# Skip List | Set 2 (Insertion)

We have already discussed the idea of Skip list and how they work in Skip List | Set 1 (Introduction). In this article, we will be discussing how to insert an element in Skip list.

Deciding nodes level

Each element in the list is represented by a node, the level of the node is chosen randomly while insertion in the list. Level does not depend on the number of elements in the node. The level for node is decided by the following algorithm –

```randomLevel()
lvl := 1
//random() that returns a random value in [0...1)
while random() < p and lvl < MaxLevel do
lvl := lvl + 1
return lvl
```

MaxLevel is the upper bound on number of levels in the skip list. It can be determined as – . Above algorithm assure that random level will never be greater than MaxLevel. Here p is the fraction of the nodes with level i pointers also having level i+1 pointers and N is the number of nodes in the list.

Node Structure

Each node carries a key and a forward array carrying pointers to nodes of a different level. A level i node carries i forward pointers indexed through 0 to i.

Insertion in Skip List

We will start from highest level in the list and compare key of next node of the current node with the key to be inserted. Basic idea is If –

1. Key of next node is less than key to be inserted then we keep on moving forward on the same level
2. 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 level 0, we will definitely find a position to insert given key. Following is the psuedo code for the insertion algorithm –

```Insert(list, searchKey)
local update[0...MaxLevel+1]
for i := list -> level downto 0 do
while x -> forward[i] -> key  forward[i]
update[i] := x
x := x -> forward[0]
lvl := randomLevel()
if lvl > list -> level then
for i := list -> level + 1 to lvl do
list -> level := lvl
x := makeNode(lvl, searchKey, value)
for i := 0 to level do
x -> forward[i] := update[i] -> forward[i]
update[i] -> forward[i] := x
```

Here update[i] holds the pointer to node at level i from which we moved down to level i-1 and pointer of node left to insertion position at level 0. Consider this example where we want to insert key 17 –

Following is the code for insertion of key in Skip list –

## C++

```// C++ code for inserting 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;

public:
SkipList(int, float);
int randomLevel();
Node* createNode(int, int);
void insertElement(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
};

// 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)
{

// 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 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";
}
};

// Display skip list level wise
void SkipList::displayList()
{
cout<<"\n*****Skip List*****"<<"\n";
for (int i=0;i<=level;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();
}
```

## Python

```# Python3 code for inserting element in skip list

import random

class Node(object):
'''
Class to implement node
'''
def __init__(self, key, level):
self.key = key

# list to hold references to node of different level
self.forward = [None]*(level+1)

class SkipList(object):
'''
Class for Skip list
'''
def __init__(self, max_lvl, P):
# Maximum level for this skip list
self.MAXLVL = max_lvl

# P is the fraction of the nodes with level
# i references also having level i+1 references
self.P = P

# create header node and initialize key to -1

# current level of skip list
self.level = 0

# create  new node
def createNode(self, lvl, key):
n = Node(key, lvl)
return n

# create random level for node
def randomLevel(self):
lvl = 0
while random.random()<self.P and \
lvl<self.MAXLVL:lvl += 1
return lvl

# insert given key in skip list
def insertElement(self, key):
# create update array and initialize it
update = [None]*(self.MAXLVL+1)

'''
start from highest level of skip list
move the current reference 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 i in range(self.level, -1, -1):
while current.forward[i] and \
current.forward[i].key < key:
current = current.forward[i]
update[i] = current

'''
reached level 0 and forward reference 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 == None or current.key != key:
# Generate a random level for node
rlevel = self.randomLevel()

'''
If random level is greater than list's current
level (node with highest level inserted in
list so far), initialize update value with reference
'''
if rlevel > self.level:
for i in range(self.level+1, rlevel+1):
self.level = rlevel

# create new node with random level generated
n = self.createNode(rlevel, key)

# insert node by rearranging references
for i in range(rlevel+1):
n.forward[i] = update[i].forward[i]
update[i].forward[i] = n

print("Successfully inserted key {}".format(key))

# Display skip list level wise
def displayList(self):
print("\n*****Skip List******")
for lvl in range(self.level+1):
print("Level {}: ".format(lvl), end=" ")
while(node != None):
print(node.key, end=" ")
node = node.forward[lvl]
print("")

# Driver to test above code
def main():
lst = 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()

main()
```

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: 3 6 12 17 25
Level 2: 6 12 17 25
Level 3: 12 17 25
```

Note: The level of nodes is decided randomly, so output may differ.
Time complexity (Average):
Time complexity (Worst):
In next article we will discuss searching and deletion in Skip List.
References

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