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Because it uses arrays of length k + 1 and n, the total space usage of the algorithm is also O(n + k). [1] For problem instances in which the maximum key value is significantly smaller than the number of items, counting sort can be highly space-efficient, as the only storage it uses other than its input and output arrays is the Count array ...
The numbers along the left edge of the triangle are the lazy caterer's sequence and the numbers along the right edge are the triangular numbers. The nth row sums to n(n 2 + 1)/2, the constant of an n × n magic square (sequence A006003 in the OEIS). Summing up the row sums in Floyd's triangle reveals the doubly triangular numbers, triangular ...
Every h 1-sorted and h 2-sorted array is also (a 1 h 1 +a 2 h 2)-sorted, for any nonnegative integers a 1 and a 2. The worst-case complexity of Shellsort is therefore connected with the Frobenius problem : for given integers h 1 ,..., h n with gcd = 1, the Frobenius number g ( h 1 ,..., h n ) is the greatest integer that cannot be represented ...
A k-cycle is a cycle that can be partitioned into k contiguous subsequences, each consisting of an increasing sequence of odd numbers, followed by a decreasing sequence of even numbers. [15] For instance, if the cycle consists of a single increasing sequence of odd numbers followed by a decreasing sequence of even numbers, it is called a 1-cycle.
The odd–even sort algorithm correctly sorts this data in passes. (A pass here is defined to be a full sequence of odd–even, or even–odd comparisons. The passes occur in order pass 1: odd–even, pass 2: even–odd, etc.) Proof: This proof is based loosely on one by Thomas Worsch. [6]
If all elements to be sorted are distinct, the expected number of comparisons performed in the average case by randomized bogosort is asymptotically equivalent to (e − 1)n!, and the expected number of swaps in the average case equals (n − 1)n!. [1] The expected number of swaps grows faster than the expected number of comparisons, because if ...
10 rem this for loop prints odd numbers from 1 to 15 20 for i = 1 to 15 step 2 30 print i 40 next i The end-loop marker specifies the name of the index variable, which must correspond to the name of the index variable at the start of the for-loop.
[1] [10] This is an internal buffer defined as s1 t s2, where s1 and s2 are each as large as the number of A and B blocks, and t contains any values immediately following s1 that are equal to the last value of s1 (thus ensuring that no value in s2 appears in s1). A second internal buffer containing √ A unique values is still used.