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Recent questions tagged algorithms
Webpage for Algorithms
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1
Cormen Edition 3 Exercise 10.1 Question 5 (Page No. 236)
Whereas a stack allows insertion and deletion of elements at only one end, and a queue allows insertion at one end and deletion at the other end, a deque (double ended queue) allows insertion and deletion at both ends. Write ... time procedures to insert elements into and delete elements from both ends of a deque implemented by an array.
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Jun 28
in
Algorithms
by
akash.dinkar12
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41.3k
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cormen
algorithms
datastructure
queues
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0
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1
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2
Cormen Edition 3 Exercise 9.1 Question 2 (Page No. 215)
Prove the lower bound of $\lceil 3n/2\rceil – 2$ comparisons in the worst case to find both the maximum and minimum of $n$ numbers. (Hint: Consider how many numbers are potentially either the maximum or minimum and investigate how a comparison affects these counts.)
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9
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cormen
algorithms
descriptive
0
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3
Cormen Edition 3 Exercise 9.1 Question 1 (Page No. 215)
Show that the second smallest of $n$ elements can be found with $n+\lceil lg\ n \rceil 2$ comparisons in the worst case. (Hint: Also find the smallest element.)
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Algorithms
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cormen
algorithms
descriptive
0
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0
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4
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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Jun 28
in
Algorithms
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akash.dinkar12
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41.3k
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29
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cormen
algorithms
sorting
bucketsort
descriptive
difficult
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0
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5
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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Jun 28
in
Algorithms
by
akash.dinkar12
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41.3k
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10
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cormen
algorithms
sorting
bucketsort
descriptive
difficult
0
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1
answer
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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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Jun 28
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Algorithms
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akash.dinkar12
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41.3k
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14
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cormen
algorithms
sorting
bucketsort
expectation
descriptive
0
votes
0
answers
7
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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Jun 28
in
Algorithms
by
akash.dinkar12
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41.3k
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10
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cormen
algorithms
sorting
bucketsort
descriptive
0
votes
1
answer
8
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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Jun 28
in
Algorithms
by
akash.dinkar12
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41.3k
points)

11
views
cormen
algorithms
sorting
bucketsort
descriptive
+1
vote
1
answer
9
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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Jun 28
in
Algorithms
by
akash.dinkar12
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41.3k
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31
views
cormen
algorithms
sorting
radixsort
descriptive
0
votes
0
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10
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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in
Algorithms
by
akash.dinkar12
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41.3k
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7
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cormen
algorithms
sorting
radixsort
descriptive
0
votes
2
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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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Algorithms
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20
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cormen
algorithms
sorting
stablesort
descriptive
0
votes
1
answer
12
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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Algorithms
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cormen
algorithms
sorting
radixsort
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0
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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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Algorithms
by
akash.dinkar12
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41.3k
points)

14
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cormen
algorithms
sorting
countingsort
descriptive
0
votes
0
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14
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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Jun 28
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Algorithms
by
akash.dinkar12
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41.3k
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5
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cormen
algorithms
sorting
countingsort
descriptive
0
votes
0
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Cormen Edition 3 Exercise 8.2 Question 2 (Page No. 196)
Prove that COUNTINGSORT is stable.
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Algorithms
by
akash.dinkar12
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41.3k
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7
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cormen
algorithms
sorting
countingsort
descriptive
0
votes
0
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16
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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Jun 28
in
Algorithms
by
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41.3k
points)

10
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cormen
algorithms
sorting
countingsort
descriptive
0
votes
0
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17
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.)
asked
Jun 28
in
Algorithms
by
akash.dinkar12
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(
41.3k
points)

10
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cormen
algorithms
sorting
descriptive
0
votes
0
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18
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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Jun 28
in
Algorithms
by
akash.dinkar12
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41.3k
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9
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cormen
algorithms
sorting
descriptive
0
votes
1
answer
19
Cormen Edition 3 Exercise 8.1 Question 2 (Page No. 194)
Obtain asymptotically tight bounds on $lg\ (n!)$ without using Stirling’s approximation. Instead, evaluate the summation $\sum_{k=1}^{n} lg\ k$.
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Jun 28
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Algorithms
by
akash.dinkar12
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10
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cormen
algorithms
asymptoticnotations
descriptive
0
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1
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20
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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Jun 28
in
Algorithms
by
akash.dinkar12
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41.3k
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14
views
cormen
algorithms
sorting
descriptive
0
votes
1
answer
21
Cormen Edition 3 Exercise 7.4 Question 6 (Page No. 185)
Consider modifying the PARTITION procedure by randomly picking three elements from the array $A$ and partitioning about their median (the middle value of the three elements). Approximate the probability of getting at worst a $\alpha$to$(1\alpha)$ split, as a function of $\alpha$ in the range $0<\alpha<1$.
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Jun 28
in
Algorithms
by
akash.dinkar12
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41.3k
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17
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cormen
algorithms
quicksort
descriptive
difficult
0
votes
1
answer
22
Cormen Edition 3 Exercise 7.4 Question 5 (Page No. 185)
We can improve the running time of quicksort in practice by taking advantage of the fast running time of insertion sort when its input is nearly sorted. Upon calling quicksort on a subarray with fewer than $k$ elements, let it simply return without ... $k$, both in theory and in practice?
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Jun 28
in
Algorithms
by
akash.dinkar12
Boss
(
41.3k
points)

