问题
Suggest an efficient way to find last position in heap satisfying the following conditions:
1) via pointers not via array
2) where we can insert or delete node
I could find it in O(n) time complexity but suggest a way which is of O(logn) or O(1) time complexity.
回答1:
I'm assuming here that you mean a binary heap.
If you know how many nodes are in the heap, you can find the last node in O(log n) time by converting the count to binary, and then following the path of bits from high to low. That is, take the left node if the bit is 0, and the right node if the bit is 1.
For example, if there are three nodes in the heap, the binary representation of the count is 11. The root is always the first node, leaving you with 1. Then you take the right branch to get the last node.
Say there are 5 nodes in the heap:
1
2 3
4 5
In binary, that's 101. So you take the root. The next digit is 0 so you take the left branch. The next digit is 1, so you take the right branch, leaving you at node 5.
If you want the next available slot, you add 1 to the count and do the same thing. So 6 would be 110. You take the root, then the right branch, and the left child of 3 is where you'd add the new node.
You can do the same kind of thing with any d-ary heap, except that instead of converting to binary you convert to base d. So if your heap nodes each have up to three children, you'd convert the count to base 3, and use essentially the same logic as above.
An alternative is to maintain a reference to the last node in the heap, updating it every time you modify the heap. Or, if you want to know where the next node would be placed, you maintain a reference to the first node that doesn't have two children. That's O(1), but requires bookkeeping on every insertion or deletion.
回答2:
I am answering my own question, There is no need to keep track of next pointer while inserting in heap (heap via pointers), even there is no need to keep track of parent, i am attaching running java code for heap, all possible method are included in it, getMin() = O(1), insert() = O(logn), extractMin = O(logn), decreasePriorityOfHead = O(logn), I have implemented it for generic code so it would be helpful to understand generic concept also.
class MinHeap<E extends Comparable<E>> {
private DoublyNode<E> root;
private int size = 0;
public DoublyNode<E> getRoot() {
return root;
}
public void setRoot(DoublyNode<E> root) {
this.root = root;
}
public int getSize() {
return size;
}
public void setSize(int size) {
this.size = size;
}
public MinHeap() {
}
public MinHeap(E data) {
this.root = new DoublyNode<E>(data);
this.size++;
}
private class NodeLevel<E extends Comparable<E>> {
private int level;
private DoublyNode<E> node;
public int getLevel() {
return level;
}
public void setLevel(int level) {
this.level = level;
}
public DoublyNode<E> getNode() {
return node;
}
public void setNode(DoublyNode<E> node) {
this.node = node;
}
public NodeLevel(DoublyNode<E> node, int level) {
this.node = node;
this.level = level;
}
}
public void insert(E data) {
if (this.size == 0) {
this.root = new DoublyNode<E>(data);
this.size++;
return;
}
DoublyNode<E> tempRoot = this.root;
Integer insertingElementPosition = this.size + 1;
char[] insertingElementArray = Integer.toBinaryString(
insertingElementPosition).toCharArray();
DoublyNode<E> newNode = new DoublyNode<E>(data);
int i;
for (i = 1; i < insertingElementArray.length - 1; i++) {
if (newNode.getData().compareTo(tempRoot.getData()) < 0) {
this.swap(newNode, tempRoot);
}
char c = insertingElementArray[i];
if (c == '0') {
tempRoot = tempRoot.getLeft();
} else {
tempRoot = tempRoot.getRight();
}
}
// newNode.setParent(tempRoot);
if (newNode.getData().compareTo(tempRoot.getData()) < 0) {
this.swap(newNode, tempRoot);
}
if (insertingElementArray[i] == '0') {
tempRoot.setLeft(newNode);
} else {
tempRoot.setRight(newNode);
}
this.size++;
}
public void swap(DoublyNode<E> node1, DoublyNode<E> node2) {
E temp = node1.getData();
node1.setData(node2.getData());
node2.setData(temp);
}
public E getMin() {
if (this.size == 0) {
return null;
}
return this.root.getData();
}
public void heapifyDownWord(DoublyNode<E> temp) {
if (temp == null) {
return;
}
DoublyNode<E> smallerChild = this.getSmallerChild(temp);
if (smallerChild == null) {
return;
}
if (smallerChild.getData().compareTo(temp.getData()) < 0) {
this.swap(temp, smallerChild);
this.heapifyDownWord(smallerChild);
}
}
public DoublyNode<E> getSmallerChild(DoublyNode<E> temp) {
if (temp.getLeft() != null && temp.getRight() != null) {
return (temp.getLeft().getData()
