What is the time complexity of Design Skiplist?

The optimal solution has a time complexity of O(log(n)).

What data structures are used to solve Design Skiplist?

This problem uses Linked List, Design. Understanding these concepts is essential for solving this Hard problem efficiently.

What are the key insights for solving Design Skiplist?

Insight 1: The core idea of Skiplist is to use multiple levels of linked lists to reduce the search space. Higher levels allow us to skip multiple nodes, speeding up search, insertion, and deletion. Insight 2: Probabilistic level assignment is crucial. Nodes are assigned a level randomly, with a probability 'P' of being included in the next higher level. This ensures a balanced structure and logarithmic time complexity.

Which companies ask LeetCode 1206. Design Skiplist in interviews?

This problem is frequently asked at Pure Storage, X, eBay.

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Design Skiplist - LeetCode 1206 Solution

Design Skiplist - Complete Solution Guide

Design Skiplist is LeetCode problem 1206, a Hard level challenge. This complete guide provides step-by-step explanations, multiple solution approaches, and optimized code in python3, java, cpp, c.

Problem Statement

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], v

Detailed Explanation

The problem asks us to design a Skiplist data structure without using any built-in libraries. A Skiplist is a probabilistic data structure that uses multiple levels of linked lists to achieve an average O(log n) time complexity for search, add, and erase operations. It's essentially a sorted linked list with multiple 'express lanes' allowing you to skip over large portions of the list during searches and modifications. The input is a series of operations (search, add, erase) performed on the Skiplist, and the output is the result of each operation (boolean for search and erase, null for add). The constraints specify the range of numbers to be added, searched, and erased, as well as the maximum number of operations that will be performed.

Solution Approach

The solution implements a Skiplist using a multi-level linked list structure. Each node in the Skiplist has a value and an array of 'forward' pointers, one for each level. The 'head' node points to the beginning of each level. The `search`, `add`, and `erase` operations all utilize a helper function `_find_predecessors` (or equivalent in other languages) to find the predecessors of the target value at each level. This allows us to quickly locate the position to search, insert, or delete.

Step-by-Step Algorithm

Step 1: **Initialization:** Create a Skiplist object with a head node, a maximum level, and a probability 'P' for level assignment. Initialize the current level of the Skiplist to 0.
Step 2: **Search:** Start from the highest level and traverse the linked list until a node with a value greater than or equal to the target is found. Move down to the next lower level and repeat the process. Continue until reaching the base level (level 0). If the node at level 0 is equal to the target, the target exists, return true. Otherwise, return false.
Step 3: **Add:** Find the predecessors of the new value at each level using `_find_predecessors`. Randomly determine the level of the new node. If the new node's level exceeds the current Skiplist level, update the Skiplist level. Create the new node and insert it into the Skiplist by updating the 'forward' pointers of the predecessors and the new node.
Step 4: **Erase:** Find the predecessors of the value to be erased at each level using `_find_predecessors`. If the value is found at level 0, remove the node by updating the 'forward' pointers of the predecessors. Adjust the Skiplist level if the highest level is now empty. Return true if the node was erased, false otherwise.
Step 5: **Random Level Generation:** The `_random_level` function determines the level of a new node probabilistically. Starting from level 1, it increases the level with probability 'P' until the maximum level is reached or a random number is greater than or equal to 'P'.

Key Insights

Insight 1: The core idea of Skiplist is to use multiple levels of linked lists to reduce the search space. Higher levels allow us to skip multiple nodes, speeding up search, insertion, and deletion.
Insight 2: Probabilistic level assignment is crucial. Nodes are assigned a level randomly, with a probability 'P' of being included in the next higher level. This ensures a balanced structure and logarithmic time complexity.
Insight 3: Maintaining an 'update' array during search is essential for efficient insertion and deletion. This array stores the predecessors of the target value at each level, allowing quick access to the insertion/deletion points.

Complexity Analysis

Time Complexity: O(log(n))

Space Complexity: O(n)

Topics

This problem involves: Linked List, Design.

Companies

Asked at: Pure Storage, X, eBay.

