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Cormen Edition 3 Exercise 12.1 Question 5 (Page No. 289)
Argue that since sorting $n$ elements takes $\Omega (n\ lgn)$ time in the worst case in the comparison model, any comparison-based algorithm for constructing a $BST$ from an arbitrary list of n elements takes $\Omega (n\ lgn)$ time in the worst case.

KUSHAGRA गुप्ता asked in Algorithms Nov 20, 2019

by KUSHAGRA गुप्ता

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Cormen Edition 3 Exercise 22.2 Question 8 (Page No. 539)
The diameter of a tree $T= (V, E)$ is defined as $max_{u,v\ \epsilon\ V}\ \delta(u,v)$, that is, the largest of all shortest-path distances in the tree. Give an efficient algorithm to compute the diameter of a tree, and analyze the running time of your algorithm.

KUSHAGRA गुप्ता asked in Algorithms Nov 12, 2019

by KUSHAGRA गुप्ता

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Cormen Edition 3 Exercise 22.2 Question 7 (Page No. 539)
There are two types of professional wrestlers: babyfaces ( good guys ) and heels ( bad guys ). Between any pair of professional wrestlers, there may or may not be a rivalry. Suppose we have n professional wrestlers and we ... between a babyface and a heel. If it is possible to perform such a designation, your algorithm should produce it.

KUSHAGRA गुप्ता asked in Algorithms Nov 12, 2019

by KUSHAGRA गुप्ता

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Cormen Edition 3 Exercise 22.2 Question 6 (Page No. 539)
Give an example of a directed graph $G=(V, E)$, a source vertex $s\ \epsilon\ V$ , and a set of tree edges $E_{\Pi}\subseteq E$ such that for each vertex $v\ \epsilon\ V$ ... set of edges $E_{\Pi}$ cannot be produced by running BFS on G, no matter how the vertices are ordered in each adjacency list.

KUSHAGRA गुप्ता asked in Algorithms Nov 12, 2019

by KUSHAGRA गुप्ता

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Cormen Edition 3 Exercise 10.2 Question 8 (Page No. 241)
Explain how to implement doubly linked lists using only one pointer value $x.np$ per item instead of the usual two (next and prev). Assume that all pointer values can be interpreted as $k$-bit integers, and define $x.np$ ... $INSERT$, and $DELETE$ operations on such a list. Also, show how to reverse such a list in $O(1)$ time.

akash.dinkar12 asked in Algorithms Jun 30, 2019

by akash.dinkar12

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Cormen Edition 3 Exercise 10.2 Question 7 (Page No. 241)
Give a $\Theta(n)$ time nonrecursive procedure that reverses a singly linked list of $n$ elements. The procedure should use no more than constant storage beyond that needed for the list itself.

akash.dinkar12 asked in Algorithms Jun 30, 2019

by akash.dinkar12

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Cormen Edition 3 Exercise 10.2 Question 6 (Page No. 241)
The dynamic-set operation $UNION$ takes two disjoint sets $S_1$ and $S_2$ as input, and it returns a set $S=S_1 \cup S_2$ consisting of all the elements of $S_1$ and $S_2$.The sets $S_1$ and $S_2$ are usually destroyed by the operation. Show how to support $UNION$ in $O(1)$ time using a suitable list data structure.

akash.dinkar12 asked in Algorithms Jun 30, 2019

by akash.dinkar12

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Cormen Edition 3 Exercise 10.2 Question 5 (Page No. 240)
Implement the dictionary operations $INSERT$, $DELETE$, and $SEARCH$ using singly linked, circular lists. What are the running times of your procedures?

akash.dinkar12 asked in Algorithms Jun 30, 2019

by akash.dinkar12

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Cormen Edition 3 Exercise 10.2 Question 4 (Page No. 240)
LIST-SEARCH’(L, k) 1 x = L.nil.next 2 while x != L.nil and x.key != k 3 x = x.next 4 return x As written, each loop iteration in the LIST-SEARCH’ procedure requires two tests: one for $x\neq L.nil$ and one for $x.key\neq k$. Show how to eliminate the test for $x\neq L.nil$ in each iteration.

akash.dinkar12 asked in Algorithms Jun 30, 2019

by akash.dinkar12

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Cormen Edition 3 Exercise 10.2 Question 3 (Page No. 240)
Implement a queue by a singly linked list $L$. The operations of $ENQUEUE$ and $DEQUEUE$ should still take $O(1)$ time.

akash.dinkar12 asked in Algorithms Jun 30, 2019

by akash.dinkar12

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Cormen Edition 3 Exercise 10.2 Question 2 (Page No. 240)
Implement a stack using a singly linked list $L$. The operations $PUSH$ and $POP$ should still take $O(1)$ time.

akash.dinkar12 asked in Algorithms Jun 30, 2019

by akash.dinkar12

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Cormen Edition 3 Exercise 10.2 Question 1 (Page No. 240)
Can you implement the dynamic-set operation $INSERT$ on a singly linked list in $O(1)$ time? How about $DELETE$?

