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The above is an approximation. The exact worst-case number of comparisons during the heap-construction phase of heapsort is known to be equal to 2n − 2s 2 (n) − e 2 (n), where s 2 (n) is the number of 1 bits in the binary representation of n and e 2 (n) is the number of trailing 0 bits. [6] [7]
But given a worst-case input, its performance degrades to O(n 2). Also, when implemented with the "shortest first" policy, the worst-case space complexity is instead bounded by O(log(n)). Heapsort has O(n) time when all elements are the same. Heapify takes O(n) time and then removing elements from the heap is O(1) time for each of the n elements.
Construction of a binary (or d-ary) heap out of a given array of elements may be performed in linear time using the classic Floyd algorithm, with the worst-case number of comparisons equal to 2N − 2s 2 (N) − e 2 (N) (for a binary heap), 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 ...
The exact value of the above (the worst-case number of comparisons during the heap construction) is known to be equal to: (), [9] [b] 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.
One implementation can be described as arranging the data sequence in a two-dimensional array and then sorting the columns of the array using insertion sort. The worst-case time complexity of Shellsort is an open problem and depends on the gap sequence used, with known complexities ranging from O(n 2) to O(n 4/3) and Θ(n log 2 n).
An adaptive sort takes advantage of this "presortedness" and runs more quickly on nearly-sorted inputs, often while still maintaining an () worst case time bound. An example is adaptive heap sort , a sorting algorithm based on Cartesian trees .
For instance, using a binary heap as a priority queue in selection sort leads to the heap sort algorithm, a comparison sorting algorithm that takes O(n log n) time. Instead, using selection sort with a bucket queue gives a form of pigeonhole sort , and using van Emde Boas trees or other integer priority queues leads to other fast integer ...
The order of growth (e.g. linear, logarithmic) of the worst-case complexity is commonly used to compare the efficiency of two algorithms. The worst-case complexity of an algorithm should be contrasted with its average-case complexity, which is an average measure of the amount of resources the algorithm uses on a random input.