Write up for day 6.

day-06/README.md

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+# Day 6 efficient solution
+
+For day 6, we have an array of lights (first booleans, then integers) on which
+we want to perform some tasks, and then know the end result. To do so, the
+easiest method is to simulate that process.
+
+All operations execute on a specific rectangle of lights. This means that we
+need to modify a potentially large number of lights at the same time. This is a
+rather costly operation.
+
+To get around this, we use run length encoding. Instead of storing a boolean
+for each light, we store, per column, a list of runs of lights that are in the
+same state. That way, the complexity of lights to flip is now proportional to
+the number of rows we need to flip.
+
+## Updating the runs
+
+When we need to update the runs for a given column, we iterate over all runs
+that are in that column. If a run is within the row range that we need to
+modify, we can set the state of the run to the proper value. You may need to
+split the run in several parts that are not all affected by the modification.
+
+As a second step, we try to combine adjacent runs that are in the same state.
+This reduces the cost of iterating over all of them significantly.
+
+## Performance analysis
+
+On my computer, the naive brute force implementation took about 5 seconds for
+both part 1 and part 2. When running the run length implementation on the first
+part, the algorithm is 20 times faster. However, on the second part, the
+algorithm is only slightly faster (3.5 seconds) because runs can have more
+values, and are thus not as easily combined.