Design Skiplist | LeetCode 1206

Hard

Linked List Design

Time: O(log(n))Space: O(n)

Solution Code

Available in:

import java.util.Random;

class Skiplist {

    private static class Node {
        int val;
        Node[] forward;

        Node(int val, int level) {
            this.val = val;
            this.forward = new Node[level];
        }
    }

    private final int MAX_LEVEL = 32;
    private final double P = 0.5;
    private Node head;
    private int level;
    private Random random;

    public Skiplist() {
        head = new Node(-1, MAX_LEVEL);
        level = 0;
        random = new Random();
    }

    private int _random_level() {
        int lvl = 1;
        while (random.nextDouble() < P && lvl < MAX_LEVEL) {
            lvl++;
        }
        return lvl;
    }

    private Node[] _find_predecessors(int target) {
        Node[] update = new Node[MAX_LEVEL];
        Node curr = head;
        for (int i = level; i >= 0; i--) {
            while (curr.forward[i] != null && curr.forward[i].val < target) {
                curr = curr.forward[i];
            }
            update[i] = curr;
        }
        return update;
    }

    public boolean search(int target) {
        Node curr = head;
        for (int i = level; i >= 0; i--) {
            while (curr.forward[i] != null && curr.forward[i].val < target) {
                curr = curr.forward[i];
            }
        }

        curr = curr.forward[0];

        return curr != null && curr.val == target;
    }

    public void add(int num) {
        Node[] update = _find_predecessors(num);

        int lvl = _random_level();

        if (lvl - 1 > level) {
            level = lvl - 1;
        }

        Node new_node = new Node(num, lvl);

        for (int i = 0; i < lvl; i++) {
            new_node.forward[i] = update[i].forward[i];
            update[i].forward[i] = new_node;
        }
    }

    public boolean erase(int num) {
        Node[] update = _find_predecessors(num);

        Node node_to_delete = update[0].forward[0];

        if (node_to_delete == null || node_to_delete.val != num) {
            return false;
        }

        for (int i = 0; i < node_to_delete.forward.length; i++) {
            if (update[i].forward[i] == node_to_delete) {
                update[i].forward[i] = node_to_delete.forward[i];
            }
        }

        while (level > 0 && head.forward[level] == null) {
            level--;
        }

        return true;
    }
}

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Problem Info

Description

Problem 1206: Design Skiplist - Detailed explanation and solution approaches.

Topics

Linked List Design

Companies

Pure StorageXeBay

Stats

Difficulty:Hard

Languages:4

Companies:3

Problem #:1206

Problem Explanation

💡

Think Before You Code

🤔 Ask Yourself

→What is the input? What output do I need?
→Can I solve a smaller version of this problem first?
→What patterns have I seen in similar problems?

✏️ Try This First

Before looking at the solution, try to solve it on paper with a simple example!

🚀 Try on LeetCode

Solution Approach

📋

Step-by-Step Solution

5 steps

1
Step 1: **Initialization:** Create a Skiplist object with a head node, a maximum level, and a probability 'P' for level assignment. Initialize the current level of the Skiplist to 0.
2
Step 2: **Search:** Start from the highest level and traverse the linked list until a node with a value greater than or equal to the target is found. Move down to the next lower level and repeat the process. Continue until reaching the base level (level 0). If the node at level 0 is equal to the target, the target exists, return true. Otherwise, return false.
3
Step 3: **Add:** Find the predecessors of the new value at each level using `_find_predecessors`. Randomly determine the level of the new node. If the new node's level exceeds the current Skiplist level, update the Skiplist level. Create the new node and insert it into the Skiplist by updating the 'forward' pointers of the predecessors and the new node.
4
Step 4: **Erase:** Find the predecessors of the value to be erased at each level using `_find_predecessors`. If the value is found at level 0, remove the node by updating the 'forward' pointers of the predecessors. Adjust the Skiplist level if the highest level is now empty. Return true if the node was erased, false otherwise.
5
Step 5: **Random Level Generation:** The `_random_level` function determines the level of a new node probabilistically. Starting from level 1, it increases the level with probability 'P' until the maximum level is reached or a random number is greater than or equal to 'P'.

