[Hard] 1206. Design Skiplist

Design a Skiplist without using any built-in libraries.

A skiplist is a data structure that takes O(log(n)) time to add, erase and search. Comparing with treap and red-black tree which has the same function and performance, the code length of Skiplist can be comparatively short and the idea behind Skiplists is just simple linked lists.

For example, we have a Skiplist containing [30,40,50,60,70,90] and we want to add 80 and 45 into it. The Skiplist works this way:

Artyom Kalinin [CC BY-SA 3.0], via Wikimedia Commons

You can see there are many layers in the Skiplist. Each layer is a sorted linked list. With the help of the top layers, add, erase and search can be faster than O(n). It can be proven that the average time complexity for each operation is O(log(n)) and space complexity is O(n).

To be specific, your design should include these functions:

bool search(int target): Return whether the target exists in the Skiplist or not.
void add(int num): Insert a value into the SkipList.
bool erase(int num): Remove a value from the Skiplist. If num does not exist in the Skiplist, do nothing and return false. If there exists multiple num values, removing any one of them is fine.

See more about Skiplist: https://en.wikipedia.org/wiki/Skip_list

Notice that duplicates may exist in the Skiplist, your code needs to handle this situation.

Examples

Example 1:

Input
["Skiplist", "add", "add", "add", "search", "add", "search", "erase", "erase", "search"]
[[], [1], [2], [3], [0], [4], [1], [0], [1], [1]]
Output
[null, null, null, null, false, null, true, false, true, false]

Explanation
Skiplist skiplist = new Skiplist();
skiplist.add(1);
skiplist.add(2);
skiplist.add(3);
skiplist.search(0);   // return False.
skiplist.add(4);
skiplist.search(1);   // return True.
skiplist.erase(0);    // return False, 0 is not in skiplist.
skiplist.erase(1);    // return True.
skiplist.search(1);   // return False, 1 has already been erased.

Constraints

0 <= num, target <= 20000
At most 50000 calls will be made to search, add, and erase.

Thinking Process

Design a Skiplist without using any built-in libraries.

Draw pointers before rewriting links.
Dummy head simplifies insert/delete at the head.
Slow/fast pointers find middle or detect cycles in one pass.

Common Approaches

Typical techniques for this pattern:

Approach	Time	Space	Notes
Iterative pointer walk (this problem)	O(n)	O(1)	Traversal, insertion
Dummy head node	O(n)	O(1)	Simplify head-edge cases
Reversal (3-pointer)	O(n)	O(1)	Reverse sublist or full list
Slow/fast pointers	O(n)	O(1)	Middle, cycle, merge lists

Solution

Time Complexity: O(log n) average for all operations
Space Complexity: O(n)

A skiplist is a probabilistic data structure that maintains multiple levels of sorted linked lists. Higher levels act as “express lanes” for faster traversal.

const int MAX_LEVEL = 32;
const double P_FACTOR = 0.25;

struct SkiplistNode {
    int val_;
    vector<SkiplistNode*> forward_;
    
    SkiplistNode(int val, int maxLevel = MAX_LEVEL) : val_(val), forward_(maxLevel, nullptr) {}
};

class Skiplist {
private:
    SkiplistNode* head_;
    int level_;

    int randomLevel() {
        int level = 1;
        while((rand() / (double)RAND_MAX) < P_FACTOR && level < MAX_LEVEL) {
            level++;
        }
        return level;
    }

public:
    Skiplist(): head_(new SkiplistNode(-1)), level_(0) {
    }
    
    bool search(int target) {
        SkiplistNode* curr = this->head_;
        for(int i = level_ - 1; i >= 0; i--) {
            while(curr->forward_[i] && curr->forward_[i]->val_ < target) {
                curr = curr->forward_[i];
            }
        }
        curr = curr->forward_[0];
        if(curr && curr->val_ == target) {
            return true;
        }
        return false;
    }
    
    void add(int num) {
        vector<SkiplistNode*> update(MAX_LEVEL, head_);
        SkiplistNode* curr = this->head_;
        
        for(int i = level_ - 1; i >= 0; i--) {
            while(curr->forward_[i] && curr->forward_[i]->val_ < num) {
                curr = curr->forward_[i];
            }
            update[i] = curr;
        }
        
        int lv = randomLevel();
        level_ = max(level_, lv);
        SkiplistNode* newNode = new SkiplistNode(num, lv);
        
        for(int i = 0; i < lv; i++) {
            newNode->forward_[i] = update[i]->forward_[i];
            update[i]->forward_[i] = newNode;
        }
    }
    
    bool erase(int num) {
        vector<SkiplistNode*> update(MAX_LEVEL, nullptr);
        SkiplistNode* curr = this->head_;
        
        for(int i = level_ - 1; i >= 0; i--) {
            while(curr->forward_[i] && curr->forward_[i]->val_ < num) {
                curr = curr->forward_[i];
            }
            update[i] = curr;
        }
        
        curr = curr->forward_[0];
        if(!curr || curr->val_ != num) {
            return false;
        }
        
        for(int i = 0; i < level_; i++) {
            if(update[i]->forward_[i] != curr) {
                break;
            }
            update[i]->forward_[i] = curr->forward_[i];
        }
        
        delete curr;
        
        while(level_ > 1 && head_->forward_[level_ - 1] == nullptr) {
            level_--;
        }
        
        return true;
    }
};

/**
 * Your Skiplist object will be instantiated and called as such:
 * Skiplist* obj = new Skiplist();
 * bool param_1 = obj->search(target);
 * obj->add(num);
 * bool param_3 = obj->erase(num);
 */

Solution Explanation

Approach: Iterative pointer walk (this problem)

Key idea: Design a Skiplist without using any built-in libraries.

