Linked Lists1 Part-B3 Linked Lists. Linked Lists2 Singly Linked List (§ 4.4.1) A singly linked list is a concrete data structure consisting of a sequence.
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Linked Lists 2
Singly Linked List (§ 4.4.1)A singly linked list is a concrete data structure consisting of a sequence of nodesEach node stores
element link to the next node
next
elem node
A B C D
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The Node Class for List Nodes(the file is source/Node.java)
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; } // 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; }}
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Inserting at the Head1. Allocate a new
node2. update new
element3. Have new node
point to old head4. Update head to
point to new node
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Removing at the Head
1. Update head to point to next node in the list
2. Allow garbage collector to reclaim the former first node
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Inserting at the Tail1. Allocate a new node2. Insert new element3. Have new node
point to null4. Have old last node
point to new node5. Update tail to point
to new node
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Removing at the Tail
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
The interface of data structure list is in List.java.
The implementation is in NodeList.java. But it uses DNode.java. Actually, it is doubly linked list.
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Stack with a Singly Linked List
We can implement a stack with a singly linked listThe top element is stored at the first node of the listThe space used is O(n) and each operation of the Stack ADT takes O(1) time
t
nodes
elements
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Queue with a Singly Linked List
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
f
r
nodes
elements
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List ADT (§ 5.2.3)
The List ADT models a sequence of positions storing arbitrary objectsIt establishes a before/after relation between positionsGeneric methods: size(), isEmpty()
Accessor methods: first(), last() prev(p), next(p)
Update methods: replace(p, e) insertBefore(p,
e), insertAfter(p, e),
insertFirst(e), insertLast(e)
remove(p)
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Doubly Linked ListA doubly linked list provides a natural implementation of the List ADTNodes implement Position and store:
element link to the previous node link to the next node
Special trailer and header nodes
prev next
elem
trailerheader nodes/positions
elements
node
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InsertionWe visualize operation insertAfter(p, X), which returns position q
A B X C
A B C
p
A B C
p
X
q
p q
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Insertion AlgorithmAlgorithm insertAfter(p,e):
Create a new node vv.setElement(e)v.setPrev(p) {link v to its predecessor}v.setNext(p.getNext()) {link v to its successor}(p.getNext()).setPrev(v){link p’s old successor to v}p.setNext(v) {link p to its new successor, v}return v {the position for the element e}
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Deletion Algorithm
Algorithm remove(p):t = p.element {a temporary variable to hold the return value}(p.getPrev()).setNext(p.getNext()) {linking out p}(p.getNext()).setPrev(p.getPrev())p.setPrev(null) {invalidating the position p}p.setNext(null)return t
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PerformanceIn the implementation of the List ADT by means of a doubly linked list The space used by a list with n
elements is O(n) The space used by each position of the
list is O(1) All the operations of the List ADT run in O(1) time
Operation element() of the Position ADT runs in O(1) time
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Terminologies
A Graph G=(V,E): V---set of vertices and E--set of edges.
Path in G: sequence v1, v2, ..., vk of vertices in V such that (vi, vi+1) is in E.
vi and vj could be the same
Simple path in G: a sequence v1, v2, ..., vk of distinct vertices in V such that (vi, vi+1) is in E.
vi and vj can not be the same
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Terminologies (continued)
Circuit: A path v1, v2, ..., vk
such that v1 = vk
.
Simple circuit: a circuit v1, v2, ..., vk,where v1=vk and vivj for any 1<i, j<k.
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Euler circuit
Input: a graph G=(V, E) Problem: is there a circuit in G that
uses each edge exactly once.Note: G can have multiple edges, .i.e.,
two or more edges connect vertices u and v.
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Story: The problem is called Konigsberg bridge problem
it asks if it is possible to take a walk in the town shown in Figure 1 (a) crossing each bridge exactly once and returning home.
solved by Leonhard Euler [pronounced OIL-er] (1736)
The first problem solved by using graph theory A graph is constructed to describe the town. (See Figure 1 (b).)
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Theorem for Euler circuit (proof is not required)
Theorem 1 (Euler’s Theorem) The graph has an Euler circuit if and only if all the vertices of a connected graph have even degree.
Proof: (if) Going through the circuit, each
time a vertex is visited, the degree is increased by 2. Thus, the degree of each vertex is even.