💡

Pro Tip for Students

Try tracing through each step with a small example on paper before coding. Understanding "why" each step happens is more important than memorizing the code!

⚡

Complexity Analysis

⏱️

Time Complexity

O(log(n))

🚀 Logarithmic - Very efficient! Halves the problem each step.

💾

Space Complexity

O(n)

📦 Linear space - Memory grows with input size.

📝

Why?

The logarithmic time complexity arises from the multi-level structure. At each level, we skip a significant portion of the list. The probabilistic level assignment ensures that the height of the Skiplist grows logarithmically with the number of elements. The space complexity is linear because, on average, each element is included in a limited number of levels. The `_find_predecessors` method allows us to efficiently navigate the Skiplist without revisiting nodes unnecessarily.

📖New to Big O? Click to learn more

Big O Notation - Quick Guide

O(1)Best - Instant

O(log n)Excellent - Binary Search

O(n)Good - One loop

O(n log n)OK - Sorting

O(n²)Slow - Nested loops

O(2ⁿ)Very Slow - Recursion

Alternative Approaches

Balanced Binary Search Tree (BST)

A balanced BST, such as an AVL tree or Red-Black tree, can also provide O(log n) time complexity for search, insert, and delete operations.

Time: O(log n)Space: O(n)

Trade-offs: While BSTs offer similar time complexity, the implementation can be more complex than a Skiplist. Skiplists, despite being probabilistic, can be easier to implement and understand. BSTs have a guaranteed log(n) performance while Skiplists have a log(n) expected performance. However, the constant factors might make a carefully tuned BST faster for specific use cases.

Hash Table with Linked List (Separate Chaining)

If only search and add operations are required and no ordering is needed, a hash table with separate chaining can offer amortized O(1) time complexity for these operations.

Time: O(1) average, O(n) worst-caseSpace: O(n)

Trade-offs: Hash tables do not support efficient ordered traversal or range queries. The erase operation requires rehashing in many implementations to maintain optimal performance. Hash tables are significantly faster than Skiplists or BSTs for search and add, but they are unsuitable if ordering or deletion is required.

✨

Key Insights

1Insight 1: The core idea of Skiplist is to use multiple levels of linked lists to reduce the search space. Higher levels allow us to skip multiple nodes, speeding up search, insertion, and deletion.
2Insight 2: Probabilistic level assignment is crucial. Nodes are assigned a level randomly, with a probability 'P' of being included in the next higher level. This ensures a balanced structure and logarithmic time complexity.
3Insight 3: Maintaining an 'update' array during search is essential for efficient insertion and deletion. This array stores the predecessors of the target value at each level, allowing quick access to the insertion/deletion points.

📚

Key Terms Explained

For Students

Linked List

Elements connected via pointers. Like a chain where each link knows the next.

Edge Cases

Edge case 1: Empty Skiplist. The `search` and `erase` operations should handle the case where the Skiplist is initially empty. The provided solution correctly initializes the Skiplist with a head node, preventing null pointer exceptions.
Edge case 2: Target value not found. The `search` operation should return false if the target value is not present in the Skiplist. The solution checks if `curr != null && curr.val == target` at the base level.
Edge case 3: Erasing a value that doesn't exist. The `erase` operation should return false and do nothing if the target value is not found. The solution checks `node_to_delete == null || node_to_delete.val != num` before proceeding with the deletion.
Edge case 4: Multiple occurrences of the same value. The `erase` function only removes one instance of the target value. The problem statement allows removing any one of them. This is correctly handled by the existing code.
Edge case 5: Adding a value when MAX_LEVEL is reached. The MAX_LEVEL serves as a hard cap to prevent unbounded memory usage. Even if randomLevel generates a level higher than the current level but less than or equal to MAX_LEVEL-1, the if (lvl - 1 > level) check ensures the global level reflects the highest node level present in the Skiplist, up to MAX_LEVEL-1.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Incorrectly calculating the random level. The probability 'P' must be used correctly to ensure logarithmic height. A common mistake is to not have a hard cap (MAX_LEVEL) which could potentially lead to very tall skiplists and increased space usage.

