Skip List | Set 3 (Searching and Deletion)
In previous article Skip List | Set 2 (Insertion) we discussed the structure of skip nodes and how to insert an element in the skip list. In this article we will discuss how to search and delete an element from skip list.
Searching an element in Skip list
Searching an element is very similar to approach for searching a spot for inserting an element in Skip list. The basic idea is if –
- Key of next node is less than search key then we keep on moving forward on the same level.
- Key of next node is greater than the key to be inserted then we store the pointer to current node i at update[i] and move one level down and continue our search.
At the lowest level (0), if the element next to the rightmost element (update[0]) has key equal to the search key, then we have found key otherwise failure. Following is the pseudo code for searching element –
Search(list, searchKey)
x := list -> header
-- loop invariant: x -> key level downto 0 do
while x -> forward[i] -> key forward[i]
x := x -> forward[0]
if x -> key = searchKey then return x -> value
else return failure
Consider this example where we want to search for key 17-
Deleting an element from the Skip list
Deletion of an element k is preceded by locating element in the Skip list using above mentioned search algorithm. Once the element is located, rearrangement of pointers is done to remove element form list just like we do in singly linked list. We start from lowest level and do rearrangement until element next to update[i] is not k. After deletion of element there could be levels with no elements, so we will remove these levels as well by decrementing the level of Skip list. Following is the pseudo code for deletion –
Delete(list, searchKey)
local update[0..MaxLevel+1]
x := list -> header
for i := list -> level downto 0 do
while x -> forward[i] -> key forward[i]
update[i] := x
x := x -> forward[0]
if x -> key = searchKey then
for i := 0 to list -> level do
if update[i] -> forward[i] ? x then break
update[i] -> forward[i] := x -> forward[i]
free(x)
while list -> level > 0 and list -> header -> forward[list -> level] = NIL do
list -> level := list -> level – 1
Consider this example where we want to delete element 6 –
Here at level 3, there is no element (arrow in red) after deleting element 6. So we will decrement level of skip list by 1. Following is the code for searching and deleting element from Skip List –
Implementation:
// C++ code for searching and deleting element in skip list
#include <bits/stdc++.h>
using namespace std;
// Class to implement node
class Node
{
public:
int key;
// Array to hold pointers to node of different level
Node **forward;
Node(int, int);
};
Node::Node(int key, int level)
{
this->key = key;
// Allocate memory to forward
forward = new Node*[level+1];
// Fill forward array with 0(NULL)
memset(forward, 0, sizeof(Node*)*(level+1));
};
// Class for Skip list
class SkipList
{
// Maximum level for this skip list
int MAXLVL;
// P is the fraction of the nodes with level
// i pointers also having level i+1 pointers
float P;
// current level of skip list
int level;
// pointer to header node
Node *header;
public:
SkipList(int, float);
int randomLevel();
Node* createNode(int, int);
void insertElement(int);
void deleteElement(int);
void searchElement(int);
void displayList();
};
SkipList::SkipList(int MAXLVL, float P)
{
this->MAXLVL = MAXLVL;
this->P = P;
level = 0;
// create header node and initialize key to -1
header = new Node(-1, MAXLVL);
};
// create random level for node
int SkipList::randomLevel()
{
float r = (float)rand()/RAND_MAX;
int lvl = 0;
while(r < P && lvl < MAXLVL)
{
lvl++;
r = (float)rand()/RAND_MAX;
}
return lvl;
};
// create new node
Node* SkipList::createNode(int key, int level)
{
Node *n = new Node(key, level);
return n;
};
// Insert given key in skip list
void SkipList::insertElement(int key)
{
Node *current = header;
// create update array and initialize it
Node *update[MAXLVL+1];
memset(update, 0, sizeof(Node*)*(MAXLVL+1));
/* start from highest level of skip list
move the current pointer forward while key
is greater than key of node next to current
Otherwise inserted current in update and
move one level down and continue search
*/
for(int i = level; i >= 0; i--)
{
while(current->forward[i] != NULL &&
current->forward[i]->key < key)
current = current->forward[i];
update[i] = current;
}
/* reached level 0 and forward pointer to
right, which is desired position to
insert key.
