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Sequences 10/14/2004 4:33 PM Singly Linked List (§ 4.4.1) A singly linked list is a concrete data structure consisting of a sequence of nodes Each node stores Linked Lists next element link to the next node node elem ∅ A © 2004 Goodrich, Tamassia Linked Lists and Vectors 1 The Node Class for List Nodes Linked Lists and Vectors 3 Inserting at the Head Linked Lists and Vectors 2 // Accessor methods: public Object getElement() { return element; } public Node getNext() { return next; } // Modifier methods: public void setElement(Object newElem) { element = newElem; } public void setNext(Node newNext) { next = newNext; } } © 2004 Goodrich, Tamassia Linked Lists and Vectors 4 1. Update head to node 2. Insert new element 3. Have new node point to old head 4. Update head to point to new node Linked Lists and Vectors D Removing at the Head 1. Allocate a new © 2004 Goodrich, Tamassia C The Node Class for List Nodes public class Node { // Instance variables: private Object element; private Node next; /** Creates a node with null references to its element and next node. */ public Node() { this(null, null); } /** Creates a node with the given element and next node. */ public Node(Object e, Node n) { element = e; next = n; } © 2004 Goodrich, Tamassia © 2004 Goodrich, Tamassia B point to next node in the list 2. Allow garbage collector to reclaim the former first node 5 © 2004 Goodrich, Tamassia Linked Lists and Vectors 6 1 Sequences 10/14/2004 4:33 PM Inserting at the Tail Removing at the Tail 1. Allocate a new 2. 3. 4. 5. Removing at the tail of a singly linked list is not efficient! There is no constant-time way to update the tail to point to the previous node node Insert new element Have new node point to null Have old last node point to new node Update tail to point to new node © 2004 Goodrich, Tamassia Linked Lists and Vectors 7 Stack with a Singly Linked List © 2004 Goodrich, Tamassia Linked Lists and Vectors Queue with a Singly Linked List We can implement a stack with a singly linked list The top element is stored at the first node of the list The space used is O(n) and each operation of the Stack ADT takes O(1) time We can implement a queue with a singly linked list The front element is stored at the first node The rear element is stored at the last node The space used is O(n) and each operation of the Queue ADT takes O(1) time r nodes nodes ∅ t f ∅ elements © 2004 Goodrich, Tamassia 8 Linked Lists and Vectors elements 9 © 2004 Goodrich, Tamassia Linked Lists and Vectors 10 The Vector ADT (§5.1) The Vector ADT extends the notion of array by storing a sequence of arbitrary objects An element can be accessed, inserted or removed by specifying its rank (number of elements preceding it) An exception is thrown if an incorrect rank is specified (e.g., a negative rank) Vectors and Array Lists © 2004 Goodrich, Tamassia Linked Lists and Vectors 11 © 2004 Goodrich, Tamassia Main vector operations: object elemAtRank(integer r): returns the element at rank r without removing it object replaceAtRank(integer r, object o): replace the element at rank with o and return the old element insertAtRank(integer r, object o): insert a new element o to have rank r object removeAtRank(integer r): removes and returns the element at rank r Additional operations size() and isEmpty() Linked Lists and Vectors 12 2 Sequences 10/14/2004 4:33 PM Applications of Vectors Array-based Vector Use an array V of size N A variable n keeps track of the size of the vector (number of elements stored) Operation elemAtRank(r) is implemented in O(1) time by returning V[r] Direct applications Sorted collection of objects (elementary database) Indirect applications Auxiliary data structure for algorithms Component of other data structures Linked Lists and Vectors © 2004 Goodrich, Tamassia V 0 1 2 13 Insertion V 0 1 2 r n V 0 1 2 r 0 1 2 o r n n 0 1 2 r n 0 1 2 n Linked Lists and Vectors 15 The space used by the data structure is O(n) size, isEmpty, elemAtRank and replaceAtRank run in O(1) time insertAtRank and removeAtRank run in O(n) time If we use the array in a circular fashion, insertAtRank(0) and removeAtRank(0) run in O(1) time In an insertAtRank operation, when the array is full, instead of throwing an exception, we can replace the array with a larger one Linked Lists and Vectors © 2004 Goodrich, Tamassia r n Linked Lists and Vectors 16 Growable Array-based Vector In the array based implementation of a Vector © 2004 Goodrich, Tamassia o r V Performance 0 1 2 V V 14 In operation removeAtRank(r), we need to fill the hole left by the removed element by shifting backward the n − r − 1 elements V[r + 1], …, V[n − 1] In the worst case (r = 0), this takes O(n) time V Linked Lists and Vectors © 2004 Goodrich, Tamassia Deletion In operation insertAtRank(r, o), we need to make room for the new element by shifting forward the n − r elements V[r], …, V[n − 1] In the worst case (r = 0), this takes O(n) time © 2004 Goodrich, Tamassia n r In a push (insertAtRank(t)) Algorithm push(o) operation, when the array if t = S.length − 1 then is full, instead of throwing A ← new array of an exception, we can size … replace the array with a for i ← 0 to t do larger one A[i] ← S[i] S←A How large should the new t←t+1 array be? S[t] ← o incremental strategy: increase the size by a constant c doubling strategy: double the size Linked Lists and Vectors © 2004 Goodrich, Tamassia 17 18 3 Sequences 10/14/2004 4:33 PM Comparison of the Strategies Incremental Strategy Analysis We replace the array k = n/c times The total time T(n) of a series of n push operations is proportional to n + c + 2c + 3c + 4c + … + kc = n + c(1 + 2 + 3 + … + k) = n + ck(k + 1)/2 Since c is a constant, T(n) is O(n + k2), i.e., O(n2) The amortized time of a push operation is O(n) We compare the incremental strategy and the doubling strategy by analyzing the total time T(n) needed to perform a series of n push operations We assume that we start with an empty stack represented by an array of size 1 We call amortized time of a push operation the average time taken by a push over the series of operations, i.e., T(n)/n © 2004 Goodrich, Tamassia Linked Lists and Vectors 19 © 2004 Goodrich, Tamassia Linked Lists and Vectors geometric series 20 Java.util package contains classes which implement useful data structures: 2 4 1 Linked Lists and Vectors java.util.Vector and java.util.ArrayList Doubling Strategy Analysis We replace the array k = log2 n times The total time T(n) of a series of n push operations is proportional to n + 1 + 2 + 4 + 8 + …+ 2k = n + 2k + 1 −1 = 2n −1 T(n) is O(n) The amortized time of a push operation is O(1) © 2004 Goodrich, Tamassia 1 ArrayList, LinkedList, Stack, HashMap,.. Vector, Hashtable: retro-fitted in the Collections hierarchy, thread-safe Java API – see for example http://www.cs.nott.ac.uk/TSG/manuals/java/JDK14/api/ Or see http://java.sun.com 8 21 © 2004 Goodrich, Tamassia Linked Lists and Vectors 22 4