Day 46: Sort K sorted array

day46/README.md

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+Question of the day: http://www.techiedelight.com/sort-k-sorted-array/
+
+Given a k-sorted array, which is an array that is almost sorted, to the
+point where each of the N elements in the array are at most `k` index
+positions away from its correct sorted order, find an efficient
+algorithm to sort the array.
+
+## Ideas
+
+Ignoring the interesting constraint, if we want to end up with a sorted
+array, we can just use an efficient sorting algorithm such as merge sort
+to sort the array and output it. An in-place merge sort would take `O(NlogN)`
+time and `O(1)` space complexity. 
+
+Using the k-sorted constraint to our advantage, we can pick a better sort,
+insertion sort which would run in `O(Nk)` time. For each element in the array,
+check its `k` nearest neighbors to find the right place to insert it.
+
+However, the best time complexity lies in a heap solution. If we start off
+by creating a min heap of size `k+1`, we can iterate through the array by
+popping the min element out of the heap and adding a new element into the
+heap from the array. For each element, we do one pop from the heap and one
+push onto the heap, both of which are `O(logk)` time complexity. The total
+time complexity would therefore be `O(Nlogk)`. The difference in runtime
+would especially be noticeable for larger values of `k`. The space required
+to maintain the heap would be `O(k)` and the space we need to store the
+resulting sorted array would come out to `O(N)`.
+
+## Code
+
+[Python](./sortKSortedArray.py)
+
+## Follow up
+

day46/sortKSortedArray.py

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+from heapq import *
+
+def sortKSortedArray(arr, k):
+    heap = arr[:k]
+    heapify(heap)
+
+    remaining = arr[k:]
+
+    result = []
+    for num in remaining:
+        # push num into the heap to maintain k+1 elements,
+        # then pop the min
+        result.append(heappushpop(heap, num))
+
+    while len(heap) > 0:
+        result.append(heappop(heap))
+
+    return result
+
+def testSortKSortedArray():
+    assert sortKSortedArray([1,4,5,2,3,7,8,6,10,9], 2) == [1,2,3,4,5,6,7,8,9,10]
+    assert sortKSortedArray([2,1,3], 1) == [1, 2, 3]
+
+def tests():
+    testSortKSortedArray()
+
+if __name__ == "__main__":
+    tests()