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jenyasd209 [6]
3 years ago
13

Write down a recurrence relation for this version of QuickSort, and solve it asymp-totically. Show your work. Assume that the ti

me it takes to find the pivot (eitherbest or worst, depending on the level of recursion) is Θ(n) for lists of lengthn.
Computers and Technology
1 answer:
torisob [31]3 years ago
5 0

Answer:As in merge sort, the time for a given recursive call on an n-element subarray is Θ(n). In merge sort, that was the time for merging, but in quicksort it's the time for partitioning.

Explanation:

Worst-case running time:

When quicksort always has the most unbalanced partitions possible, then the original call takes cncnc, n time for some constant ccc, the recursive call on n-1n−1n, minus, 1 elements takes c(n-1)c(n−1)c, left parenthesis, n, minus, 1, right parenthesis time, the recursive call on n-2n−2n, minus, 2 elements takes c(n-2)c(n−2)c, left parenthesis, n, minus, 2, right parenthesis time, and so on. cn+c(n−1)+c(n−2)+⋯+2c=c(n+(n−1)+(n−2)+⋯+2)

=c((n+1)(n/2)−1)

The last line is because 1 + 2 + 3 +...... n is the arithmetic series

Best-case running time:

Quicksort's best case occurs when the partitions are as evenly balanced as possible: their sizes either are equal or are within 1 of each other. The former case occurs if the subarray has an odd number of elements and the pivot is right in the middle after partitioning, and each partition has (n-1)/ 2 elements. The latter case occurs if the subarray has an even number n of elements and one partition has n/2 elements with the other having n/2-1.In either of these cases, each partition has at most n/2 elements.

Using big-Θ notation, we get the same result as for merge sort: Θ(nlog2n)

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Determine whether or not the following pairs of predicates are unifiable. If they are, give the most-general unifier and show th
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Answer:

a) P(B,A,B), P(x,y,z)

=> P(B,A,B) , P(B,A,B}  

Hence, most general unifier = {x/B , y/A , z/B }.

b. P(x,x), Q(A,A)  

No mgu exists for this expression as any substitution will not make P(x,x), Q(A, A) equal as one function is of P and the other is of Q.

c. Older(Father(y),y), Older(Father(x),John)

Thus , mgu ={ y/x , x/John }.

d) Q(G(y,z),G(z,y)), Q(G(x,x),G(A,B))

=> Q(G(x,x),G(x,x)), Q(G(x,x),G(A,B))  

This is not unifiable as x cannot be bound for both A and B.

e) P(f(x), x, g(x)), P(f(y), A, z)    

=> P(f(A), A, g(A)), P(f(A), A, g(A))  

Thus , mgu = {x/y, z/y , y/A }.

Explanation:  

Unification: Any substitution that makes two expressions equal is called a unifier.  

a) P(B,A,B), P(x,y,z)  

Use { x/B}  

=> P(B,A,B) , P(B,y,z)  

Now use {y/A}  

=> P(B,A,B) , P(B,A,z)  

Now, use {z/B}  

=> P(B,A,B) , P(B,A,B}  

Hence, most general unifier = {x/B , y/A , z/B }  

b. P(x,x), Q(A,A)  

No mgu exists for this expression as any substitution will not make P(x,x), Q(A, A) equal as one function is of P and the other is of Q  

c. Older(Father(y),y), Older(Father(x),John)  

Use {y/x}  

=> Older(Father(x),x), Older(Father(x),John)  

Now use { x/John }  

=> Older(Father(John), John), Older(Father(John), John)  

Thus , mgu ={ y/x , x/John }  

d) Q(G(y,z),G(z,y)), Q(G(x,x),G(A,B))  

Use { y/x }  

=> Q(G(x,z),G(z,x)), Q(G(x,x),G(A,B))

Use {z/x}  

=> Q(G(x,x),G(x,x)), Q(G(x,x),G(A,B))  

This is not unifiable as x cannot be bound for both A and B  

e) P(f(x), x, g(x)), P(f(y), A, z)  

Use {x/y}  

=> P(f(y), y, g(y)), P(f(y), A, z)  

Now use {z/g(y)}  

P(f(y), y, g(y)), P(f(y), A, g(y))  

Now use {y/A}  

=> P(f(A), A, g(A)), P(f(A), A, g(A))  

Thus , mgu = {x/y, z/y , y/A }.

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In order to derive the probability of stock outs, divide the total value of the stock outs by the number of requests demanded. The resulting figure must then be multiplied by 100.

<h3>What is a stock out?</h3>

In business, a stock out refers to a condition where in a certain item or items are no longer available in stock.

The formula can be sated simply as:

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Thus Number of Stock outs = Total probability of stock outs * total number of demand requests.

<h3>What is the formula for the Total Cost?</h3>

The formula for Total Cost is given as:

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Explanation:

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