0.9 Exercises  (Page 8/5)

Exercises

(From Introduction to Algorithms, Second Edition. MIT Press, ISBN: 0262032937)

Chapter 2. linked lists

Exercises 2.1

Implement a stack using a singly linked list L. The operations PUSH and POP should still take O(1) time.

Exercises 2.2

Implement a queue by a singly linked list L. The operations ENQUEUE and DEQUEUE should still take O(1) time.

Exercises 2.3

The dynamic-set operation UNION takes two disjoint sets S1 and S2 as input, and it returns a set S = S1 _ S2 consisting of all the elements of S1 and S2. The sets S1 and S2 are usually destroyed by the operation. Show how to support UNION in O(1) time using a suitable list data structure.

Exercises 2.4

Explain how to implement doubly linked lists using only one pointer value np[x] per item

instead of the usual two (next and prev). Assume that all pointer values can be interpreted as k-bit integers, and define np[x] to be np[x]= next[x] XOR prev[x], the k-bit "exclusive-or" of next[x] and prev[x]. (The value NIL is represented by 0.) Be sure to describe what information is needed to access the head of the list. Show how to implement the SEARCH, INSERT, and DELETE operations on such a list. Also show how to reverse such a list in O(1) time.

Chapter 3. stack and queue

Exercises 3.1

Using Figure above as a model, illustrate the result of each operation in the sequence PUSH(S, 4), PUSH(S, 1), PUSH(S, 3), POP(S), PUSH(S, 8), and POP(S) on an initially empty stack S stored in array S[1 _ 6].

Exercises 3.2

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.

Exercises 3.3

Using Figure above as a model, illustrate the result of each operation in the sequence

ENQUEUE(Q, 4), ENQUEUE(Q, 1), ENQUEUE(Q, 3), DEQUEUE(Q), ENQUEUE(Q, 8), and DEQUEUE(Q) on an initially empty queue Q stored in array Q[1 _ 6].

Exercises 3.4

Rewrite ENQUEUE and DEQUEUE to detect underflow and overflow of a queue.

Exercises 3.5

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 four O(1)-time procedures to insert elements into and delete elements from both ends of a deque constructed from an array.

Exercises 3.6

Show how to implement a queue using two stacks. Analyze the running time of the queue operations.

Exercises 3.7.

Show how to implement a stack using two queues. Analyze the running time of the stack operations.

Chapter 4. designing algorithms

Exercises 4.1.

Using Figure below as a model, illustrate the operation of merge sort on the array A = _3, 41, 52, 26, 38, 57, 9, 49_.

Exercises 4.2.

Rewrite the MERGE procedure so that it does not use sentinels, instead 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.

Exercises 4.3

Insertion sort can be expressed as a recursive procedure as follows. In order to sort A[1 _ n], we recursively sort A[1 _ n -1]and then insert A[n] into the sorted array A[1 _ n - 1]. Write a recurrence for the running time of this recursive version of insertion sort.