Problem Description

You are provided with a sorted array named arr, which consists of the number 1 and other prime numbers. All elements in this array are unique. Alongside this, you are given an integer k. Taking every combination of elements i and j from the array—where the index i is less than the index j—you form fractions of the form arr[i] / arr[j]. Your task is to find the kth smallest fraction among all the possible fractions you can create this way. The result should be returned as an array with two elements: [arr[i], arr[j]], representing the numerator and denominator of the determined fraction, respectively.

Intuition

To solve this problem, we leverage a min-heap to efficiently track and sort fractions according to their value. The min-heap is a data structure that allows us to always have access to the smallest element. By pushing all possible fractions formed by the first element of the array as the numerator and all other elements as denominators into the heap, we get all the smallest possible fractions with 1 / arr[j].

The min-heap is initialized with tuples containing the fraction arr[i] / arr[j], as well as the indices i and j. This initial population of the heap starts with i fixed at 0 and j ranging over all valid indices, which ensures that all the smallest fractions are considered first.

Once the heap is populated, the following steps are taken:

1. We pop out the smallest element from the heap.
2. Since each fraction arr[i] / arr[j] is formed by considering all the possible j for a particular i, we need to consider the next possible fraction for the current smallest i. This means if arr[i] / arr[j] was the smallest, we now need to consider arr[i + 1] / arr[j], ensuring i + 1 < j to maintain our fraction condition i < j.
3. We then push the new fraction arr[i + 1] / arr[j] into the heap.
4. This process is repeated k - 1 times because every pop operation retrieves the smallest fraction at the moment, and we want the kth smallest.

After repeating this process k - 1 times, the top of the heap contains the kth smallest fraction. We then return this fraction as [numerator, denominator] using the indices stored in the heap tuple to access elements from the arr.

The reason we don't initialize the heap with all possible fractions is because it would be inefficient. Since the array is sorted, the smallest fractions are formed with the smallest denominator. Hence, we initially consider these and incrementally add fractions with larger numerators, leveraging the heap's sorting property to efficiently find the kth smallest fraction.

Learn more about Binary Search, Sorting and Heap (Priority Queue) patterns.

Solution Approach

The solution involves multiple concepts, primarily heap data structure operations and basic arithmetic. Let's walk through the implementation:

• Initialize the Heap: The first step involves initializing a min-heap with tuples. Each tuple contains three elements:

  1. The value of the fraction arr[i] / arr[j].
  2. The index i of the numerator.
  3. The index j of the denominator.

  Using Python's heapq module, we create a min-heap because it allows us to easily push and pop the smallest elements. The heap is populated with the reciprocal of all elements of arr starting from the second element because the list of primes is sorted and fractions with 1 as a numerator will be the smallest.

• Heapify: The heapify function converts the list into a heap structure. This step is essential to maintain the heap properties after the initial insertion of elements.

• Iteration and Heap Operations: The core logic of the heap manipulation happens inside a loop that runs k-1 times. This loop represents the iteration to find the kth smallest element:

  for _ in range(k - 1):
      frac, i, j = heappop(h)
      if i + 1 < j:
          heappush(h, (arr[i + 1] / arr[j], i + 1, j))

  In each iteration, we:

  • Extract (heappop) the smallest element from the heap. The smallest element corresponds to the currently smallest fraction.
  • Check if we can form a new fraction by incrementing the numerator's index i. If i + 1 < j, it means there is another fraction to consider.
  • If the new fraction can be formed, push (heappush) the new fraction arr[i + 1] / arr[j] along with its indices back into the heap.

• Extracting the Result: After the loop has completed k-1 iterations, the smallest element remaining on top of the heap is the kth smallest fraction. We extract the indices i and j from the top of the heap and use them to return the result as [arr[i], arr[j]].

• Return the kth Smallest Fraction: Finally, the indices from the tuple that's at the top of the heap after k-1 pops represent the kth smallest fraction, and the final return statement return [arr[h[0][1]], arr[h[0][2]]] fetches the numerator and denominator from the arr.

In summary, the algorithm efficiently keeps track of the potential kth smallest fraction at each step by using a min-heap to ensure that the smallest possible fraction is always available for comparison, without the need to calculate and store all possible fractions at the beginning. It's an elegant solution that combines heap operations with the sorted property of the input array to provide an efficient answer.

Example Walkthrough

Let's go through an example to illustrate the solution approach. Assume we have the sorted array arr containing prime numbers, which looks like this: [1, 2, 3, 5], and we want to find the kth smallest fraction where k=3.

The initial step is to initialize a min-heap and populate it with fractions having 1 as the numerator. So initially, our heap looks like this, containing the value of the fraction and the indices (numerator index, denominator index):

• [(0.5, 0, 1), (0.333..., 0, 2), (0.2, 0, 3)]

We run the heapify process to ensure it's a valid min-heap (although in this case, it's already a min-heap because we started with the smallest possible fractions):

After the heapify process, our heap remains the same (it's already in heap order):

• [(0.5, 0, 1), (0.333..., 0, 2), (0.2, 0, 3)]

Now we want to find the 3rd smallest fraction. We need to perform k-1 operations on the heap.

1. First iteration:

   • We pop out the smallest fraction, which is (0.2, 0, 3) corresponding to fraction 1/5.
   • Next, we check whether we can form a new fraction by incrementing the numerator's index. However, since there is not an index i + 1 for i=0 that is less than j=3, we do not push a new fraction to the heap.
   • The heap after the first pop is: [(0.333..., 0, 2), (0.5, 0, 1)].

2. Second iteration:

   • We pop the next smallest fraction, which is (0.333..., 0, 2) corresponding to the fraction 1/3.
   • We verify if a new fraction can be formed with i+1 where i=0 before j=2, but since 1 < 2, we cannot increment i to get a valid new fraction while keeping i < j.
   • The heap after the second pop is: [(0.5, 0, 1)].

Since we've done k-1 operations (here k=3, so we did 2 operations), the top of the heap now contains the 3rd smallest fraction. We now have the smallest fraction on top of the heap as (0.5, 0, 1) which corresponds to the fraction 1/2.

The result for k=3 would be [arr[0], arr[1]] which translates to [1, 2]. Thus, the 3rd smallest fraction formed by elements from arr is 1/2.

By following this process, we efficiently find the kth smallest fraction without creating a full list of fractions at the beginning and instead only maintaining a heap of the smallest fractions at any given time, minimizing memory usage and computation.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the given code is governed by the following factors:

1. Heap Construction: The list comprehension creates a heap with an initial size of n-1, where n is the length of the input array arr. The heapify function has a time complexity of O(n).

2. Heap Operations: The main loop runs (k - 1) times because it pops the smallest element from the heap and potentially pushes a new element onto the heap. Each heappop and heappush operation has a time complexity of O(log n).

Thus, the total time complexity is given by the initial heapification, O(n), plus the k iterations of heap operations, each of which is O(log n), resulting in O(n + klog n).

Space Complexity

The space complexity of the given code depends on:

1. Heap Space: The heap size is at most n-1, where n is the length of the input array arr.
2. No Additional Space: No additional space other than the heap is used that grows with input size.

Hence, the space complexity is O(n) since that is the space used by the heap.

Learn more about how to find time and space complexity quickly using problem constraints.