Create sort_a_partially_sorted_array.md

Author: Joseph Luce

Diff for: real_interview_questions/Google/sort_a_partially_sorted_array.md

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+# Question
+Given an array of positive integers (possibly with duplicates) such that the numbers have been sorted only by 28 most significant bits. Sort the array completely.
+
+Example 1:
+
+Input: [0, 15, 12, 17, 18, 19, 33, 32]
+Output: [0, 12, 15, 17, 18, 19, 32, 33]
+
+```
+Explanation:
+The integers in their binary representation are:
+ 0 = 0000 0000 0000 0000 0000 0000 0000 0000
+15 = 0000 0000 0000 0000 0000 0000 0000 1111
+12 = 0000 0000 0000 0000 0000 0000 0000 1100
+17 = 0000 0000 0000 0000 0000 0000 0001 0001
+18 = 0000 0000 0000 0000 0000 0000 0001 0010
+19 = 0000 0000 0000 0000 0000 0000 0001 0011
+33 = 0000 0000 0000 0000 0000 0000 0010 0001
+32 = 0000 0000 0000 0000 0000 0000 0010 0000
+
+In sorted order:
+ 0 = 0000 0000 0000 0000 0000 0000 0000 0000
+12 = 0000 0000 0000 0000 0000 0000 0000 1100
+15 = 0000 0000 0000 0000 0000 0000 0000 1111
+17 = 0000 0000 0000 0000 0000 0000 0001 0001
+18 = 0000 0000 0000 0000 0000 0000 0001 0010
+19 = 0000 0000 0000 0000 0000 0000 0001 0011
+32 = 0000 0000 0000 0000 0000 0000 0010 0000
+33 = 0000 0000 0000 0000 0000 0000 0010 0001
+```
+
+```
+Example 2:
+
+Input: [100207, 100205, 100204, 100206, 100203]
+Output: [100203, 100204, 100205, 100206, 100207]
+Explanation:
+The integers in their binary representation are:
+100207 = 0000 0000 0000 0001 1000 0111 0110 1111
+100205 = 0000 0000 0000 0001 1000 0111 0110 1101
+100204 = 0000 0000 0000 0001 1000 0111 0110 1100
+100206 = 0000 0000 0000 0001 1000 0111 0110 1110
+100203 = 0000 0000 0000 0001 1000 0111 0110 1011
+
+In sorted order:
+100203 = 0000 0000 0000 0001 1000 0111 0110 1011
+100204 = 0000 0000 0000 0001 1000 0111 0110 1100
+100205 = 0000 0000 0000 0001 1000 0111 0110 1101
+100206 = 0000 0000 0000 0001 1000 0111 0110 1110
+100207 = 0000 0000 0000 0001 1000 0111 0110 1111
+```
+
+Expected O(n) time solution.
+
+## Solution
+
+- Runtime: O(N)
+- Space: O(1), assuming result doesn't use space
+- N = Number of elements in array
+
+The idea is to use bucket sort, since the array is already partially sorted by the first 28 bits.
+During this time, we will place the numbers into 1 of 16 buckets based on the last 4 bits.
+We can just iterate from left to right until the first 28 bits are different.
+Then we can save the results we have in the buckets and reset.
+With this approach, we will never need more than 16 buckets at a given time.
+
+```
+def sort_partial_sorted_28b(nums):
+    if len(nums) == 0:
+        return nums
+    mask = ~0 << 4
+    curr_28b = nums[0] & mask 
+    buckets = [0] * 16 # 16 buckets -> n occurances
+    results = list()
+    for num in nums:
+        if (num & mask) != curr_28b: # start sort
+            for bucket, occurance in enumerate(buckets):
+                for o in range(occurance):
+                    results.append(curr_28b | bucket)
+            curr_28b = num & mask # set to next 28 bit group
+            buckets = [0] * 16 # reset
+        # add to buckets
+        buckets[num & 15] += 1
+    for bucket, occurance in enumerate(buckets):
+        for o in range(occurance):
+            results.append(curr_28b | bucket)
+    return results
+```