*/
current = current->forward[0];
/* if current is NULL that means we have reached
to end of the level or current's key is not equal
to key to insert that means we have to insert
node between update[0] and current node */
if (current == NULL || current->key != key)
{
// Generate a random level for node
int rlevel = randomLevel();
/* If random level is greater than list's current
level (node with highest level inserted in
list so far), initialize update value with pointer
to header for further use */
if(rlevel > level)
{
for(int i=level+1;i<rlevel+1;i++)
update[i] = header;
// Update the list current level
level = rlevel;
}
// create new node with random level generated
Node* n = createNode(key, rlevel);
// insert node by rearranging pointers
for(int i=0;i<=rlevel;i++)
{
n->forward[i] = update[i]->forward[i];
update[i]->forward[i] = n;
}
cout<<"Successfully Inserted key "<<key<<"\n";
}
};
// Delete element from skip list
void SkipList::deleteElement(int key)
{
Node *current = header;
// create update array and initialize it
Node *update[MAXLVL+1];
memset(update, 0, sizeof(Node*)*(MAXLVL+1));
/* start from highest level of skip list
move the current pointer forward while key
is greater than key of node next to current
Otherwise inserted current in update and
move one level down and continue search
*/
for(int i = level; i >= 0; i--)
{
while(current->forward[i] != NULL &&
current->forward[i]->key < key)
current = current->forward[i];
update[i] = current;
}
/* reached level 0 and forward pointer to
right, which is possibly our desired node.*/
current = current->forward[0];
// If current node is target node
if(current != NULL and current->key == key)
{
/* start from lowest level and rearrange
pointers just like we do in singly linked list
to remove target node */
for(int i=0;i<=level;i++)
{
/* If at level i, next node is not target
node, break the loop, no need to move
further level */
if(update[i]->forward[i] != current)
break;
update[i]->forward[i] = current->forward[i];
}
// Remove levels having no elements
while(level>0 &&
header->forward[level] == 0)
level--;
cout<<"Successfully deleted key "<<key<<"\n";
}
};
// Search for element in skip list
void SkipList::searchElement(int key)
{
Node *current = header;
/* start from highest level of skip list
move the current pointer forward while key
is greater than key of node next to current
Otherwise inserted current in update and
move one level down and continue search
*/
for(int i = level; i >= 0; i--)
{
while(current->forward[i] &&
current->forward[i]->key < key)
current = current->forward[i];
}
/* reached level 0 and advance pointer to
right, which is possibly our desired node*/
current = current->forward[0];
// If current node have key equal to
// search key, we have found our target node
if(current and current->key == key)
cout<<"Found key: "<<key<<"\n";
};
// Display skip list level wise
void SkipList::displayList()
{
cout<<"\n*****Skip List*****"<<"\n";
for(int i=0;i<=level;i++)
{
Node *node = header->forward[i];
cout<<"Level "<<i<<": ";
while(node != NULL)
{
cout<<node->key<<" ";
node = node->forward[i];
}
cout<<"\n";
}
};
// Driver to test above code
int main()
{
// Seed random number generator
srand((unsigned)time(0));
// create SkipList object with MAXLVL and P
SkipList lst(3, 0.5);
lst.insertElement(3);
lst.insertElement(6);
lst.insertElement(7);
lst.insertElement(9);
lst.insertElement(12);
lst.insertElement(19);
lst.insertElement(17);
lst.insertElement(26);
lst.insertElement(21);
lst.insertElement(25);
lst.displayList();
//Search for node 19
lst.searchElement(19);
//Delete node 19
lst.deleteElement(19);
lst.displayList();
}
/*package whatever //do not write package name here */
import java.io.*;
// Java code for inserting element in skip list
class GFG {
// Class to implement node
static class Node {
int key;
// Array to hold pointers to node of different level
Node forward[];
Node(int key, int level)
{
this.key = key;
// Allocate memory to forward
forward = new Node[level + 1];
}
};
// Class for Skip list
static class SkipList {
// Maximum level for this skip list
int MAXLVL;
// P is the fraction of the nodes with level
// i pointers also having level i+1 pointers
float P;
// current level of skip list
int level;
// pointer to header node
Node header;
SkipList(int MAXLVL, float P)
{
this.MAXLVL = MAXLVL;
this.P = P;
level = 0;
// create header node and initialize key to -1
header = new Node(-1, MAXLVL);
}
int randomLevel()
{
float r = (float)Math.random();
int lvl = 0;
while (r < P && lvl < MAXLVL) {
lvl++;
r = (float)Math.random();
}
return lvl;
}
Node createNode(int key, int level)
{
Node n = new Node(key, level);
return n;
}
// Insert given key in skip list
void insertElement(int key)
{
Node current = header;
// create update array and initialize it
Node update[] = new Node[MAXLVL + 1];
/* start from highest level of skip list
move the current pointer forward while
key is greater than key of node next to
current Otherwise inserted current in update
and move one level down and continue search
*/
for (int i = level; i >= 0; i--) {
while (current.forward[i] != null && current.forward[i].key < key)
current = current.forward[i];
update[i] = current;
}
/* reached level 0 and forward pointer to
right, which is desired position to
insert key.
