811312A Data Structures and Algorithms, , Exercise 1 Solutions

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1 811312A Data Structures and Algorithms, , Exercise 1 Solutions Topics of this exercise are stacks, queues, and s. Cormen 3 rd edition, chapter 10. Task 1.1 Assume that L is a containing 1000 items. For each of the four types of s in the following table, what is the worst-case number of basic operations for each procedure mentioned? We assume that in a doubly linked, a tail pointer is maintained in addition to the head pointer, but in a singly linked, only head pointer is maintained. Here, x means a pointer to a node, and k means a key that is searched from the. Each of the following is counted as one operation: moving to next node, comparing key and changing a link. SEARCH(L,k) INSERT(L,x) DELETE(L,x) MINIMUM(L) MAXIMUM(L) singly linked Sorted, singly linked doubly linked Sorted, doubly linked Solution. Following table shows approximate number of operations, when contains 1000 nodes. singly linked Sorted, singly linked doubly linked Sorted, doubly linked SEARCH(L,k) ca ca ca ca INSERT(L,x) 2 ca ca DELETE(L,x) ca ca MINIMUM(L) ca ca MAXIMUM(L) ca ca ca Comments. Assume that the is sorted in increasing order. Searching. To find a given key in a of any kind requires in the worst case about 1000 comparisons and moving to next node. Binary search is not possible in a even if it is sorted. Insertion. If the is not sorted, we can insert in front, which takes a constant time. In a singly linked, we must update the next pointer of the added node and update the head of the. In a doubly linked, we must update also the previous pointers. If the is sorted, we must seek the right position, which takes in the worst case about 1000 comparisons and moving to next node. Deletion. From a doubly linked, one can delete an element in constant time, because an element contains a link to both previous and next element in the. Hence, these two can be linked to each other using two pointer assignments (and comparisons for nil

2 pointer). On the other hand, in a singly linked, an element contains link to its successor only. Thus we need to search the predecessor, which takes about 2000 operations in the worst case (by each node, comparison and moving). The actual deletion takes only few operations in this case also. Minimum. In ordered s the element is at the head of the, hence can be found with one operation. When using unordered s, one must iterate the entire to find the minimum. Maximum. In ordered s, the element is at the end of the. Doubly linked s have a pointer to the tail, allowing getting the maximum in one operation. Sometimes also singly linked s are implemented with pointer to the tail; in this case, the maximum could be found in constant time also here. Task 1.2 Write a pseudo code program to reverse the direction of a given singly-linked L. Use a stack as an aid. The pointer to the head of L is L.head and if x is a node in the L, the field x.next points to the node following x in L. Stack operations are given in the next task. Solution. Assume that we have a stack, which is useful when reversing orders, at our disposal. Iterate the, push the elements into the stack, pop out the elements in reversed order, take care that the new head of the is the former last element and the links are reversed as well. Below is pseudo code for the algorithm. One can also easily implement the algorithm without using stack. REVERSE(L) 1. if L.head!= NIL 2. STACK S 3. x = L.head 4. while x!=nil 5. S.PUSH(x) 6. x = x.next 7. L.head = S.POP() 8. prev = L.head 9. while!s.empty() 10. x = S.POP() 11. prev.next = x 12. prev = x 13. prev.next = NIL

3 Task 1.3 Show that the following is true: a) it is possible to implement a stack using two queues; b) it is possible to implement a queue using two stacks. Stack and queue operations are as follows: Stack S S.EMPTY() True, if S empty. False otherwise S.PUSH(x) Inserts x on top of S S.POP() Removes an element from the top of S and returns it Queue Q Q.EMPTY() True, if Q empty. False otherwise Q. ENQUEUE(x) Inserts x to the back of Q Q. DEQUEUE() Removes an element from the front of Q (i.e. the first inserted element) and returns it Solution. Stack operations are EMPTY, PUSH, and POP. Stack is a last-in-first-out container. Queue operations are EMPTY, ENQUEUE, and DEQUEUE. Queue is a first-in-first-out container. a) Here we have to implement stack operations using two queues Q1 and Q2. Let us keep Q2 empty and use it only as an aid when pushing an element into the stack. When we push an element, we use Q2 to insert the new element into the tail of Q1 so it can be directly removed. Hence we get the pseudocode #STACK WITH TWO QUEUES Q1 AND Q2 EMPTY() 1. return Q1.EMPTY() POP() 1. return Q1.DEQUEUE() PUSH(x) 1. while(!q1.empty()) 2. y = Q1.DEQUEUE() 3. Q2.ENQUEUE(y) 4. Q1.ENQUEUE(x) 5. while(!q2.empty()) 6. y = Q2.DEQUEUE() 7. Q1.ENQUEUE(y) Operations EMPTY and POP are constant time, but PUSH depends on the number of elements in the stack.

4 b) Here we have to implement queue operations using two stacks S1 and S2. Let us keep S2 empty and use it only as an aid when we enqueue an element. When we enqueue we insert the new element, with the aid of S2 to the bottom of S1. Thus the new element will be last to remove. Hence we get the pseudocode #QUEUE WITH TWO STACKS S1 AND S2 EMPTY() 1. return S1.EMPTY() DEQUEUE() 1. return S1.POP() ENQUEUE(x) 1. while(!s1.empty()) 2. y = S1.POP() 3. S2.PUSH(y) 4. S1.PUSH(x) 5. while(!s2.empty()) 6. y = S2.POP() 7. S1.PUSH(y) Operations EMPTY and DEQUEUE are constant time, but ENQUEUE depends on the number of elements in the queue.

5 Task 1.4. A queue to a service desk can hold at most 10 customers. Write a program (in C or Python) to add customers to queue and remove customers from queue one by one in the arriving order. The customers are numbered by increasing consecutive numbers. If the queue contains 10 customers, it is not possible to add a customer. Apply queue data structure in your program. From below, you can download a base file to start from. Solution. The solution programs in C and Python are linked below. In each program, the queue is a cyclic buffer implemented with an array according to Cormen s book. In C program, the struct queue represents the data structure. The functions enqueue() and dequeue() use the aforementioned queue. The former function returns 0 if inserting an element would cause overflow; the latter function returns -1 (which cannot be a customer number) in case of underflow. The main program deals with the return values. In Python program, the data structure is a class with the size of the queue as an additional attribute compared to C program. The operations enqueue() and dequeue() are functions, not methods of the Queue class, which could also be possible (and, in fact, more consistent with object-oriented programming paradigm). Otherwise, the operation is quite similar with the C program. With either language, one could naturally implement the program by using an array as a local variable in the main program.