Smallest Range Covering Elements from K Lists

Smallest Range Covering Elements from K Lists

What is this problem about?

The "Smallest Range Covering Elements from K Lists" interview question is a "HARD" level challenge. You are given k sorted lists of integers. You need to find the smallest range [a, b] that includes at least one number from each of the k lists. If there are multiple ranges of the same size, the one with the smaller a is preferred. This "Smallest Range Covering Elements from K Lists coding problem" is a variation of the multi-way merge problem.

Why is this asked in interviews?

Tech giants like Microsoft, Google, and LinkedIn ask this to test your mastery of priority queues (heaps) and the "sliding window" concept across multiple data streams. It evaluates your ability to track the minimum and maximum of a set of numbers that is constantly changing. It's a high-stakes problem that requires a very efficient O(N log K) solution (where N is the total number of elements).

Algorithmic pattern used

The primary pattern is the "Heap (Priority Queue) and Greedy interview pattern".

1. Put the first element of each of the k lists into a Min-Heap.
2. Keep track of the current max_value among all elements in the heap.
3. The current range is [min_heap_top, max_value].
4. To try and find a smaller range, pop the minimum element from the heap.
5. Push the next element from the same list that the popped element belonged to.
6. Update the max_value and re-calculate the range.
7. Stop when you cannot push any more elements from one of the lists.

Example explanation

Lists: L1:[4, 10], L2:[1, 5], L3:[7, 12]

1. Initial Heap: {1(L2), 4(L1), 7(L3)}. min=1, max=7. Range [1, 7].
2. Pop 1 from L2. Next in L2 is 5.
3. New Heap: {4(L1), 5(L2), 7(L3)}. min=4, max=7. Range [4, 7] (Smaller!).
4. Pop 4 from L1. Next in L1 is 10.
5. New Heap: {5(L2), 7(L3), 10(L1)}. min=5, max=10. Range [5, 10].
6. Pop 5 from L2. No more in L2. Stop. Smallest range: [4, 7].

Common mistakes candidates make

A common error is not tracking the current max_value correctly, which is necessary to define the range. Another mistake is using an inefficient O(N*K) search instead of a Min-Heap. Candidates also sometimes struggle to correctly identify which list to pull the next element from after a pop operation.

Interview preparation tip

Whenever you need to process multiple sorted lists simultaneously, think of a Min-Heap. This "Smallest Range Covering Elements from K Lists interview question" is a direct extension of the "Merge K Sorted Lists" algorithm.