*/
current = current.forward[0];
/* if current is NULL that means we have reached
to end of the level or current's key is not
equal to key to insert that means we have to
insert node between update[0] and current node
*/
if (current == null || current.key != key) {
// Generate a random level for node
int rlevel = randomLevel();
// If random level is greater than list's
// current level (node with highest level
// inserted in list so far), initialize
// update value with pointer to header for
// further use
if (rlevel > level) {
for (int i = level + 1; i < rlevel + 1;
i++)
update[i] = header;
// Update the list current level
level = rlevel;
}
// create new node with random level
// generated
Node n = createNode(key, rlevel);
// insert node by rearranging pointers
for (int i = 0; i <= rlevel; i++) {
n.forward[i] = update[i].forward[i];
update[i].forward[i] = n;
}
System.out.println(
"Successfully Inserted key " + key);
}
}
// Display skip list level wise
void displayList()
{
System.out.println("\n*****Skip List*****");
for (int i = 0; i <= level; i++) {
Node node = header.forward[i];
System.out.print("Level " + i + ": ");
while (node != null) {
System.out.print(node.key + " ");
node = node.forward[i];
}
System.out.println();
}
}
void deleteElement(int key){
Node current = header;
// create update array and initialize it
Node[] update = new Node[MAXLVL+1];
/* start from highest level of skip list
move the current pointer forward while
key is greater than key of node next to
current Otherwise inserted current in update
and move one level down and continue search
*/
for (int i = level; i >= 0; i--) {
while (current.forward[i] != null && current.forward[i].key < key)
current = current.forward[i];
update[i] = current;
}
/* reached level 0 and forward pointer to
right, which is desired position to
delete key.
*/
current=current.forward[0];
if(current!=null && current.key==key){
// delete node by rearranging pointers
for (int i = 0; i < level; i++) {
if (update[i].forward[i] != current)
break;
update[i].forward[i] = current.forward[i];
}
//if it was the only node in that level and you deleted it, delete the level
while(level>0 && header.forward[level] ==null){
level --;
}
System.out.println("Successfully deleted key "+ key);
}
}
void Search(int key){
Node current = header;
/* start from highest level of skip list
move the current pointer forward while
key is greater than key of node next to
current and move one level down and continue search
*/
for (int i = level; i >=0 ; i--) {
while (current.forward[i] != null && current.forward[i].key < key) {
current = current.forward[i];
}
}
current = current.forward[0];
// current has to be the key if it is present
if(current !=null && current.key==key) {
System.out.println("Key found");
}
else{
System.out.println("Key not found");
}
}
}
// Driver to test above code
public static void main(String[] args)
{
// create SkipList object with MAXLVL and P
SkipList lst = new SkipList(3, 0.5f);
lst.insertElement(3);
lst.insertElement(6);
lst.insertElement(7);
lst.insertElement(9);
lst.insertElement(12);
lst.insertElement(19);
lst.insertElement(17);
lst.insertElement(26);
lst.insertElement(21);
lst.insertElement(25);
lst.displayList();
lst.deleteElement(9);
lst.deleteElement(12);
lst.displayList();
lst.Search(9);
}
}
import random
class Node:
def __init__(self, key, level):
self.key = key
self.forward = [None] * (level + 1)
class SkipList:
def __init__(self, max_level, p):
self.MAXLVL = max_level
self.P = p
self.level = 0
self.header = self.create_node(-1, self.MAXLVL)
def random_level(self):
lvl = 0
while random.random() < self.P and lvl < self.MAXLVL:
lvl += 1
return lvl
def create_node(self, key, level):
return Node(key, level)
