Smallest Range Covering Elements from K Lists

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from queue import PriorityQueue
from typing import List

class Solution:
    def smallestRange(self, nums: List[List[int]]) -> List[int]:
        k = len(nums)
        pq = PriorityQueue()
        max_num = float('-inf')

        # Add the first element of each list to the priority queue
        # and keep track of the maximum value among the first elements
        for i in range(k):
            pq.put((nums[i][0], i, 0))
            max_num = max(max_num, nums[i][0])

        best_range = float('inf')
        best_start, best_end = -1, -1

        # Continue until all elements are processed
        while pq.qsize() == k:
            val, row, col = pq.get()
            current_range = max_num - val

            # Update the best range if the current one is smaller
            if current_range < best_range:
                best_range = current_range
                best_start = val
                best_end = max_num

            # Move to the next element in the same row
            if col + 1 < len(nums[row]):
                next_val = nums[row][col + 1]
                pq.put((next_val, row, col + 1))
                max_num = max(max_num, next_val)

        return [best_start, best_end]

10 Prerequisite LeetCode Problems

“632. Smallest Range Covering Elements from K Lists” covers heaps, pointers, and interval handling. Here are 10 problems before attempting this one:

“21. Merge Two Sorted Lists”: This is a simple problem to understand merging two sorted lists.
“88. Merge Sorted Array”: Here, you have to merge two sorted arrays, which is a step up from merging sorted lists.
“23. Merge k Sorted Lists”: This is a natural progression from merging two sorted lists or arrays. It introduces the concept of a priority queue (or heap) for efficient merging.
“239. Sliding Window Maximum”: This problem introduces the concept of a sliding window, which is a key aspect of the problem at hand.
“295. Find Median from Data Stream”: This problem requires you to manipulate a data structure to keep track of a dynamically changing median, similar to how you must keep track of the smallest range in the given problem.
“347. Top K Frequent Elements”: This problem requires using a heap to solve a problem efficiently, a skill that is required in the given problem.
“378. Kth Smallest Element in a Sorted Matrix”: This problem gives you practice with dealing with sorted structures and using a heap for efficiency.
“480. Sliding Window Median”: This problem is a combination of the “sliding window” and “median finding” problems, bringing you closer to the requirements of the given problem.
“581. Shortest Unsorted Continuous Subarray”: This problem requires you to find a range in an array, similar to the given problem.
“759. Employee Free Time”: This problem requires merging several sorted lists of intervals, which is a good lead-in to the problem at hand.

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class Solution:
    def smallestRange(self, nums: List[List[int]]) -> List[int]:
        n = len(nums)
        pq = []
        ma = 0

        for i in range(n):            
            heappush(pq, (nums[i][0] , i, 0))
            ma = max(ma , nums[i][0])
        
        ans = [pq[0][0] , ma]
        while True:
            _,i,j = heappop(pq)

            if j == len(nums[i])-1:
                break

            next_num = nums[i][j+1]
            ma = max( ma , next_num)
            heappush(pq,(next_num, i , j+1))

            if ma-pq[0][0] < ans[1] - ans[0]:
                ans= [pq[0][0], ma]
        return ans

Problem Classification

The problem falls under the category of “Algorithms and Data Structures”, more specifically, it deals with “Array Manipulation” and “Priority Queues or Heaps”. The problem involves dealing with multiple sorted arrays and trying to find a specific range in those arrays.

‘What’ Components:

You are given ‘k’ lists of sorted integers in non-decreasing order.
You need to find the smallest range that includes at least one number from each of the ‘k’ lists.
A range [a, b] is considered smaller than another range [c, d] if (b - a) < (d - c), or if a < c when (b - a) == (d - c).

This is an “Optimization” problem, as the task is to find an optimal range (smallest in this case) that satisfies the given conditions. It also involves elements of “Searching” as you have to look through ‘k’ lists to find this range. Given the multiple lists involved and the requirement to find a smallest range across these lists, this problem could also be seen as a “Multi-List Manipulation” problem.

