Day 16 part 2 method 2 optimized

Author: jcisio

day16/README.md

@@ -125,4 +125,16 @@ It took me like 5 minutes to write this:
 ```
 And it takes forever trying to finish the first iteration. When I look at the code, the complexity analyse above is wrong, it actually needs `100*(L/2)^2 = 10^15` additions to finish.
 
-Lesson learnt: *sometimes we can just let the computer do the computing, but sometimes maths is helpful to solve impossible computing problem*.
+Lesson learnt: *sometimes we can just let the computer do the computing, but sometimes maths is helpful to solve impossible computing problem*.
+
+PS: now I got two stars, I went take a look on subreddit, and found that my method above is far from optimized. Each iteration could be written as:
+```python
+    for _ in range(N_ITERATION):
+        output[-1] = input[-1]
+        for j in range(L2-2, -1, -1):
+            output[j] = (output[j+1] + input[j]) % 10
+        input = output.copy()
+```
+We reduced the complexity of each iteration from O(L^2) to O(L), and PyPy finished in 0.57s instead of 3.6s (but CPython is slower: 6.4s). We can see this trick is many place, to do sub array sum in O(1) (with O(L) preparation time) instead of O(L) without preparation.
+
+A user, bla2, cleverly [remarked](https://www.reddit.com/r/adventofcode/comments/ebxz7f/2019_day_16_part_2_visualization_and_hardmode/) that we can apply this trick to calculate the full message. The complexity per iteration is now O(LlogL) instead of O(L). With `L = 6.5*10^6` and 100 iteration, Python will be two slow and his C implementation finished in 50s.

day16/d16.py

@@ -40,15 +40,16 @@ def part2_2(input):
         sys.exit('Oh my, I have no idea deal with that offset.')
 
     input = list(map(int, (input*N_REPEAT)[offset:]))
-    output = input.copy()
-    for i in range(N_ITERATION):
-        print('Iteration', i, len(input))
-        for j in range(len(input)):
-            output[j] = sum(input[j:]) % 10
+    L2 = len(input)
+    output = [0]*L2
+    for _ in range(N_ITERATION):
+        output[-1] = input[-1]
+        for j in range(L2-2, -1, -1):
+            output[j] = (output[j+1] + input[j]) % 10
         input = output.copy()
-    return ''.join(map(str, output))
+    return ''.join(map(str, output))[0:8]
 
 
 print('Part 1:', get_output(input, N_ITERATION)[0:8])
 print('Part 2:', part2(input))
-#print('Part 2:', part2_2(input))
+print('Part 2:', part2_2(input))