✗Common mistake 2: Failing to maintain the 'update' array correctly. The `_find_predecessors` function must accurately identify the predecessors at each level for insertion and deletion to work properly. Incorrect handling can lead to broken links and incorrect search results.

✗Common mistake 3: Forgetting to update the Skiplist's level after adding or erasing a node. This can lead to the Skiplist not using the higher levels effectively, degrading performance.

✗Common mistake 4: Not properly freeing memory in C implementation. Memory leaks can occur if dynamically allocated `Node`s and `update` arrays are not freed after use, especially during `skiplistErase` and `skiplistFree`.

✅

Learning Tip

Every mistake is a learning opportunity! When you make an error, understand why it happened before moving on.

🌍

Where is This Used in Real Life?

💼Real-world scenario 1: Concurrent data structures. Skiplists are often used in concurrent data structures because their structure allows for relatively easy lock-free or fine-grained locking schemes compared to balanced trees.

🏦Real-world scenario 2: Databases and indexing. Skiplists can be used as an alternative to B-trees or other indexing structures in databases. They offer good performance for search, insertion, and deletion operations, and are relatively simple to implement.

🎮Real-world scenario 3: In-memory key-value stores. Some in-memory key-value stores use Skiplists to manage sorted data efficiently.

📱Real-world scenario 4: Implementing priority queues. Though heaps are more common, skiplists could be adapted for priority queues, especially where concurrent access is important.

💡Understanding real-world applications helps you see the "why" behind algorithms. It's not just about passing interviews—these concepts power the apps and websites you use every day!

Author's Notes

💡 Key Insight

The dummy node technique prevents edge cases. For cycle detection, Floyd's algorithm is the standard. Always handle empty list and single node cases.

🎯 Interview Tip

Hard problems often combine multiple techniques. Break the problem into sub-problems.

⚠️ Watch Out

Losing reference to nodes when modifying pointers. Use temporary variables to save references.

Written by Saswata Mondal

Software Engineer • Algorithms Enthusiast • Interview Coach

👨‍💻

About the Author

👨‍💻

Saswata MondalAuthor400+ LC Problems

B.Tech CSE, IEM Kolkata | LeetCode Rating: 1672 (Top 15%)

Certified: IBM Java Developer, NPTEL Programming in Java

Last updated: October 5, 2024LeetCode Profile GitHub

This solution has been crafted based on my experience solving 400+ LeetCode problems. Each explanation is designed to help you understand the underlying concepts, not just memorize the code.

Community Tips & Insights

The key insight here is recognizing the pattern. Once you see it, the solution becomes straightforward.

@AlgoMasterJul 13

I was asked this exact problem at my Google interview. The interviewer wanted me to optimize space complexity.

@InterviewProJul 6

❓

Frequently Asked Questions

Why use a dummy node in linked list problems?

A dummy node simplifies edge cases where the head might change (insertion/deletion at head) or when building a new list. Initialize dummy.next = head, work with dummy.next throughout, and return dummy.next at the end to handle all cases uniformly.

How do I approach hard algorithm problems?

Hard problems often require: 1) Breaking down into subproblems, 2) Combining multiple advanced techniques, 3) Careful optimization of brute force. Start with a working solution (even O(n²) or worse), then optimize. Draw examples, identify patterns, and think about what makes this problem unique.

How should I practice algorithm problems effectively?

Focus on understanding patterns rather than memorizing solutions. After solving a problem, review optimal solutions and understand the intuition. Group similar problems to recognize patterns. Practice explaining your approach out loud. Review problems after a few days to test retention.

What is the optimal approach for "Design Skiplist"?

For this hard-level Linked List problem, the key is to identify the core pattern. Analyze the input constraints to determine acceptable time complexity. Consider whether design techniques can optimize the solution. Always handle edge cases and validate your approach with examples before coding.

🤖 Need help with this problem?