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2. Time Complexity: Heapify a single node takes O(log N) time complexity where N is the total number of Nodes. 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. It doesn't need any extra storage and that makes it good for situations where array size is large. To delete this root, all heap implementations have a O(log(n)) time complexity. Its typical implementation is not stable, but can be made stable (See this) Time Complexity: Time complexity of heapify is O(Logn). 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). 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). 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. 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) Search in a heap, as it is, will need O(N) time. Time complexity of createAndBuildHeap() is O(n) and overall time complexity of Heap Sort is O(nLogn). 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. Hence, Heapify takes different time for each node, which is . 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. Sort a nearly sorted (or K sorted) array 2. You are correct: it's $\Theta(n)$ in the worst case. The heart of the Heap data structure is Heapify algortihm. 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. Effectively a heap sort. Ok O(1) is only for retrieving the root of the heap. Heap is a popular tree-based data structure. In reality, building a heap takes O(n) time depending on the implementation which can be seen here. A common operation in a heap is to insert a new node. every height level set of nodes is full except at the bottom level. Applications of HeapSort 1. Yes you are right about the best-case running time. Time Complexity: O(logn). 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. 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