.compareTo(temp.getRight().getData()) < 0) ? temp.getLeft()
: temp.getRight();
} else if (temp.getLeft() != null) {
return temp.getLeft();
} else {
return temp.getRight();
}
}
public E extractMin() {
if (this.root == null) {
return null;
}
E temp = this.root.getData();
if (this.root.getLeft() == null && this.root.getRight() == null) {
this.root = null;
this.size--;
return temp;
}
DoublyNode<E> parentOfLastData = this.getParentOfLastData();
if (parentOfLastData.getRight() != null) {
this.root.setData(parentOfLastData.getRight().getData());
parentOfLastData.setRight(null);
} else {
this.root.setData(parentOfLastData.getLeft().getData());
parentOfLastData.setLeft(null);
}
this.heapifyDownWord(this.root);
return temp;
}
public DoublyNode<E> getParentOfLastData() {
if (this.size == 0) {
return null;
}
DoublyNode<E> tempRoot = this.root;
Integer insertingElementPosition = this.size;
char[] insertingElementArray = Integer.toBinaryString(
insertingElementPosition).toCharArray();
int i;
for (i = 1; i < insertingElementArray.length - 1; i++) {
char c = insertingElementArray[i];
if (c == '0') {
tempRoot = tempRoot.getLeft();
} else {
tempRoot = tempRoot.getRight();
}
}
return tempRoot;
}
public DoublyNode<E> getParentOfLastEmptyPosition() {
if (this.size == 0) {
return null;
}
DoublyNode<E> tempRoot = this.root;
Integer insertingElementPosition = this.size + 1;
char[] insertingElementArray = Integer.toBinaryString(
insertingElementPosition).toCharArray();
System.out.println(insertingElementArray.toString());
int i;
for (i = 1; i < insertingElementArray.length - 1; i++) {
char c = insertingElementArray[i];
if (c == '0') {
tempRoot = tempRoot.getLeft();
} else {
tempRoot = tempRoot.getRight();
}
}
return tempRoot;
}
public void print() {
if (this.root == null) {
System.out.println("Heap via pointer is empty!");
return;
}
System.out.println("\n Heap via pointer is:- ");
Queue<NodeLevel<E>> dataQueue = new Queue<NodeLevel<E>>();
Queue<Space> spaceQueue = new Queue<Space>();
dataQueue.enQueue(new NodeLevel<E>(this.root, 1));
int heightOfTree = this.getHeightOfHeap();
Double powerHeghtBST = Math.pow(heightOfTree, 2);
spaceQueue.enQueue(new Space(powerHeghtBST.intValue(), false));
while (!dataQueue.isEmpty()) {
Space space = spaceQueue.deQueue();
NodeLevel<E> nodeLevel = dataQueue.deQueue();
while (space.isNullSpace()) {
space.printNullSpace();
spaceQueue.enQueue(space);
space = spaceQueue.deQueue();
}
space.printFrontSpace();
System.out.print(nodeLevel.getNode().getData().printingData());
space.printBackSpace();
if (nodeLevel.getNode().getLeft() != null) {
dataQueue.enQueue(new NodeLevel<E>(nodeLevel.getNode()
.getLeft(), nodeLevel.getLevel() + 1));
spaceQueue.enQueue(new Space(space.getSpaceSize() / 2, false));
} else {
spaceQueue.enQueue(new Space(space.getSpaceSize() / 2, true));
}
if (nodeLevel.getNode().getRight() != null) {
dataQueue.enQueue(new NodeLevel<E>(nodeLevel.getNode()
.getRight(), nodeLevel.getLevel() + 1));
spaceQueue.enQueue(new Space(space.getSpaceSize() / 2, false));
} else {
spaceQueue.enQueue(new Space(space.getSpaceSize() / 2, true));
}
if (!dataQueue.isEmpty()
&& nodeLevel.getLevel() + 1 == dataQueue.getFrontData()
.getLevel()) {
System.out.println("\n");
}
}
}
public int getHeightOfHeap() {
if (this.size == 0) {
return 0;
}
Double height = Math.log(this.size) / Math.log(2) + 1;
return height.intValue();
}
public void changePriorityOfHeapTop(E data) {
if (this.root == null) {
return;
}
this.root.setData(data);
this.heapifyDownWord(this.root);
}
}
interface Comparable<T> extends java.lang.Comparable<T> {
/**
* this methos returns a string of that data which to be shown during
* printing tree
*
* @return
*/
public String printingData();
}
public class PracticeMainClass {
public static void main(String[] args) {
MinHeap<Student> minHeap1 = new MinHeap<Student>();
minHeap1.insert(new Student(50, "a"));
minHeap1.insert(new Student(20, "a"));
minHeap1.insert(new Student(60, "a"));
minHeap1.insert(new Student(30, "a"));
minHeap1.insert(new Student(40, "a"));
minHeap1.insert(new Student(70, "a"));
minHeap1.insert(new Student(10, "a"));
minHeap1.insert(new Student(55, "a"));
minHeap1.insert(new Student(35, "a"));
minHeap1.insert(new Student(45, "a"));
minHeap1.print();
minHeap1.getMin();
minHeap1.print();
System.out
.println("\nminimum is:- " + minHeap1.getMin().printingData());
minHeap1.print();
System.out.println("\nminimum is:- "
+ minHeap1.extractMin().printingData());
minHeap1.print();
minHeap1.changePriorityOfHeapTop(new Student(75, "a"));
