3.4 Discussion and Exercises

Both singly-linked and doubly-linked lists are folklore, having been used in programs for over 40 years. They are discussed, for example, by Knuth [46, Sections 2.2.3-2.2.5]. Even the SEList data structure seems to be a well-known data structures exercise.

Exercise 3..1   Why is it not possible, in an SLList to use a dummy node to avoid all the special cases that occur in the operations $\mathtt{push(x)}$, $\mathtt{pop()}$, $\mathtt{add(x)}$, and $\mathtt{remove()}$?

Exercise 3..2   Design and implement an SLList method, $\mathtt{secondLast()}$, that returns the second-last element of an SLList. Do this without using the member variable, $\mathtt{n}$, that keeps track of the size of the list.

Exercise 3..3   Describe and implement the List operations $\mathtt{get(i)}$, $\mathtt{set(i,x)}$, $\mathtt{add(i,x)}$ and $\mathtt{remove(i)}$ on an SLList. Each of these operations should run in $O(1+\ensuremath{\mathtt{i}})$ time.

Exercise 3..4   Design and implement an SLList method, $\mathtt{reverse()}$ that reverses the order of elements in an SLList. This method should run in $O(\ensuremath{\mathtt{n}})$ time, should not use recursion, should not use any secondary data structures, and should not create any new nodes.

Exercise 3..5   Design and implement SLList and DLList methods called $\mathtt{checkSize()}$. These methods walk through the list and count the number of nodes to see if this matches the value, $\mathtt{n}$, stored in the list. These methods return nothing, but throw an exception if the size they compute does not match the value of $\mathtt{n}$.

Exercise 3..6   Without referring to this chapter, try to recreate the code for the $\mathtt{addBefore(w)}$ operation, that creates a node, $\mathtt{u}$, and adds it just before the node $\mathtt{w}$ in a DLList. If your code does not exactly match the code given in this book it may still be correct. Test it and see if it works.

The next few exercises involve performing manipulations on DLLists. These should all be done without allocating any new nodes or temporary arrays. More specifically, they can all be done only by changing the $\mathtt{prev}$ and $\mathtt{next}$ values of existing nodes.

Exercise 3..7   Write a DLList method $\mathtt{isPalindrome()}$ that returns $\mathtt{true}$ if the list is a palindrome, i.e., the element at position $\mathtt{i}$ is equal to the element at position $\ensuremath{\mathtt{n}}-i-1$ for all $i\in\{0,\ldots,\ensuremath{\mathtt{n}}-1\}$. Your code should run in $O(\ensuremath{\mathtt{n}})$ time.

Exercise 3..8   Implement a method $\mathtt{rotate(r)}$ that ``rotates'' a DLList so that list item $\mathtt{i}$ becomes list item $(\ensuremath{\mathtt{i}}+\ensuremath{\mathtt{r}})\bmod \ensuremath{\mathtt{n}}$. This method should run in $O(1+\min\{\ensuremath{\mathtt{r}},\ensuremath{\mathtt{n}}-\ensuremath{\mathtt{r}}\})$ time and should not modify any nodes in the list.

Exercise 3..9   Write a method, $\mathtt{truncate(i)}$, that truncates a DLList at position $\mathtt{i}$. After the execution of this method, the size of the list is $\mathtt{i}$ and it contains only the elements at indices $0,\ldots,\ensuremath{\mathtt{i}}-1$. The return value is another DLList that contains the elements at indices $\ensuremath{\mathtt{i}},\ldots,\ensuremath{\mathtt{n}}-1$. This method should run in $O(\min\{\ensuremath{\mathtt{i}},\ensuremath{\mathtt{n}}-\ensuremath{\mathtt{i}}\})$ time.

Exercise 3..10   Write a DLList method, $\mathtt{absorb(l2)}$, that takes as an argument a DLList, $\mathtt{l2}$, empties it and appends its contents, in order, to the receiver. For example, if $\mathtt{l1}$ contains $a,b,c$ and $\mathtt{l2}$ contains $d,e,f$, then after calling $\mathtt{l1.absorb(l2)}$, $\mathtt{l1}$ will contain $a,b,c,d,e,f$ and $\mathtt{l2}$ will be empty.

