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Cormen Edition 3 Exercise 8.4 Question 5 (Page No. 204)
A probability distribution function $P(x)$ for a random variable $X$ is defined by $P(x) =Pr\{X\leq x\}$.Suppose that we draw a list of $n$ random variables $X_1,X_2,…,X_n$ from a continuous probability distribution function $P$ that is computable in $O(1)$ time. Give an algorithm that sorts these numbers in linear averagecase time.
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Cormen Edition 3 Exercise 8.4 Question 4 (Page No. 204)
We are given $n$ points in the unit circle, $P_i=(x_i,y_i)$, such that $0<x_i^2+y_i^2<1$ for $i=1,2, .,n$.Suppose that the points are uniformly distributed; that is, the probability of finding a point in ... the origin. (Hint: Design the bucket sizes in BUCKETSORT to reflect the uniform distribution of the points in the unit circle.)
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Cormen Edition 3 Exercise 8.4 Question 3 (Page No. 204)
Let $X$ be a random variable that is equal to the number of heads in two flips of a fair coin. What is $E[X^2]$? What is $E^2[X]$?
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Cormen Edition 3 Exercise 8.4 Question 2 (Page No. 204)
Explain why the worstcase running time for bucket sort is $\Theta(n^2)$. What simple change to the algorithm preserves its linear averagecase running time and makes its worstcase running time $O(n\ lg\ n)$?
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Cormen Edition 3 Exercise 8.4 Question 1 (Page No. 204)
BUCKETSORT(A) 1 let B[0...n1] be a new array 2 n = A.length 3 for i  0 to n  1 4 make B[i] an empty list 5 for i = 1 to n 6 insert A[i] into list B[nA[i]] 7 for i = 0 to n  1 8 sort list B[i] with ... ,B[n1] together in order illustrate the operation of BUCKETSORT on the array $A=\langle .79,.13,.16,.64,.39,.20,.89,.53,.71,.42\rangle$
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Cormen Edition 3 Exercise 8.3 Question 4 (Page No. 200)
Show how to sort $n$ integers in the range $0$ to $n^31$ in $O(n)$ time.
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Cormen Edition 3 Exercise 8.3 Question 3 (Page No. 200)
Use induction to prove that radix sort works. Where does your proof need the assumption that the intermediate sort is stable?
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Cormen Edition 3 Exercise 8.3 Question 2 (Page No. 200)
Which of the following sorting algorithms are stable: insertion sort, merge sort, heapsort, and quicksort? Give a simple scheme that makes any sorting algorithm stable. How much additional time and space does your scheme entail?
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Cormen Edition 3 Exercise 8.3 Question 1 (Page No. 199)
RADIXSORT(A, d) 1 for i = 1 to d 2 use a stable sort to sort array A on digit i illustrate the operation of RADIXSORT on the following list of English words: COW, DOG, SEA, RUG, ROW, MOB, BOX, TAB, BAR, EAR, TAR, DIG, BIG, TEA, NOW, FOX.
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Cormen Edition 3 Exercise 8.2 Question 4 (Page No. 197)
Describe an algorithm that, given $n$ integers in the range $0$ to $k$ preprocesses its input and then answers any query about how many of the $n$ integers fall into the range $[a..b]$ in $O(1)$ time.Your algorithm should use $\Theta(n+k)$ preprocessing time.
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Cormen Edition 3 Exercise 8.2 Question 3 (Page No. 196)
Suppose that we were to rewrite the for loop header in line $10$ of the COUNTINGSORT as 10 for j = 1 to A.length Show that the algorithm still works properly. Is the modified algorithm stable?
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Cormen Edition 3 Exercise 8.2 Question 2 (Page No. 196)
Prove that COUNTINGSORT is stable.
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Cormen Edition 3 Exercise 8.2 Question 1 (Page No. 196)
COUNTINGSORT(A, B, k) 1 let C[0, ,k] be a new array 2 for i = 0 to k 3 C[i] = 0 4 for j = 1 to A.length 5 C[A[j]] = C[A[j]] + 1 6 // C[i] now contains the number of elements equal to i . 7 for i =1 to k 8 C[i] = C[ ... j] 12 C[A[j]] = C[A[j]]  1 illustrate the operation of COUNTINGSORT on the array $A=\langle 6,0,2,0,1,3,4,6,1,3,2 \rangle $
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Cormen Edition 3 Exercise 8.1 Question 4 (Page No. 194)
Suppose that you are given a sequence of $n$ elements to sort.The input sequence consists of $n/k$ subsequences, each containing $k$ elements.The elements in a given subsequence are all smaller than the elements in the ... of the sorting problem. (Hint: It is not rigorous to simply combine the lower bounds for the individual subsequences.)