Language Agnostic Coding Drills

Understanding the Problem Statement - Given k lists of sorted integers, the task is to find the smallest range that includes at least one number from each of the k lists. The range [a, b] is smaller than range [c, d] if b-a < d-c or a < c if b-a == d-c.
List and Array Manipulation - Practice creating, manipulating, and iterating through lists and arrays. This is foundational in many coding problems, and it’s necessary here to iterate through the list of lists (nums) and to add elements to and remove elements from the heap.
Priority Queues - Learn how to implement a priority queue using a heap data structure. Understand how to push elements onto the heap and pop them off. In Python, the heapq module provides heap operations, which can be used to create a priority queue.
Heap Manipulation - The heap data structure is used to keep track of the minimum and maximum elements in a collection, or to sort a collection. The heap used in this code is a min-heap, which is a complete binary tree where the key at the root is minimum among all other keys. The heap is used to keep track of the current minimum element, which is the first element of the current smallest range.
Loops and Conditionals - Understand how to use loops to iterate through collections and how to use conditional statements to make decisions. Here, a while-loop is used to repeatedly pop the minimum element from the heap and push the next element from the same list. The loop breaks when it encounters the end of one of the lists.
Tuple Manipulation - Learn how to create, access and manipulate tuples. In this problem, each element in the heap is a tuple containing the number, its list index, and its element index within the list.
Math Operations - Practice performing basic math operations. The maximum and minimum numbers in the current range are tracked, and these are used to calculate the size of the current range and compare it to the smallest range found so far.

These are the steps to solve the problem:

Push the first element of each list into a min-heap along with its list index and its element index within the list. Track the maximum element.
Pop the heap to get the current smallest number and its indices. If the range from this number to the current maximum number is smaller than the smallest range found so far, update the smallest range.
Push the next number from the same list into the heap and update the maximum number if necessary. If we’ve reached the end of a list, we’ve found the smallest possible range that includes at least one number from each list, so we can stop.
Repeat the above steps until we’ve exhausted one of the lists.
Return the smallest range found.

Targeted Drills in Python

Understanding the Problem Statement
For this drill, simply practice reading problem statements and understanding what they are asking for. There’s no specific Python code for this drill, as it’s more about comprehension than coding.

List and Array Manipulation

Create and manipulate a list of lists. Here’s an example of how to do it in Python:

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nums = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
for i in range(len(nums)):
    for j in range(len(nums[i])):
        print(nums[i][j], end=' ')
    print()

Priority Queues

Learn how to create a priority queue in Python using the heapq module:

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import heapq
queue = []
heapq.heappush(queue, (2, 'A'))
heapq.heappush(queue, (1, 'B'))
heapq.heappush(queue, (3, 'C'))
while queue:
    print(heapq.heappop(queue))

Heap Manipulation

Understand how to manipulate heaps. Here’s how to create a min heap and perform operations on it:

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import heapq
h = []
heapq.heappush(h, (1, 'one'))
heapq.heappush(h, (3, 'three'))
heapq.heappush(h, (2, 'two'))
print('Smallest:', h[0])
print('Pop:', heapq.heappop(h))
print('After Pop:', h)

Loops and Conditionals

Practice using loops and conditionals. Here’s a simple example:

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for i in range(10):
    if i % 2 == 0:
        print(i, "is even")
    else:
        print(i, "is odd")

Tuple Manipulation

Learn how to manipulate tuples in Python. For example:

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t = (1, 'one', True)
print(t[0])
print(t[1])
print(t[2])

Math Operations

Practice performing basic math operations. Here’s a simple example:

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a = 10
b = 3
print('Add:', a + b)
print('Subtract:', a - b)
print('Multiply:', a * b)
print('Divide:', a / b)

After mastering these drills, you should be able to understand the given code better and implement it as well.