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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 shortestpath distances in the tree. Give an efficient algorithm to compute the diameter of a tree, and analyze the running time of your algorithm.
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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.
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Algorithms
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Kushagra गुप्ता
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5
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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.
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graphalgorithms
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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.
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Jun 30
in
Algorithms
by
akash.dinkar12
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89
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cormen
datastructure
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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.
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Jun 30
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Algorithms
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33
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Cormen Edition 3 Exercise 10.2 Question 6 (Page No. 241)
The dynamicset 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.
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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?
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22
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Cormen Edition 3 Exercise 10.2 Question 4 (Page No. 240)
LISTSEARCH’(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 LISTSEARCH’ 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.
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1
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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.
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Jun 30
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Algorithms
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19
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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.
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12
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Cormen Edition 3 Exercise 10.2 Question 1 (Page No. 240)
Can you implement the dynamicset operation $INSERT$ on a singly linked list in $O(1)$ time? How about $DELETE$?
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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.
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Jun 28
in
Algorithms
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akash.dinkar12
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64
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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.
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queues
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14
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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Algorithms
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40
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queues
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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.
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Algorithms
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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 $Q[1...6]$.
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29
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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.
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25
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Cormen Edition 3 Exercise 10.1 Question 1 (Page No. 235)
STACKEMPTY(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 STACKEMPTY(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 $S[1...6]$
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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.)
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20
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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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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58
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cormen
algorithms
sorting
bucketsort
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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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bucketsort
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difficult
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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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27
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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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bucketsort
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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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radixsort
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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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radixsort
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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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