Skip List | Set 3 (Searching and Deletion)

14 Jun, 2022
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.
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 – 
C++
Python
// C++ code for searching and deleting element in skip list #include <bits/stdc++.h>using namespace std; // Class to implement nodeclass 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 listclass 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 nodeint 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 nodeNode* SkipList::createNode(int key, int level){    Node *n = new Node(key, level);    return n;}; // Insert given key in skip listvoid 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 listvoid 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 listvoid 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 wisevoid 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 codeint 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();}
# 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        self.header = self.createNode(self.MAXLVL, -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)        current = self.header         '''        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            to header for further use            '''            if rlevel > self.level:                for i in range(self.level+1, rlevel+1):                    update[i] = self.header                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)        current = self.header         '''        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 possibly 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.header.forward[self.level] == None):                self.level -= 1            print("Successfully deleted {}".format(search_key))     def searchElement(self, key):        current = self.header         '''        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 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:            print("Found key ", key)      # Display skip list level wise    def displayList(self):        print("\n*****Skip List******")        head = self.header        for lvl in range(self.level+1):            print("Level {}: ".format(lvl), end=" ")            node = head.forward[lvl]            while(node != None):                print(node.key, end=" ")                node = node.forward[lvl]            print("") # Driver to test above codedef 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 Worst case:
Access – O(n)
Search – O(n)
Insert – O(n)
Space – O(nlogn
Delete – O(n)
References ftp://ftp.cs.umd.edu/pub/skipLists/skiplists.pdf This article is contributed by Atul Kumar. If you like GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.
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