Question
In: Advanced Math
On a circular array with n positions, we wish to place the integers 1, 2, ......
On a circular array with n positions, we wish to place the integers 1, 2, ... r in order, clockwise, such that consecutive integers, including the pair (r,1) are not in adjacent positions on the array. Arrangements obtained by rotation are considered the same. In how many ways can this be done? Give a combinatorial proof.
Solutions
Colby Messinger answered 2 years agoRelated Solutions
1. We have an array A of size n. There are only positive integers in this...
1. We have an array A of size n. There are only positive
integers in this array. Note that the array may have integers that
are not distinct, and it could be any array of positive integers in
creation (I like the numbers found the decimal expansion of π for
instance). When possible provide the exact complexity of the
algorithm. If it’s not possible explain the O/Ω/Θ complexity.
Design an efficient algorithm to find the maximum difference
between any two...
1. We have an array A of size n. There are only positive integers in this...
1. We have an array A of size n. There are only positive
integers in this array. Note that the array may have integers that
are not distinct, and it could be any array of positive integers in
creation (I like the numbers found the decimal expansion of π for
instance). When possible provide the exact complexity of the
algorithm. If it’s not possible explain the O/Ω/Θ complexity.
Design an efficient algorithm to find the maximum difference
between any two...
Consider the problem of sorting an array A[1, ..., n] of integers. We presented an O(n...
Consider the problem of sorting an array A[1, ..., n] of
integers. We presented an O(n log n)-time algorithm in class and,
also, proved a lower bound of Ω(n log n) for any comparison-based
algorithm.
2. Give an efficient sorting algorithm for an array C[1, ..., n]
whose elements are taken from the set {1, 2, 3, 4, 5}.
Consider the problem of sorting an array A[1, ..., n] of integers. We presented an O(n...
Consider the problem of sorting an array A[1, ..., n] of
integers. We presented an O(n log n)-time algorithm in class and,
also, proved a lower bound of Ω(n log n) for any comparison-based
algorithm.
3. Give an efficient sorting algorithm for an array D[1, ..., n]
whose elements are distinct (D[i] ̸= D[j], for every i ̸= j ∈ {1,
..., n}) and are taken from the set {1, 2, ..., 2n}.
Consider the problem of sorting an array A[1, ..., n] of integers. We presented an O(n...
Consider the problem of sorting an array A[1, ..., n] of
integers. We presented an O(n log n)-time algorithm in class and,
also, proved a lower bound of Ω(n log n) for any comparison-based
algorithm.
1. Give an efficient sorting algorithm for a boolean1 array
B[1, ..., n].
2. Give an efficient sorting algorithm for an array C[1,...,n]
whose elements are taken from the set
{1,2,3,4,5}.
3. Give an efficient sorting algorithm for an array D[1, ...,
n] whose elements...
We have an array A of size n. There are only positive integers in this array....
We have an array A of size n. There are only positive integers
in this array. Note that the array may have integers that are not
distinct, and it could be any array of positive integers in
creation (I like the numbers found the decimal expansion of π for
instance). When possible provide the exact complexity of the
algorithm. If it’s not possible explain the O/Ω/Θ complexity. a.
Design an efficient algorithm to find the maximum difference
between any two...
We have an array A of size n. There are only positive integers in this array....
We have an array A of size n. There are only positive integers
in this array. Note that the array may have integers that are not
distinct, and it could be any array of positive integers in
creation (I like the numbers found the decimal expansion of π for
instance). When possible provide the exact
complexity of the algorithm. If it’s not possible explain the O/Ω/Θ
complexity. a. Design an efficient algorithm to find the maximum
difference between any two...
An array A[0..n - 2] contains n-1 integers from 1 to n in increasing order. (Thus...
An array A[0..n - 2] contains n-1 integers from 1 to n in
increasing order. (Thus one integer in this range is missing.)
Design an algorithm in (Theta(log n)) to find the missing integer.
Your algorithm should be given in pseudo code. For
example, the array A could be {1, 2, 3, 4, 6, 7, 8, 9, 10} in which
5 is missing.
Let iqsort(A, 1, n) be an algorithm that sorts an array A with n integers. It...
Let iqsort(A, 1, n) be an algorithm that sorts an array A with n
integers. It works as follows:
iqsort(A, p, q){
if p ≥ q, return;
r=partition(A, p, q);
//run quick sort on the low part
quicksort(A, p, r − 1);
//run insert sort on the high part
insertsort(A, r + 1, q); }
Compute the best-case, worst-case, and average-case complexities
of iqsort.
Suppose we are given a set ? containing 2? integers, and we wish to partition it...
Suppose we are given a set ? containing 2? integers, and we wish
to partition it into two sets ?1 and ?2 so that |?1 | = |?2 | = ?
and so that the sum of the numbers in ?1 is as close as possible to
the sum of those in ?2. Let the neighborhood ? be determined by all
possible interchanges of two integers between ?1 and ?2. Is ?
exact?
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