How the code works:

Draw pointers before rewriting links.
Dummy head simplifies insert/delete at the head.
Slow/fast pointers find middle or detect cycles in one pass.

Operation	Time (Average)	Time (Worst)	Space
`search(target)`	O(log n)	O(n)	O(1)
`add(num)`	O(log n)	O(n)	O(1)
`erase(num)`	O(log n)	O(n)	O(1)
Overall	O(log n)	O(n)	O(n)

Note: Worst case occurs when all nodes are at level 0 (degenerates to linked list), but probability is extremely low.

Example Walkthrough

Adding elements: [1, 2, 3, 4]

Initial: head -> null (all levels)

add(1):  head -> [1] (level 0, 1)
         Random level = 1

add(2):  head -> [1] -> [2] (level 0)
         head -> [1] -> [2] (level 1, if level 1)
         Random level determines height

add(3):  head -> [1] -> [2] -> [3] (level 0)
         Higher levels may skip some nodes

add(4):  head -> [1] -> [2] -> [3] -> [4] (level 0)

Search operation:

search(3):
Level 2: head -> ... (skip if 3 not at this level)
Level 1: head -> ... -> [2] -> ... (move right while < 3)
Level 0: head -> ... -> [2] -> [3] (found!)

Complexity

| Operation | Time (Average) | Time (Worst) | Space | |———–|—————|————–|——-| | search(target) | O(log n) | O(n) | O(1) | | add(num) | O(log n) | O(n) | O(1) | | erase(num) | O(log n) | O(n) | O(1) | | Overall | O(log n) | O(n) | O(n) |

Note: Worst case occurs when all nodes are at level 0 (degenerates to linked list), but probability is extremely low.

Key Design Choices

MAX_LEVEL = 32: Maximum height of any node
- For 2^32 elements, expected height is ~32
- Provides good balance between space and time
P_FACTOR = 0.25: Probability of increasing level
- Lower values = fewer high-level nodes = less space
- Standard value is 0.5, but 0.25 works well
Head node with value -1: Sentinel node simplifies edge cases
- Always exists, never deleted
- Points to first real node at each level
Dynamic level management: level_ tracks current maximum
- Increases when adding high-level nodes
- Decreases when erasing removes highest level

Common Mistakes

Empty skiplist: level_ = 0, all forward_ pointers are null
Duplicate values: Multiple nodes can have same value
Erase non-existent: Returns false, no changes
Single element: Works correctly with one node
All same values: Handles multiple nodes with same value
Not updating level_: Must track current maximum level
Wrong update array initialization: Should initialize to head_ for add()
Not decreasing level_: After erase, check if highest level is empty
Memory leaks: Must delete erased nodes
Index out of bounds: Check update[i]->forward_[i] before accessing

Why Skiplist?

Advantages:

Simple implementation: Easier than balanced trees
Probabilistic balance: No complex rebalancing needed
Cache friendly: Better than trees for sequential access
Concurrent operations: Easier to make thread-safe

Disadvantages:

Space overhead: Each node stores multiple pointers
Worst case O(n): Can degenerate (very unlikely)
Random number generation: Requires good RNG

Key Takeaways

Pattern: Iterative pointer walk (this problem)
Draw pointers before rewriting links.
Dummy head simplifies insert/delete at the head.

References

LC 1206: Design Skiplist on LeetCode
LeetCode Discuss — LC 1206: Design Skiplist
LeetCode Editorial (may require premium)

432. All O`one Data Structure - Complex data structure design
146. LRU Cache - Cache design
460. LFU Cache - Frequency-based cache
380. Insert Delete GetRandom O(1) - Randomized data structure

Pattern Recognition

This problem demonstrates the “Probabilistic Data Structure” pattern:

Use randomization to achieve balance
Multiple levels for faster traversal
Maintain sorted order at each level
Trade space for time complexity

Similar concepts:

Bloom filters (probabilistic membership)
Treap (randomized BST)
Skip lists (this problem)

Robina Li

[Hard] 1206. Design Skiplist

Examples

Constraints

Thinking Process

Common Approaches

Solution

Solution Explanation

Example Walkthrough

Complexity

Key Design Choices

Common Mistakes

Why Skiplist?

Advantages:

Disadvantages:

Key Takeaways

References

Pattern Recognition

Related Posts

Recent Posts

Examples

Constraints

Thinking Process

Common Approaches

Solution

Solution Explanation

Example Walkthrough

Complexity

Key Design Choices

Common Mistakes

Why Skiplist?

Advantages:

Disadvantages:

Key Takeaways

References

Related Problems

Pattern Recognition

Related Posts

Recent Posts