Friday, September 20, 2013

Shortest Interval in k-sorted list - Hard

Given k-sorted array, find the minimum interval such that there is at least one element from each array within the interval. Eg. [1,10,14],[2,5,10],[3,40,50]. Output : 1-3

To solve this problem, we perform a k-way merge as described here. At each point of 'popping' an element from an array. We keep track of the minimum and maximum head element (the first element) from all the k-lists. The minimum and maximum will obviously contain the rest of the header elements of the k-arrays. As we keep doing this, we find the smallest interval (max - min). That will be our solution. Here's a pictorial working of the algorithm.

And here's the Python code. Time Complexity : O(nlogk)

Merge k-sorted lists - Medium

Problem - Given k-sorted lists, merge them into a single sorted list.
A daft way of doing this would be to copy all the list into a new array and sorting the new array. ie O(n log(n))

The naive method would be to simply perform k-way merge similar to the auxiliary method in Merge Sort. But that is reduces the problem to a minimum selection from a list of k-elements. The Complexity of this algorithm is an abysmal O(nk).

Here's how it looks in Python. We maintain an additional array called Index[1..k] to maintain the head of each list.

We improve upon this by optimizing the minimum selection process by using a familiar data structure, the Heap! Using a MinHeap, we extract the minimum element from a list and then push the next element from the same list into the heap, until all the list get exhausted.

This reduces the Time complexity to O(nlogk) since for each element we perform O(logk) operations on the heap. An important implementation detail is we need to keep track of the origins of the elements of the heap, ie the list it came from and it's index. Therefore, I use a 3 member tuple (item, list-index, item-index) and push it onto the heap.

Fortunately for us, Python has the heapq module for implementing a heap. So here's the code.

The entire code can be found here.

Find Increasing Triplet Subsequence - Medium

Problem - Given an integer array A[1..n], find an instance of i,j,k where 0 < i < j < k <= n and A[i] < A[j] < A[k].

Let's start with the obvious solution, bruteforce every three element combination until we find a solution.

Let's try to reduce this by searching left and right for each element, we search left for an element smaller than the current one, towards the right an element greater than the current one. When we find an element that has both such elements, we found the solution. The complexity of this solution is O(n^2).
To reduce this even further, One can simply apply the longest increasing subsequence algorithm and break when we get an LIS of length 3. But the best algorithm that can find an LIS is O(nlogn) with O( n ) space.

An O(nlogn) algorithm seems like an overkill! Can this be done in linear time?

The Algorithm:
We iterate over the array and keep track of two things.
  • The minimum value iterated over (min)
  • The minimum increasing pair (a,b) where min <= a < b
When we encounter an element, say v,  that is greater than the increasing pair we return (a,b,v).

Here's my Python implementation to make my point.

Until next time, Cheers!