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Cormen Edition 3 Exercise 8.1 Question 3 (Page No. 194)
Show that there is no comparison sort whose running time is linear for at least half of the $n!$ inputs of length $n$.What about a fraction of $1/n$ inputs of length $n$? What about a fraction $1/2^n$?
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Cormen Edition 3 Exercise 8.1 Question 1 (Page No. 193)
What is the smallest possible depth of a leaf in a decision tree for a comparison sort?
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Cormen Edition 3 Exercise 7.1 Question 2 (Page No. 174)
What value of $q$ does PARTITION return when all elements in the array $A[p..r]$ have the same value? Modify PARTITION so that $q=\lfloor(p+r)/2 \rfloor$ when all elements in the array $A[p..r]$ have the same value.
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Cormen Edition 3 Exercise 7.1 Question 1 (Page No. 173)
PARTITION(A,p,r) 1 x = A[r] 2 i = p – 1 3 for j = p to r – 1 4 if A[j] <= x 5 i = i + 1 6 exchange A[i] with A[j] 7 exchange A[i+1] with A[r] 8 return i + 1 illustrate the operation of PARTITION on the array $A=\langle 13,19,9,5,12,8,7,4,21,2,6,11\rangle$
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Cormen Edition 3 Exercise 2.3 Question 4 (Page No. 38)
We can express the insertion sort as a recursive procedure as follows.In order to sort $A[1\dots n]$, we recursively sort $A[1 \dots n1]$ and then insert $A[n]$ into the sorted array $A[1 \dots n1]$. Write a recurrence for the running time of this recursive version of insertion sort.
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Cormen Edition 3 Exercise 2.3 Question 2 (Page No. 37)
Rewrite the MERGE procedure so that it does not use sentinels, instead of stopping once either array $L$ or $R$ has had all its elements copied back to $A$ and then copying the remainder of the other array back into $A$.
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Cormen Edition 3 Exercise 2.3 Question 1 (Page No. 37)
Using Figure $2.4$ as a model, illustrate the operation of merge sort on the array $A=\langle 3,41,52,26,38,57,9,49 \rangle $
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Cormen Edition 3 Exercise 2.1 Question 2 (Page No. 22)
Rewrite the INSERTIONSORT procedure to sort into nonincreasing instead of nondecreasing order.
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Discrete mathematics 7th ed by Kenneth Rosen,chapter3:Algorithm
Should not there be a second condition stating i = j1 in While loop's conditional statement,if not then it seems to me while loop will be a infinite loop..
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#CLRS #Algorithm Doubt about randomized QuickSort.
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#Algorithms QuickSort Algorithm Doubt regarding pivot and analysis.
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Made Easy Test Series: AlgorithmSorting
An array $A$ of size n is known to be sorted except for the first $k$ elements and the last $k$ elements, where $k$ is a constant. Which of the following algorithms will be the best choice for sorting the array $A?$ $a)$ ... sorts part by part using pivot. So, why not will it be answer?? How do we know it is asking for almost sorted array??
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Mimimum number of comparison to sort 13 elements/numbers for any comparison based sorting algorithm?
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#Algorithms Quicksort VS Mergesort? Which is a faster sorting algorithm
I did Google and found out that Quicksort is better then Mergesort, but my question is which is faster among both?
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Total number of function calls in Merge sort Algorithm
In Merge sort Algorithm when I took input array of size 2 and I got 4 function calls as including original function call with which I call MS algorithm i.e. MS (1,2) and which in turn calls two recursive function calls to merge ... function calls. So, how can I analyze the total number of function calls when input array size is n? thank you!
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made easy test series:algorithms,sorting
why not merge sort?we don’t swap in merge sort,we just create auxillary arrays and merge them by changing elements in the original array.should we consider that as a swap?
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