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Proof of Theorem 1: (only if)
We give way to find an Euler circuit for a graph in which every vertex has an even degree.
Since each node v has even degree, when we first enter v, there is an unused edge that can be used to get out v.
The only exception is when v is a starting node. Then we get a circuit (may not contain all edges in
G) If every node in the circuit has no unused edge, all
the edges in G have been used since G is connected.
Otherwise, we can construct another circuit, merge the two circuits and get a larger circuit. In this way, every edge in G can be used.
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An example for Theorem 1:
after merge
f
a
d
b
g
h
j
i
c
e
1
4
32
87
6 5
10
9
12
11
13
a b
c
1
3 2
d
feb
4
7 6
5
d
fea b
c
15 4
32
7 6
g
h
j
i
c
e
1213
8
910
11
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An efficient algorithm for Euler circuit
1. Starting with any vertex u in G, take an unused edge (u,v) (if there is any) incident to u2. Do Step 1 for v and continue the process until v has no
unused edge. (a circuit C is obtained) 3. If every node in C has no unused edge, stop. 4. Otherwise, select a vertex, say, u in C, with some unused edge incident to u and do Steps 1 and 2 until
another circuit is obtained. 5. Merge the two circuits obtained to form one circuit 6. Continue the above process until every edge in G is used.
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Euler Path
A path which contains all edges in a graph G is called an Euler path of G.
Corollary: A graph G=(V,E) which
has an Euler path has 2 vertices of odd degree.
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Proof of the Corollary
Suppose that a graph which has an Euler path starting at u and ending at v, where uv.
Creating a new edge e joining u and v, we have an Euler circuit for the new graph G’=(V, E{e}).
From Theorem 1, all the vertices in G’ have even degree. Remove e. Then u and v are the only vertices of odd degree
in G. (Nice argument, not required for exam.)
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Representations of Graphs
Two standard ways Adjacency-list representation
Space required O(|E|) Adjacency-matrix representation
Space required O(n2).
Depending on problems, both representations are useful.
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Adjacency-list representation
Let G=(V, E) be a graph. V– set of nodes (vertices) E– set of edges.
For each uV, the adjacency list Adj[u] contains all nodes in V that are adjacent to u.
21
5 4
3
2
1
2
2
4
5 /
5
4 /
5
1
3 4 /
2 /
3 /
1
2
3
4
5
(a) (b)
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Adjacency-matrix representation
Assume that the nodes are numbered 1, 2, …, n. The adjacency-matrix consists of a |V||V| matrix
A=(aij) such that
aij= 1 if (i,j) E, otherwise aij= 0.
21
5 4
3
(a)
0 1 0 0 1 1 0 1 1 1
0 1 0 1 0
0 1 1 0 1
1 1 0 1 0
1 2 3 4 5
1
2
3
4
5
(c)
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Implementation of Euler circuit algorithm (Not required)
Data structures: Adjacency matrix Also, we have two lists to store the circuits
One for the circuit produced in Steps 1-2. One for the circuit produced in Step 4 We can merge the two lists in O(n) time.
In Step 1: when we take an unused edge (u, v), this edge is deleted from the adjacency matrix.
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Implementation of Euler circuit algorithm
In Step 2: if all cells in the column and row of v is 0, v has no unused edge.
1. Testing whether v has no unused edge.2. A circuit (may not contain all edges) is
obtained if the above condition is true.
In Step 3: if all the element’s in the matrix are 0, stop.
In step 4: if some elements in the matrix is not 0, continue.
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Summary of Euler circuit algorithm
Design a good algorithm needs two parts
1. Theorem, high level part2. Implementation: low level part.
Data structures are important. We will emphasize both parts.
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Summary (Subject to change) Understand singly linked list
How to create a list insert at head, insert at tail, remove at head and
remove at tail. Should be able to write program using singly linked
list We will have chance to practice this.
Know the concept of doubly linked list. No time to write program about this.
Euler Circuit Understand the ideas No need for the implementation.
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My Questions: (not part of the lecture) Have you learn recursive call?
A function can call itself. .
Example: f(n)=n!=n×(n-1)×(n-2)×…×2×1 and 0!=1.It can also be written as f(n)=n×f(n-1) and
f(0)=1.Java code:
Public static int recursiveFactorial(int n) { if (n==0) return 1; else return n*recursiveFactorial(n-1);}
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