Cracking the Coding Interview, 4 Edition - 150 Programming Interview Questions and Solutions

Solutions to Chapter 20 | Hard
20.6 Describe an algorithm to find the largest 1 million numbers in 1 billion numbers. Assume
that the computer memory can hold all one billion numbers.
SOLUTION
pg 91
Approach 1: Sorting
Sort the elements and then take the first million numbers from that. Complexity is O(n log n).
Approach 2: Max Heap
1. Create a Min Heap with the first million numbers.
2. For each remaining number, insert it in the Min Heap and then delete the minimum
value from the heap.
3. The heap now contains the largest million numbers.
4. This algorithm is O(n log m), where m is the number of values we are looking for.
Approach 3: Selection Rank Algorithm (if you can modify the original array)
Selection Rank is a well known algorithm in computer science to find the ith smallest (or
largest) element in an array in expected linear time. The basic algorithm for finding the ith
smallest elements goes like this:
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Pick a random element in the array and use it as a ‘pivot’. Move all elements smaller than
that element to one side of the array, and all elements larger to the other side.
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If there are exactly i elements on the right, then you just find the smallest element on
that side.
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Otherwise, if the right side is bigger than i, repeat the algorithm on the right. If the right
side is smaller than i, repeat the algorithm on the left for i – right.size().
Given this algorithm, you can either:
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Tweak it to use the existing partitions to find the largest i elements (where i = one million).
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Or, once you find the ith largest element, run through the array again to return all elements
greater than or equal to it.
This algorithm has expected O(n) time.
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