2. Therefore, building the entire Heap will take N heapify operations and the total time complexity will be O(N*logN). Time Complexity: Heapify a single node takes O(log N) time complexity where N is the total number of Nodes. here is the pseudocode for Max-Heapify algorithm A is an array , index starts with 1. and i points to root of tree. Line-3 of Build-Heap runs a loop from the index of the last internal node (heapsize/2) with height=1, to the index of root(1) with height = lg(n). Before looking into Heap Sort, let's understand what is Heap and how it helps in sorting. Suppose you're looking for something that's no bigger than the smallest value in a max-heap. Now your new sorted array can be searched through in O(logN) time. For finding the Time Complexity of building a heap, we must know the number of nodes having height h. In reality, building a heap takes O(n) time depending on the implementation which can be seen here. The heart of the Heap data structure is Heapify algortihm. Its typical implementation is not stable, but can be made stable (See this) Time Complexity: Time complexity of heapify is O(Logn). 1. Heap sort has the best possible worst case running time complexity of O(n Log n). It doesn't need any extra storage and that makes it good for situations where array size is large. And for the worst-case running time, you are also right that this is Theta(lg n) and the reason why is that your heap is always assumed to be BALANCED, i.e. Yes you are right about the best-case running time. You are correct: it's $\Theta(n)$ in the worst case. Heap sort is an in-place algorithm. Supplement: Maybe the complexity isn't that, in fact I don't know the time complexity of heappush() and heappop() # O(k+(n-k)lgk) time, min-heap def findKthLargest(self, nums, k): heap = [] for num in nums: heapq.heappush(heap, num) for _ in xrange(len(nums)-k): heapq.heappop(heap) return heapq.heappop(heap) here i am going to explain using Max_heap. Time Complexity: O(logn). For example the python heapq module implements a heap with an array, and all the time the first element of the array is the root of the heap. In this tutorial, we’ll discuss how to insert a new node into the heap.We’ll also present the time complexity analysis of the insertion process. But if you can take the hit of one time pre-processing of popping out all the elements sequentially in an array, you'll get a sorted array in O(N.logN). Sort a nearly sorted (or K sorted) array 2. Time complexity of createAndBuildHeap() is O(n) and overall time complexity of Heap Sort is O(nLogn). Ok O(1) is only for retrieving the root of the heap. Heap is a popular tree-based data structure. The max-heap property (that the value of every node is at least as big as everything in the subtree below it) gives you no useful information and you must check both subtrees of every node. This is equal to the height of the complete binary tree. every height level set of nodes is full except at the bottom level. Heap is a complete binary tree and in the worst case we start at the root and come down to the leaf. Applications of HeapSort 1. Search in a heap, as it is, will need O(N) time. Effectively a heap sort. To delete this root, all heap implementations have a O(log(n)) time complexity. Also, the siftDown version of heapify has O(n) time complexity, while the siftUp version given below has O(n log n) time complexity due to its equivalence with inserting each element, one at a time, into an empty heap. A common operation in a heap is to insert a new node. 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