Its typical implementation is not stable, but can be made stable (See this) Time Complexity: Time complexity of heapify is O(Logn). Time Complexity: Heapify a single node takes O(log N) time complexity where N is the total number of Nodes. Heap sort has the best possible worst case running time complexity of O(n Log n). In reality, building a heap takes O(n) time depending on the implementation which can be seen here. 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. It doesn't need any extra storage and that makes it good for situations where array size is large. For finding the Time Complexity of building a heap, we must know the number of nodes having height h. This is equal to the height of the complete binary tree. Search in a heap, as it is, will need O(N) time. Therefore, building the entire Heap will take N heapify operations and the total time complexity will be O(N*logN). Applications of HeapSort 1. Heap is a complete binary tree and in the worst case we start at the root and come down to the leaf. Hence, Heapify takes different time for each node, which is . To delete this root, all heap implementations have a O(log(n)) time complexity. Before looking into Heap Sort, let's understand what is Heap and how it helps in sorting. Now your new sorted array can be searched through in O(logN) time. 2. A common operation in a heap is to insert a new node. Ok O(1) is only for retrieving the root of the heap. every height level set of nodes is full except at the bottom level. Yes you are right about the best-case running time. 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. Sort a nearly sorted (or K sorted) array 2. 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. You are correct: it's \$\Theta(n)\$ in the worst case. 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). Time complexity of createAndBuildHeap() is O(n) and overall time complexity of Heap Sort is O(nLogn). 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. here is the pseudocode for Max-Heapify algorithm A is an array , index starts with 1. and i points to root of tree. Suppose you're looking for something that's no bigger than the smallest value in a max-heap. Time Complexity: O(logn). 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. 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). Heap is a popular tree-based data structure. 1. Effectively a heap sort. The heart of the Heap data structure is Heapify algortihm. Heap sort is an in-place algorithm. 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. Complexity where n is the pseudocode for Max-Heapify algorithm a is an array, index starts 1.... 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