akash.dinkar12 asked in Algorithms Jun 30, 2019

by akash.dinkar12

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2 answers

13

Cormen Edition 3 Exercise 10.1 Question 7 (Page No. 236)
Show how to implement a stack using two queues. Analyze the running time of the stack operations.

akash.dinkar12 asked in Algorithms Jun 28, 2019

by akash.dinkar12

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Cormen Edition 3 Exercise 10.1 Question 6 (Page No. 236)
Show how to implement a queue using two stacks. Analyze the running time of the queue operations.

akash.dinkar12 asked in Algorithms Jun 28, 2019

by akash.dinkar12

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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.

akash.dinkar12 asked in Algorithms Jun 28, 2019

by akash.dinkar12

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Cormen Edition 3 Exercise 10.1 Question 4 (Page No. 235)
Rewrite ENQUEUE and DEQUEUE to detect underflow and overflow of a queue.

akash.dinkar12 asked in Algorithms Jun 28, 2019

by akash.dinkar12

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Cormen Edition 3 Exercise 10.1 Question 3 (Page No. 235)
ENQUEUE(Q, x) 1 Q[Q.tail] = x 2 if Q.tail == Q.length 3 Q.tail = 1 4 else Q.tail = Q.tail + 1 DEQUEUE(Q) 1 x = Q[Q.head] 2 if Q.head == Q.length 3 Q.head = 1 4 else Q.head = Q.head + 1 5 return x illustrate the ... (Q,4),ENQUEUE(Q,1),ENQUEUE(Q,3),DEQUEUE(Q),ENQUEUE(Q,8),DEQUEUE(Q) on an initially empty queue $Q$ stored in array $Q[1...6]$.

akash.dinkar12 asked in Algorithms Jun 28, 2019

by akash.dinkar12

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18

Cormen Edition 3 Exercise 10.1 Question 2 (Page No. 235)
Explain how to implement two stacks in one array $A[1...n]$ in such a way that neither stack overflows unless the total number of elements in both stacks together is $n$.The $PUSH$ and $POP$ operations should run in $O(1)$ time.

akash.dinkar12 asked in Algorithms Jun 28, 2019

by akash.dinkar12

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Cormen Edition 3 Exercise 10.1 Question 1 (Page No. 235)
STACK-EMPTY(S) 1 if S.top == 0 2 return TRUE 3 else return FALSE PUSH(S , x) 1 S.top = S.top + 1 2 S[S.top] = x POP(S) 1 if STACK-EMPTY(S) 2 error underflow 3 else S.top = S.top - 1 4 return S[S.top + 1] illustrate the result of ... $S$ stored in array $S[1...6]$

akash.dinkar12 asked in Algorithms Jun 28, 2019

by akash.dinkar12

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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.)

akash.dinkar12 asked in Algorithms Jun 28, 2019

by akash.dinkar12

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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.)

akash.dinkar12 asked in Algorithms Jun 28, 2019

by akash.dinkar12

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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.

akash.dinkar12 asked in Algorithms Jun 28, 2019

by akash.dinkar12

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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 BUCKET-SORT to reflect the uniform distribution of the points in the unit circle.)

akash.dinkar12 asked in Algorithms Jun 28, 2019

by akash.dinkar12

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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 $E^2[X]$?

akash.dinkar12 asked in Algorithms Jun 28, 2019

by akash.dinkar12

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Cormen Edition 3 Exercise 8.4 Question 2 (Page No. 204)
Explain why the worst-case running time for bucket sort is $\Theta(n^2)$. What simple change to the algorithm preserves its linear average-case running time and makes its worst-case running time $O(n\ lg\ n)$?

akash.dinkar12 asked in Algorithms Jun 28, 2019

by akash.dinkar12

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Cormen Edition 3 Exercise 8.4 Question 1 (Page No. 204)
BUCKET-SORT(A) 1 let B[0...n-1] 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[n-1] together in order illustrate the operation of BUCKET-SORT on the array $A=\langle .79,.13,.16,.64,.39,.20,.89,.53,.71,.42\rangle$

akash.dinkar12 asked in Algorithms Jun 28, 2019

by akash.dinkar12

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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 $n^3-1$ in $O(n)$ time.

akash.dinkar12 asked in Algorithms Jun 28, 2019

by akash.dinkar12

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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?

akash.dinkar12 asked in Algorithms Jun 28, 2019

by akash.dinkar12

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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?

akash.dinkar12 asked in Algorithms Jun 28, 2019

by akash.dinkar12

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Cormen Edition 3 Exercise 8.3 Question 1 (Page No. 199)
RADIX-SORT(A, d) 1 for i = 1 to d 2 use a stable sort to sort array A on digit i illustrate the operation of RADIX-SORT on the following list of English words: COW, DOG, SEA, RUG, ROW, MOB, BOX, TAB, BAR, EAR, TAR, DIG, BIG, TEA, NOW, FOX.

akash.dinkar12 asked in Algorithms Jun 28, 2019

by akash.dinkar12

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