{}
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def new(%__MODULE__{} = set), do: set
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def new(a..b//1) when a <= b, do: %__MODULE__{ranges: [a..b//1]}
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def new(b..a//-1) when b > a, do: %__MODULE__{ranges: [a..b//1]}
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def new(list), do: %__MODULE__{ranges: list |> Enum.sort_by(& &1.first) |> squash()}
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def min(%__MODULE__{ranges: [a.._ | _]}), do: a
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def max(%__MODULE__{ranges: ranges}) do
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_..a = List.last(ranges)
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a
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end
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def gaps(%__MODULE__{ranges: ranges}) do
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gaps =
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ranges
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|> Enum.chunk_every(2, 1, :discard)
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|> Enum.flat_map(fn [_..a, b.._] -> [(a + 1)..(b - 1)] end)
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|> squash()
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%__MODULE__{ranges: gaps}
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end
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def empty?(%__MODULE__{ranges: ranges}), do: ranges == []
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def has?(%__MODULE__{ranges: ranges}, value), do: Enum.any?(ranges, &(value in &1))
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def length(%__MODULE__{ranges: ranges}),
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do: Enum.reduce(ranges, 0, fn r, acc -> acc + Range.size(r) end)
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def to_list(%__MODULE__{ranges: ranges}),
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do: Enum.flat_map(ranges, & &1)
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def remove(%__MODULE__{ranges: ranges}, int) when is_integer(int) do
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new_ranges =
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Enum.flat_map(ranges, fn a..b ->
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cond do
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a == int -> [(int + 1)..b]
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b == int -> [a..(int - 1)]
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int in a..b -> [a..(int - 1), (int + 1)..b]
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true -> [a..b]
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end
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end)
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%__MODULE__{ranges: new_ranges}
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end
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def remove(%__MODULE__{ranges: ranges}, c..d//1) when c <= d do
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new_ranges =
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Enum.flat_map(ranges, fn a..b ->
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cond do
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a in c..d and b in c..d -> []
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a < c and b in c..d -> [a..(c - 1)]
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a in c..d and b > d -> [(d + 1)..b]
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c in a..b and d in a..b -> [a..(c - 1), (d + 1)..b]
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true -> [a..b]
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end
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end)
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%__MODULE__{ranges: new_ranges}
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end
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def remove(%__MODULE__{} = set, %__MODULE__{ranges: ranges}),
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do: Enum.reduce(ranges, set, &remove(&2, &1))
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def remove(%__MODULE__{} = set, list),
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do: Enum.reduce(list, set, &remove(&2, &1))
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def add(set, int) when is_integer(int), do: add(set, int..int//1)
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def add(set, a..b//-1), do: add(set, b..a//1)
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def add(%__MODULE__{} = set, a..b//1) when a > b, do: set
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def add(%__MODULE__{ranges: ranges}, a..b//1 = range) when a <= b do
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ranges =
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ranges
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|> insert_sorted(range)
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|> squash()
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%__MODULE__{ranges: ranges}
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end
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defp insert_sorted([], val), do: [val]
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defp insert_sorted([a.._ = x | rest], b.._ = y) when a < b,
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do: [x | insert_sorted(rest, y)]
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defp insert_sorted(rest, y),
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do: [y | rest]
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defp squash([]), do: []
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defp squash([_] = list), do: list
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defp squash([a..b, c..d | rest]) when b >= c - 1,
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do: squash([a..max(b, d) | rest])
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defp squash([x | rest]), do: [x | squash(rest)]
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end
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```
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<!-- livebook:{"output":true} -->
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```
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{:module, RangeSet, <<70, 79, 82, 49, 0, 0, 37, ...>>, {:squash, 1}}
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```
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```elixir
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RangeSet.new([6..10, -8..26])
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```
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<!-- livebook:{"output":true} -->
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```
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%RangeSet{ranges: [-8..26]}
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```
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Our list can be change into list of radii of the circles (in Taxicab metric these will be visually squares), so then using inequality for circle
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$$
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r \ge d(C, a)
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$$
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where $d(a, b)$ is distance function for given metric, we can compute possible ranges of values at given line in constant time. In Taxicab metric $d$ is defined as:
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$$
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d(a, b) = |a_x - b_x| + |a_y - b_y|
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$$
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So after substitution at the above inequality we get:
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$$
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r \ge |C_x - a_x| + |C_y - a_y|
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$$
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In the task $a_y = const$ and $C = const$, so we need to solve only for $a_x$. So we can define $d_y = |C_y - a_y| = const$ as well.
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That gives us:
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$$
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|a_x - C_x| \le r - d_y
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$$
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$$
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\begin{align*}
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\therefore&\ -r + d_y &\le&\ a_x - C_x &\le&\ r - d_y \\
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\therefore&\ -r + d_y + C_x &\le&\ a_x &\le&\ r - d_y + C_x
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\end{align*}
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$$
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```elixir
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defmodule Beacons do
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def covered_at_line(circles, a_y) do
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Enum.reduce(circles, RangeSet.new(), fn {{c_x, c_y}, r}, set ->
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d_y = abs(c_y - a_y)
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RangeSet.add(set, (-r + d_y + c_x)..(r - d_y + c_x)//1)
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end)
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end
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end
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```
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<!-- livebook:{"output":true} -->
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```
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{:module, Beacons, <<70, 79, 82, 49, 0, 0, 9, ...>>, {:covered_at_line, 2}}
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```
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## Task 1
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```elixir
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a_y = 2_000_000
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beacons_at_line =
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Enum.flat_map(points, fn %{beacon: {x, y}} -> if y == a_y, do: [x], else: [] end)
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circles
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|> Beacons.covered_at_line(a_y)
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|> RangeSet.remove(beacons_at_line)
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|> RangeSet.length()
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```
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<!-- livebook:{"output":true} -->
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```
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5108096
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```
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## Task 2
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```elixir
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range = 4_000_000..0//-1
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# range = 0..4_000_000
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{y, [x]} =
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range
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|> Enum.find_value(fn y ->
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covered = Beacons.covered_at_line(circles, y)
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gaps = RangeSet.gaps(covered)
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if not RangeSet.empty?(gaps) do
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{y, RangeSet.to_list(gaps)}
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end
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end)
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x * 4_000_000 + y
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```
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<!-- livebook:{"output":true} -->
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```
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10553942650264
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```