Move 2015 out of the way.

2015/day-17/README.md

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+# Day 17 efficient solution
+
+Day 17 is an instance of the subset sum problem. This problem asks whether for
+a (multi)set of integers *V*, there is a non-empty subset of integers summing up to
+exactly *s*. This problem is NP-complete.
+
+The brute force approach of this is trying every possible set in the powerset
+of *V* to see if they match. This is inpractical however, because a powerset of
+a set of size *n* contains 2<sup>2</sup> sets.
+
+In the exercise, we have 20 buckets, and 2<sup>20</sup> is still
+brute-forcable. There is a smarter approach.
+
+We split the list of buckets in two lists of (approximately) the same size. We
+then take the powersets of those two lists and compute the sum for each entry.
+This leaves us with a total of 2<sup>n / 2 + 1</sup> entries. We then sort both
+sublists on the total value of each entry.
+
+Finally, we iterate of the first list, and use binary search to see whether
+there is an appropriately sized entry in the second list. This gives us a final
+complexity of *n* times 2<sup>*n*/2</sup>, allowing the solution to be computed
+instantly.
+
+The algorithm above can be modified to find all combinations, not just one, in
+time proportional to the number of solutions. This is implemented in the final
+program.

2015/day-17/input.txt

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+33
+14
+18
+20
+45
+35
+16
+35
+1
+13
+18
+13
+50
+44
+48
+6
+24
+41
+30
+42

2015/day-17/solution.py

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+from __future__ import print_function, division
+import fileinput
+from collections import defaultdict
+import bisect
+
+def value(buckets, choice):
+    total = 0
+    for value in buckets:
+        if choice % 2 == 1:
+            total += value
+
+        choice //= 2
+
+    return total
+
+def ones(x):
+    n = 0
+    while x > 0:
+        if x % 2:
+            n += 1
+
+        x //= 2
+
+    return n
+
+def partition(a_list):
+    pivot = len(a_list) // 2
+
+    return a_list[:pivot], a_list[pivot:]
+
+def partitionList(buckets):
+    result = [(value(buckets, x), ones(x)) for x in range(1 << len(buckets))]
+    result.sort()
+    return result
+
+buckets = []
+
+for line in fileinput.input():
+    buckets.append(int(line))
+
+partition1, partition2 = partition(buckets)
+
+values1 = partitionList(partition1)
+values2 = partitionList(partition2)
+
+possible = defaultdict(lambda: 0)
+
+i = 0
+
+target = 150
+
+for entry in values1:
+
+    i = bisect.bisect_left(values2, (target - entry[0], 0))
+
+    while i < len(values2) and entry[0] + values2[i][0] == target:
+        possible[entry[1] + values2[i][1]] += 1
+        i += 1
+
+print("Total possibilities:", sum(possible.values()))
+print("Minimal possibilities:", possible[min(possible.keys())])
+