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 –

- Key of next node is less than key to be inserted 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 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] x := list -> header 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 update[i] := list -> header 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; // pointer to header node Node *header; 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 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"; } }; // 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(); }

## 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 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)) # 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 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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