Heap

Merge K Sorted Lists

hard

DESCRIPTION (inspired by Leetcode.com)

Given k linked lists, each sorted in ascending order, in a list lists, write a function to merge the input lists into one sorted linked list.

Example 1:

Inputs:

lists = [[3,4,6],[2,3,5],[-1,6]]
3 -> 4 -> 6
2 -> 3 -> 5
-1 -> 6

Output:

[-1,2,3,3,4,5,6,6]

-1 -> 2 -> 3 -> 3 -> 4 -> 5 -> 6 -> 6

Code Editor

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Explanation

This problem can be solved using a min-heap of size k.

The min-heap will always store the next unmerged element from each of the k sorted arrays. The element with the smallest value sits at the root of the heap. We build the merged array by repeatedly popping the root from the heap. Each time we pop from the heap, we also push the next element from the same array (if one exists) onto the heap.

When the heap is empty, all elements from all arrays have been merged, and we can return the merged result.

Step 1: Initialize the heap

The first step is to initialize the heap with the first element from each of the k arrays. We iterate over each array and push a tuple containing three values onto the heap:

The element's value
The array's index (which array it came from)
The element's index within that array (starting at 0)

Visualization


from heapq import heappush, heappop
def mergeKLists(lists):
  if not lists:
    return None
  
  non_empty = [head for head in lists if head]
  if not non_empty:
    return None
  
  heap = []
  for head in non_empty:
    heappush(heap, (head.val, id(head), head))
  
  dummy = ListNode(0)
  current = dummy
  
  while heap:
    val, _, node = heappop(heap)
    current.next = node
    current = current.next
    
    if node.next:
      heappush(heap, (node.next.val, id(node.next), node.next))
  
  return dummy.next

initialize heap

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Step 2: Build the merged array

With the heap initialized, we can now build the merged result array. We start by creating an empty result array where we'll append elements in sorted order.

Visualization


from heapq import heappush, heappop
def mergeKLists(lists):
  if not lists:
    return None
  
  non_empty = [head for head in lists if head]
  if not non_empty:
    return None
  
  heap = []
  for head in non_empty:
    heappush(heap, (head.val, id(head), head))
  
  dummy = ListNode(0)
  current = dummy
  
  while heap:
    val, _, node = heappop(heap)
    current.next = node
    current = current.next
    
    if node.next:
      heappush(heap, (node.next.val, id(node.next), node.next))
  
  return dummy.next

add -1 to heap

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Now, we repeatedly pop the root of the heap, which gives us the element with the smallest value among the unmerged elements from the k arrays. We append this value to our result array.

Visualization


from heapq import heappush, heappop
def mergeKLists(lists):
  if not lists:
    return None
  
  non_empty = [head for head in lists if head]
  if not non_empty:
    return None
  
  heap = []
  for head in non_empty:
    heappush(heap, (head.val, id(head), head))
  
  dummy = ListNode(0)
  current = dummy
  
  while heap:
    val, _, node = heappop(heap)
    current.next = node
    current = current.next
    
    if node.next:
      heappush(heap, (node.next.val, id(node.next), node.next))
  
  return dummy.next

initialize result array

0 / 1

To ensure that our heap always contains the next unmerged element from each array, after popping an element, we check if there's a next element in the same array. If there is, we push a tuple with the next element's value, the same array index, and the incremented element index onto the heap.

Visualization


from heapq import heappush, heappop
def mergeKLists(lists):
  if not lists:
    return None
  
  non_empty = [head for head in lists if head]
  if not non_empty:
    return None
  
  heap = []
  for head in non_empty:
    heappush(heap, (head.val, id(head), head))
  
  dummy = ListNode(0)
  current = dummy
  
  while heap:
    val, _, node = heappop(heap)
    current.next = node
    current = current.next
    
    if node.next:
      heappush(heap, (node.next.val, id(node.next), node.next))
  
  return dummy.next

add -1 to result

0 / 1

When the heap is empty, all elements from all arrays have been merged, and we can return the result array.

Visualization


from heapq import heappush, heappop
def mergeKLists(lists):
  if not lists:
    return None
  
  non_empty = [head for head in lists if head]
  if not non_empty:
    return None
  
  heap = []
  for head in non_empty:
    heappush(heap, (head.val, id(head), head))
  
  dummy = ListNode(0)
  current = dummy
  
  while heap:
    val, _, node = heappop(heap)
    current.next = node
    current = current.next
    
    if node.next:
      heappush(heap, (node.next.val, id(node.next), node.next))
  
  return dummy.next

add 6 to result

0 / 1

Solution

Try these examples:

Visualization


from heapq import heappush, heappop
def mergeKLists(lists):
  if not lists:
    return None
  
  non_empty = [head for head in lists if head]
  if not non_empty:
    return None
  
  heap = []
  for head in non_empty:
    heappush(heap, (head.val, id(head), head))
  
  dummy = ListNode(0)
  current = dummy
  
  while heap:
    val, _, node = heappop(heap)
    current.next = node
    current = current.next
    
    if node.next:
      heappush(heap, (node.next.val, id(node.next), node.next))
  
  return dummy.next

merge k sorted arrays

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What is the time complexity of this solution?

O(N + Q)

O(4ⁿ)

O(2ⁿ)

O(n * log k)

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