minHeap1.print();
}
}
class DoublyNode<E extends Comparable<E>> {
private E data;
private DoublyNode<E> left;
private DoublyNode<E> right;
// private DoublyNode<E> parent;
public DoublyNode() {
}
public DoublyNode(E data) {
this.data = data;
}
public E getData() {
return data;
}
public void setData(E data) {
this.data = data;
}
public DoublyNode<E> getLeft() {
return left;
}
public void setLeft(DoublyNode<E> left) {
this.left = left;
}
public DoublyNode<E> getRight() {
return right;
}
public void setRight(DoublyNode<E> right) {
this.right = right;
}
// public DoublyNode<E> getParent() {
// return parent;
// }
// public void setParent(DoublyNode<E> parent) {
// this.parent = parent;
// }
}
class Space {
private boolean isNullSpace = false;
private String frontSpace;
private String backSpace;
private String nullSpace;
private int spaceSize;
public boolean isNullSpace() {
return isNullSpace;
}
public void setNullSpace(boolean isNullSpace) {
this.isNullSpace = isNullSpace;
}
public int getSpaceSize() {
return spaceSize;
}
public void setSpaceSize(int spaceSize) {
this.spaceSize = spaceSize;
}
public Space(int spaceSize, boolean isNullSpace) {
this.isNullSpace = isNullSpace;
this.spaceSize = spaceSize;
if (spaceSize == 0) {
this.frontSpace = "";
this.backSpace = "";
this.nullSpace = " ";
} else if (spaceSize == 1) {
this.frontSpace = " ";
this.backSpace = "";
this.nullSpace = " ";
} else if (spaceSize == 2) {
this.frontSpace = " ";
this.backSpace = "";
this.nullSpace = " ";
} else {
this.frontSpace = String.format("%" + (spaceSize) + "s", " ");
this.backSpace = String.format("%" + (spaceSize - 2) + "s", " ");
this.nullSpace = String.format("%" + 2 * (spaceSize) + "s", " ");
}
}
public void printFrontSpace() {
System.out.print(this.frontSpace);
}
public void printBackSpace() {
System.out.print(this.backSpace);
}
public void printNullSpace() {
System.out.print(this.nullSpace);
}
}
class Queue<E> {
private Node<E> front;
private Node<E> rear;
private int queueSize = 0;
public Queue() {
}
public Queue(E data) {
this.front = new Node(data);
this.rear = this.front;
}
public void enQueue(E data) {
if (this.rear == null) {
this.rear = new Node(data);
this.front = this.rear;
} else {
Node newNode = new Node(data);
this.rear.setNext(newNode);
this.rear = newNode;
}
this.queueSize++;
}
public E deQueue() {
E returnValue;
if (this.front == null) {
return null;
} else if (this.front == this.rear) {
returnValue = this.front.getData();
this.front = null;
this.rear = null;
} else {
returnValue = this.front.getData();
this.front = this.front.getNext();
}
this.queueSize--;
return returnValue;
}
public void print() {
Node temp = this.front;
System.out.print("\n Queue is:- ");
if (temp == null) {
System.out.println(" Empty! ");
}
while (temp != null) {
System.out.print(temp.getData() + ",");
temp = temp.getNext();
}
}
public int getQueueSize() {
return queueSize;
}
public E getFrontData() {
if (this.front == null) {
System.out.println("queue is empty!");
return null;
}
return this.front.getData();
}
public E getRearData() {
if (this.rear == null) {
System.out.println("queue is empty!");
return null;
}
return this.rear.getData();
}
public boolean isEmpty() {
return this.front == null;
}
}
class Node<E> {
private E data;
private Node next;
public Node(E data) {
this.data = data;
}
public E getData() {
return data;
}
public void setData(E data) {
this.data = data;
}
public Node getNext() {
return next;
}
public void setNext(Node next) {
this.next = next;
}
}
class Student implements Comparable<Student> {
private int id;
private String name;
@Override
public int compareTo(Student student) {
if (this.id == student.id) {
return 0;
} else if (this.id < student.id) {
return -1;
} else {
return 1;
}
}
public Student(int id, String name) {
this.id = id;
this.name = name;
}
public int getId() {
return id;
}
public void setId(int id) {
this.id = id;
}
public String getName() {
return name;
}
public void setName(String name) {
this.name = name;
}
@Override
public String printingData() {
// String printingData = "{ id: "+this.id+" name: "+this.name+" }";
String printingData = String.valueOf(this.id);
return printingData;
}
}
Output of this code is:-
Heap via pointer is:-
10
30 20
35 40 70 60
55 50 45
Heap via pointer is:-
10
30 20
35 40 70 60
55 50 45
minimum is:- 10
Heap via pointer is:-
10
30 20
35 40 70 60
55 50 45
minimum is:- 10
Heap via pointer is:-
20
30 45
35 40 70 60
55 50
Heap via pointer is:-
30
35 45
50 40 70 60
55 75
来源：https://stackoverflow.com/questions/40586802/heap-data-structure-via-pointers