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where s 2 (n) is the sum of all digits of the binary representation of n and e 2 (n) is the exponent of 2 in the prime factorization of n. The average case is more complex to analyze, but it can be shown to asymptotically approach 1.8814 n − 2 log 2 n + O (1) comparisons.
Even levels are for example 0, 2, 4, etc, and odd levels are respectively 1, 3, 5, etc. We assume in the next points that the root element is at the first level, i.e., 0. Example of Min-max heap. Each node in a min-max heap has a data member (usually called key) whose value is used to determine the order of the node in the min-max heap.
Example of a binary max-heap with node keys being integers between 1 and 100. In computer science, a heap is a tree-based data structure that satisfies the heap property: In a max heap, for any given node C, if P is the parent node of C, then the key (the value) of P is greater than or equal to the key of C.
The heapsort algorithm can be divided into two phases: heap construction, and heap extraction. The heap is an implicit data structure which takes no space beyond the array of objects to be sorted; the array is interpreted as a complete binary tree where each array element is a node and each node's parent and child links are defined by simple arithmetic on the array indexes.
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[1] [2] Deleting an element (most often used in the special case of deleting the minimum element) works in () amortized time, where is the size of the heap. [2] This means that starting from an empty data structure, any sequence of a insert and decrease-key operations and b delete-min operations would take O ( a + b log n ...
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