11
views
cormen
algorithms
quicksort
descriptive
0
votes
1
answer
23
Cormen Edition 3 Exercise 7.4 Question 4 (Page No. 184)
Show that RANDOMIZEDQUICKSORT’s expected running time is $\Omega(n\ lg\ n)$.
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Jun 28
in
Algorithms
by
akash.dinkar12
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(
41.3k
points)

38
views
cormen
algorithms
quicksort
timecomplexity
descriptive
0
votes
1
answer
24
Cormen Edition 3 Exercise 7.4 Question 3 (Page No. 184)
Show that the expression $q^2 +(nq1)^2$ achieves a maximum over $q=0,1,\dots ,n1$ when $q=0$ or $q=n1$.
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Jun 28
in
Algorithms
by
akash.dinkar12
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(
41.3k
points)

11
views
cormen
algorithms
quicksort
descriptive
0
votes
0
answers
25
Cormen Edition 3 Exercise 7.4 Question 2 (Page No. 184)
Show that quicksort’s bestcase running time is $\Omega(n\ lg\ n)$.
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Jun 28
in
Algorithms
by
akash.dinkar12
Boss
(
41.3k
points)

18
views
cormen
algorithms
quicksort
timecomplexity
descriptive
0
votes
0
answers
26
Cormen Edition 3 Exercise 7.4 Question 1 (Page No. 184)
Show that in the recurrence $T(n)=\max_{0<q\leq n1} (T(q)+T(nq1))+\Theta(n)$ $T(n)=\Omega(n^2)$
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Jun 28
in
Algorithms
by
akash.dinkar12
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(
41.3k
points)

16
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cormen
algorithms
recurrence
descriptive
0
votes
1
answer
27
Cormen Edition 3 Exercise 7.3 Question 2 (Page No. 180)
RANDOMIZEDQUICKSORT(A, p, r) 1 if p < r 2 q = RANDOMIZEDPARTITION(A, p, r) 3 RANDOMIZEDQUICKSORT(A, p, q  1) 4 RANDOMIZEDQUICKSORT(A, q + 1, r) RANDOMIZEDPARTITION(A, p, r) 1 i = RANDOM(p, r) 2 ... made to the random number generator RANDOM in the worst case? How about in the best case? Give your answer in terms of $\Theta$ notation.
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Jun 28
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Algorithms
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12
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cormen
algorithms
quicksort
descriptive
0
votes
1
answer
28
Cormen Edition 3 Exercise 7.3 Question 1 (Page No. 180)
Why do we analyze the expected running time of a randomized algorithm and not its worstcase running time?
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Jun 28
in
Algorithms
by
akash.dinkar12
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9
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cormen
algorithms
quicksort
descriptive
0
votes
1
answer
29
Cormen Edition 3 Exercise 7.2 Question 6 (Page No. 179)
Argue that for any constant $0<\alpha\leq 1/2$, the probability is approximately $12\alpha$ that on a random input array, PARTITION produces a split more balanced than $1\alpha$ to $\alpha$.
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Jun 27
in
Algorithms
by
akash.dinkar12
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(
41.3k
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9
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cormen
algorithms
quicksort
descriptive
difficult
0
votes
0
answers
30
Cormen Edition 3 Exercise 7.2 Question 5 (Page No. 178)
Suppose that the splits at every level of quicksort are in the proportion $1\alpha$ to $\alpha$, where $0<\alpha\leq1/2$ is a constant. Show that the minimum depth of a leaf in the recursion tree is approximately $lg\ n /lg\ \alpha$ and the maximum depth is approximately $lg\ n / lg\ (1\alpha)$.(Don’t worry about integer roundoff.)
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Jun 27
in
Algorithms
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views
cormen
algorithms
quicksort
descriptive
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