# 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 –

1. Key of next node is less than search key 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 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)
-- 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]
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 –

## C++

```// 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;

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
};

// 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";
}
};

// Delete element from skip list
void SkipList::deleteElement(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 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 &&
level--;
cout<<"Successfully deleted key "<<key<<"\n";
}
};

// Search for element in skip list
void SkipList::searchElement(int key)
{

/*    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++)
{
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();
}
```

## Python

```# Python3 code for searching and deleting 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))

def deleteElement(self, search_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 < search_key):
current = current.forward[i]
update[i] = current

'''
reached level 0 and advance reference to
right, which is prssibly our desired node
'''
current = current.forward[0]

# If current node is target node
if current != None and current.key == search_key:

'''
start from lowest level and rearrange references
just like we do in singly linked list
to remove target node
'''
for i in range(self.level+1):

'''
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(self.level>0 and\
self.level -= 1
print("Successfully deleted {}".format(search_key))

def searchElement(self, key):

'''
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]

# reached level 0 and advance reference to
# right, which is prssibly 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:
print("Found key ", 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()

# Search for node 19
lst.searchElement(19)

# Delete node 19
lst.deleteElement(19)
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 17 19 21 26
Level 2: 17 19 21
Found key: 19
Successfully deleted key 19

*****Skip List*****
Level 0: 3 6 7 9 12 17 21 25 26
Level 1: 3 17 21 26
Level 2: 17 21
```

Time complexity of both searching and deletion is same –
Time complexity (Average):
Time complexity (Worst):

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