def insert_element(self, key):
current = self.header
update = [None] * (self.MAXLVL + 1)
for i in range(self.level, -1, -1):
while current.forward[i] and current.forward[i].key < key:
current = current.forward[i]
update[i] = current
current = current.forward[0]
if not current or current.key != key:
rlevel = self.random_level()
if rlevel > self.level:
for i in range(self.level + 1, rlevel + 1):
update[i] = self.header
self.level = rlevel
new_node = self.create_node(key, rlevel)
for i in range(rlevel + 1):
new_node.forward[i] = update[i].forward[i]
update[i].forward[i] = new_node
print(f"Successfully Inserted key {key}")
def delete_element(self, key):
current = self.header
update = [None] * (self.MAXLVL + 1)
for i in range(self.level, -1, -1):
while current.forward[i] and current.forward[i].key < key:
current = current.forward[i]
update[i] = current
current = current.forward[0]
if current and current.key == key:
for i in range(self.level + 1):
if update[i].forward[i] != current:
break
update[i].forward[i] = current.forward[i]
while self.level > 0 and not self.header.forward[self.level]:
self.level -= 1
print(f"Successfully deleted key {key}")
def search_element(self, key):
current = self.header
for i in range(self.level, -1, -1):
while current.forward[i] and current.forward[i].key < key:
current = current.forward[i]
current = current.forward[0]
if current and current.key == key:
print(f"Found key: {key}")
else:
print(f"Key {key} not found")
def display_list(self):
print("\n*****Skip List*****")
for i in range(self.level + 1):
node = self.header.forward[i]
print(f"Level {i}: ", end="")
while node:
print(node.key, end=" ")
node = node.forward[i]
print()
# Driver code
if __name__ == "__main__":
random.seed()
lst = SkipList(3, 0.5)
lst.insert_element(3)
lst.insert_element(6)
lst.insert_element(7)
lst.insert_element(9)
lst.insert_element(12)
lst.insert_element(19)
lst.insert_element(17)
lst.insert_element(26)
lst.insert_element(21)
lst.insert_element(25)
lst.display_list()
lst.search_element(19)
lst.delete_element(19)
lst.display_list()
using System;
using System.Collections.Generic;
class Node
{
// Class to implement node
public int Key;
public Node[] Forward;
public Node(int key, int level)
{
Key = key;
Forward = new Node[level + 1];
}
}
class SkipList
{
// Class for Skip list
private const int MaxLevel = 3; // Maximum level for this skip list
private readonly double P; // The fraction of nodes with level
// i references also having level i+1 references
private Node Header; // Header node
private int Level; // Current level of skip list
public SkipList(double p)
{
P = p;
Header = new Node(-1, MaxLevel);
Level = 0;
}
// Create new node
private Node CreateNode(int level, int key)
{
return new Node(key, level);
}
// Generate random level for a node
private int RandomLevel()
{
int level = 0;
while (new Random().NextDouble() < P && level < MaxLevel)
{
level++;
}
return level;
}
// Insert given key into skip list
public void InsertElement(int key)
{
// Create an update array and initialize it
var update = new Node[MaxLevel + 1];
var current = Header;
// Start from the highest level of skip list
// Move the current reference forward while the key
// is greater than the key of the node next to current
// Otherwise, insert current into update and move one level down and continue search
for (var i = Level; i >= 0; i--)
{
while (current.Forward[i] != null && current.Forward[i].Key < key)
{
current = current.Forward[i];
}
update[i] = current;
}
// Reached level 0 and forward reference to the right, which is the desired position to insert the key.
current = current.Forward[0];
// If current is null or its key is not equal to the key to insert, insert the node between update[0] and current node.