Exercise 3..11   Write a method $\mathtt{deal()}$ that removes all the elements with odd-numbered indices from a DLList and return a DLList containing these elements. For example, if $\mathtt{l1}$, contains the elements $a,b,c,d,e,f$, then after calling $\mathtt{l1.deal()}$, $\mathtt{l1}$ should contain $a,c,e$ and a list containing $b,d,f$ should be returned.

Exercise 3..12   Write a method, $\mathtt{reverse()}$, that reverses the order of elements in a DLList.

Exercise 3..13   This exercises walks you through an implementation of the merge sort algorithm for sorting a DLList, as discussed in Section 11.1.1. In your implementation, perform comparisons between elements using the $\mathtt{compareTo(x)}$ method so that the resulting implementation can sort any DLList containing elements that implement the Comparable interface.

1. Write a DLList method called $\mathtt{takeFirst(l2)}$. This method takes the first node from $\mathtt{l2}$ and appends it to the the receiving list. This is equivalent to $\mathtt{add(size(),l2.remove(0))}$, except that it should not create a new node.
2. Write a DLList static method, $\mathtt{merge(l1,l2)}$, that takes two sorted lists $\mathtt{l1}$ and $\mathtt{l2}$, merges them, and returns a new sorted list containing the result. This causes $\mathtt{l1}$ and $\mathtt{l2}$ to be emptied in the proces. For example, if $\mathtt{l1}$ contains $a,c,d$ and $\mathtt{l2}$ contains $b,e,f$, then this method returns a new list containing $a,b,c,d,e,f$.
3. Write a DLList method $\mathtt{sort()}$ that sorts the elements contained in the list using the merge sort algorithm. This recursive algorithm works as following:
   1. If the list contains 0 or 1 elements then there is nothing to do. Otherwise,
   2. Split the list into two approximately equal length lists $\mathtt{l1}$ and $\mathtt{l2}$ using the $\mathtt{truncate(size()/2)}$ method;
   3. Recursively sort $\mathtt{l1}$;
   4. Recursively sort $\mathtt{l2}$; and, finally,
   5. Merge $\mathtt{l1}$ and $\mathtt{l2}$ into a single sorted list.

The next few exercises are more advanced and require a clear understanding of what happens to the minimum value stored in a Stack or Queue as items are added and removed.

Exercise 3..14   Design and implement a MinStack data structure that can store comparable elements and supports the stack operations $\mathtt{push(x)}$, $\mathtt{pop()}$, and $\mathtt{size()}$, as well as the $\mathtt{min()}$ operation, which returns the minimum value currently stored in the data structure. All operations should run in constant time.

Exercise 3..15   Design an implement a MinQueue data structure that can store comparable elements and supports the queue operations $\mathtt{add(x)}$, $\mathtt{remove()}$, and $\mathtt{size()}$, as well as the $\mathtt{min()}$ operation, which returns the minimum value currently stored in the data structure. All operations should run in constant amortized time.

Exercise 3..16   Design an implement a MinDeque data structure that can store comparable elements and supports the queue operations $\mathtt{addFirst(x)}$, $\mathtt{addLast(x)}$ $\mathtt{removeFirst()}$, $\mathtt{removeLast()}$ and $\mathtt{size()}$, as well as the $\mathtt{min()}$ operation, which returns the minimum value currently stored in the data structure. All operations should run in constant amortized time.

The next exercises are designed to test the reader's understanding of the implementation an analysis of the space-efficient SEList:

Exercise 3..17   Prove that, if an SEList is used like a Stack (so that the modifications are done using $\ensuremath{\mathtt{push(x)}}\equiv \ensuremath{\mathtt{add(size(),x)}}$ and $\ensuremath{\mathtt{pop()}}\equiv \ensuremath{\mathtt{remove(size()-1)}}$) then these operations run in constant amortized time, independent of the value of $\mathtt{b}$.

Exercise 3..18   Design an implement of a version of an SEList that supports all the Deque operations in constant amortized time per operation, independent of the value of $\mathtt{b}$.