if (current == null || current.Key != key)
{
// Generate a random level for the node
var randomLevel = RandomLevel();
// If the random level is greater than the list's current level, initialize update values with references to the header for further use
if (randomLevel > Level)
{
for (var i = Level + 1; i <= randomLevel; i++)
{
update[i] = Header;
}
Level = randomLevel;
}
// Create a new node with the random level generated
var newNode = CreateNode(randomLevel, key);
// Insert the node by rearranging references
for (var i = 0; i <= randomLevel; i++)
{
newNode.Forward[i] = update[i].Forward[i];
update[i].Forward[i] = newNode;
}
Console.WriteLine($"Successfully inserted key {key}");
}
}
// Delete an element with the given search key
public void DeleteElement(int searchKey)
{
// Create an update array and initialize it
var update = new Node[MaxLevel + 1];
var current = Header;
// Start from the highest level of skip list
// Move the current reference forward while the key
// is greater than the key of the node next to current
// Otherwise, insert current into update and move one level down and continue search
for (var i = Level; i >= 0; i--)
{
while (current.Forward[i] != null && current.Forward[i].Key < searchKey)
{
current = current.Forward[i];
}
update[i] = current;
}
// Reached level 0 and advanced reference to the right, which is possibly our desired node
current = current.Forward[0];
// If the current node is the target node
if (current != null && current.Key == searchKey)
{
// Start from the lowest level and rearrange references
// just like we do in a singly linked list to remove the target node
for (var i = 0; i <= Level; i++)
{
// If at level i, the next node is not the target node, break the loop
if (update[i].Forward[i] != current)
{
break;
}
update[i].Forward[i] = current.Forward[i];
}
// Remove levels having no elements
while (Level > 0 && Header.Forward[Level] == null)
{
Level--;
}
Console.WriteLine($"Successfully deleted {searchKey}");
}
}
// Search for an element with the given key
public void SearchElement(int key)
{
var current = Header;
// Start from the highest level of skip list
// Move the current reference forward while the key
// is greater than the key of the node next to current
// Otherwise, insert current into update and move one level down and continue search
for (var i = Level; i >= 0; i--)
{
while (current.Forward[i] != null && current.Forward[i].Key < key)
{
current = current.Forward[i];
}
}
// Reached level 0 and advanced reference to the right, which is possibly our desired node
current = current.Forward[0];
// If the current node has a key equal to the search key, we have found our target node
if (current != null && current.Key == key)
{
Console.WriteLine($"Found key {key}");
}
}
// Display the skip list level-wise
public void DisplayList()
{
Console.WriteLine("\n***** Skip List ******");
var head = Header;
for (var lvl = 0; lvl <= Level; lvl++)
{
Console.Write($"Level {lvl}: ");
var node = head.Forward[lvl];
while (node != null)
{
Console.Write(node.Key + " ");
node = node.Forward[lvl];
}
Console.WriteLine();
}
}
}
class Program
{
public static void Main(string[] args)
{
var lst = new SkipList(0.5);
lst.InsertElement(3);
lst.InsertElement(6);
lst.InsertElement(7);
lst.InsertElement(9);
lst.InsertElement(12);
lst.InsertElement(19);
lst.InsertElement(17);
lst.InsertElement(26);
lst.InsertElement(21);
lst.InsertElement(25);
lst.DisplayList();
// Search for node 19
lst.SearchElement(19);
// Delete node 19
lst.DeleteElement(19);
lst.DisplayList();
}
}
// Node class
class Node {
constructor(key, level) {
this.key = key; // Key of the node
this.forward = new Array(level + 1); // Array to hold pointers to node of different level
}
}
// SkipList class
class SkipList {
constructor(MAXLVL, P) {
this.MAXLVL = MAXLVL; // Maximum level for this skip list
this.P = P; // P is the fraction of the nodes with level i pointers also having level i+1 pointers
this.level = 0; // Current level of skip list
this.header = new Node(-1, MAXLVL); // Pointer to header node
}
// Function to generate random level for node
randomLevel() {
let lvl = 0;
while (Math.random() < this.P && lvl < this.MAXLVL) {
lvl++;
}
return lvl;
}
// Function to create a new node
createNode(key, level) {
return new Node(key, level);
}
// Function to insert element into the skip list
insertElement(key) {
let current = this.header;
let update = new Array(this.MAXLVL + 1);
// Start from highest level of skip list
for (let i = this.level; i >= 0; i--) {
// Move the current pointer forward while key is greater than key of node next to current
while (current.forward[i] !== undefined && current.forward[i].key < key)
current = current.forward[i];
update[i] = current;
}
// Reached level 0 and forward pointer to right, which is desired position to insert key
current = current.forward[0];
// If current is NULL that means we have reached to end of the level or current's key is not equal to key to insert
if (current === undefined || current.key !== key) {
// Generate a random level for node
let rlevel = this.randomLevel();
// If random level is greater than list's current level (node with highest level inserted in list so far), initialize update value with pointer to header for further use
if (rlevel > this.level) {
for (let i = this.level + 1; i < rlevel + 1; i++)
update[i] = this.header;
// Update the list current level
this.level = rlevel;
}
// Create new node with random level generated
let n = this.createNode(key, rlevel);
// Insert node by rearranging pointers
for (let i = 0; i <= rlevel; i++) {
n.forward[i] = update[i].forward[i];
update[i].forward[i] = n;
}
console.log("Successfully Inserted key " + key);
}
}
// Function to delete element from the skip list
deleteElement(key) {
let current = this.header;
let update = new Array(this.MAXLVL + 1);
// Start from highest level of skip list
for (let i = this.level; i >= 0; i--) {
// Move the current pointer forward while key is greater than key of node next to current
while (current.forward[i] !== undefined && current.forward[i].key < key)
current = current.forward[i];
update[i] = current;
}
// Reached level 0 and forward pointer to right, which is desired position to delete key
current = current.forward[0];
// If current is not NULL and current's key is equal to key to delete
if (current !== undefined && current.key === key) {
// Delete node by rearranging pointers
for (let i = 0; i <= this.level; i++) {
if (update[i].forward[i] !== current)
break;
update[i].forward[i] = current.forward[i];
}
// If it was the only node in that level and you deleted it, delete the level
while (this.level > 0 && this.header.forward[this.level] === undefined) {
this.level--;
}
console.log("Successfully deleted key " + key);
}
}
// Function to search for an element in the skip list
search(key) {
let current = this.header;
// Start from highest level of skip list
for (let i = this.level; i >= 0; i--) {
// Move the current pointer forward while key is greater than key of node next to current
while (current.forward[i] !== undefined && current.forward[i].key < key) {
current = current.forward[i];
}
}
// Move to right node
current = current.forward[0];
// If current is not NULL and current's key is equal to key to search
if (current !== undefined && current.key === key) {
console.log("Key found");
} else {
console.log("Key not found");
}
}
// Function to display the skip list
displayList() {
console.log("\n*****Skip List*****");
for (let i = 0; i <= this.level; i++) {
let node = this.header.forward[i];
let str = "Level " + i + ": ";
while (node !== undefined) {
str += node.key + " ";
node = node.forward[i];
}
console.log(str);
}
}
}
// Driver code to test above
let lst = new SkipList(3, 0.5);
lst.insertElement(3);
lst.insertElement(6);
lst.insertElement(7);
lst.insertElement(9);
lst.insertElement(12);
lst.insertElement(19);
lst.insertElement(17);
lst.insertElement(26);
lst.insertElement(21);
lst.insertElement(25);
lst.displayList();
lst.deleteElement(9);
lst.deleteElement(12);
lst.displayList();
lst.search(9);
Output
Successfully Inserted key 3 Successfully Inserted key 6 Successfully Inserted key 7 Successfully Inserted key 9 Successfully Inserted key 12 Successfully Inserted key 19 Successfully Inserted key 17 Successfully Inserted key 26 Successfully Inserted key 21 Successfully Inserted key 25 *****Skip List***** Level 0: 3 6 7 9 12 17 19 21 25 26 Level 1: 6 9 19 26 Level 2: 19 Found key: 19 Successfully deleted key 19 *****Skip List***** Level 0: 3 6 7 9 12 17 21 25 26 Level 1: 6 9 26
Time complexity of both searching and deletion is same – Time complexity Worst case:
- Access – O(n)
- Search – O(n)
- Insert – O(n)
- Space – O(nlogn)
- Delete – O(n)
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