#1206Hard~50 min

Design Skiplist

Time O(logn), on average · Space O(n) · Official statement on LeetCode

Solutions

// Time:  O(logn) on average for each operation
// Space: O(n)

// see proof in references:
// 1. https://kunigami.blog/2012/09/25/skip-lists-in-python/
// 2. https://opendatastructures.org/ods-cpp/4_4_Analysis_Skiplists.html
// 3. https://brilliant.org/wiki/skip-lists/
class Skiplist {
private:
    class SkipNode {
    public:
        SkipNode() : SkipNode(0, -1) {
            
        }
    
        SkipNode(int level, int num)
          : num(num)
          , nexts(level) {
            
        }

        int num;
        vector<SkipNode *> nexts;
    };

public:
    Skiplist()
      : gen_((random_device())())
      , len_(0)
      , head_(new SkipNode()) {
        
    }
    
    ~Skiplist() {
        if (head_->nexts.empty()) {
            return;
        }
        auto curr = head_->nexts[0];
        while (curr) {
            auto next = curr->nexts[0];
            delete curr;
            curr = next;
        }
    }
    
    bool search(int target) const {
        return find(target, find_prev_nodes(target)) != nullptr;
    }
    
    void add(int num) {
        auto node = new SkipNode(random_level(), num);
        if (head_->nexts.size() < node->nexts.size()) {
            head_->nexts.resize(node->nexts.size());
        }
        auto prevs = find_prev_nodes(num);
        for (int i = 0; i < node->nexts.size(); ++i) {
            node->nexts[i] = prevs[i]->nexts[i];
            prevs[i]->nexts[i] = node;
        }
        ++len_;
    }
    
    bool erase(int num) {
        auto prevs = find_prev_nodes(num);
        auto curr = find(num, prevs);
        if (!curr) {
            return false;
        }
        --len_;
        for (int i = curr->nexts.size() - 1; i >= 0; --i) {
            prevs[i]->nexts[i] = curr->nexts[i];
            if (!head_->nexts[i]) {
                head_->nexts.pop_back();
            }
        }
        delete curr;
        return true;
    }

    int size() const {
        return len_;
    }
    
private:
    SkipNode *find(int num, const vector<SkipNode *>& prevs) const {
        if (!prevs.empty()) {
            auto candidate = prevs[0]->nexts[0];
            if (candidate && candidate->num == num) {
                return candidate;
            }
        }
        return nullptr;
    }
    
    vector<SkipNode *> find_prev_nodes(int num) const {
        vector<SkipNode *> prevs(head_->nexts.size());
        auto curr = head_;
        for (int i = head_->nexts.size() - 1; i >= 0; --i) {
            while (curr->nexts[i] && curr->nexts[i]->num < num) {
                curr = curr->nexts[i];
            }
            prevs[i] = curr;
        }
        return prevs;
    }
    
    int random_level() {
        static const int P_NUMERATOR = 1;
        static const int P_DENOMINATOR = 2;  // P = 1/4 in redis implementation
        static const int MAX_LEVEL = 32;  // enough for 2^32 elements
        int level = 1;
        while (uniform_int_distribution<int>{1, P_DENOMINATOR}(gen_) <= P_NUMERATOR &&
               level < MAX_LEVEL) {
            ++level;
        }
        return level;
    }
    
    void print_list() const {
        for (int i = head_->nexts.size() - 1; i >= 0; --i) {
            auto curr = head_->nexts[i];
            cout << curr->num;
            curr = curr->nexts[i];
            while (curr) {
                cout << "->" << curr->num;
                curr = curr->nexts[i];
            }
            cout << endl;
        }
    }

    default_random_engine gen_;
    int len_;
    SkipNode *head_;
};

// Time:  O(logn) on average for each operation
// Space: O(n)
// smart pointer version (a little bit slower)
class Skiplist2 {
private:
    class SkipNode {
    public:
        SkipNode() : SkipNode(0, -1) {
            
        }
    
        SkipNode(int level, int num)
          : num(num)
          , nexts(level) {
            
        }

        int num;
        vector<shared_ptr<SkipNode>> nexts;
    };

public:
    Skiplist2()
      : gen_((random_device())())
      , len_(0)
      , head_(make_shared<SkipNode>()) {
        
    }
    
    bool search(int target) const {
        return find(target, find_prev_nodes(target)) != nullptr;
    }
    
    void add(int num) {
        auto node = make_shared<SkipNode>(random_level(), num);
        if (head_->nexts.size() < node->nexts.size()) {
            head_->nexts.resize(node->nexts.size());
        }
        auto prevs = find_prev_nodes(num);
        for (int i = 0; i < node->nexts.size(); ++i) {
            node->nexts[i] = prevs[i]->nexts[i];
            prevs[i]->nexts[i] = node;
        }
        ++len_;
    }
    
    bool erase(int num) {
        auto prevs = find_prev_nodes(num);
        auto curr = find(num, prevs);
        if (!curr) {
            return false;
        }
        --len_;
        for (int i = curr->nexts.size() - 1; i >= 0; --i) {
            prevs[i]->nexts[i] = curr->nexts[i];
            if (!head_->nexts[i]) {
                head_->nexts.pop_back();
            }
        }
        return true;
    }

    int size() const {
        return len_;
    }

private:
    shared_ptr<SkipNode> find(int num, const vector<shared_ptr<SkipNode>>& prevs) const {
        if (!prevs.empty()) {
            auto candidate = prevs[0]->nexts[0];
            if (candidate && candidate->num == num) {
                return candidate;
            }
        }
        return nullptr;
    }
    
    vector<shared_ptr<SkipNode>> find_prev_nodes(int num) const {
        vector<shared_ptr<SkipNode>> prevs(head_->nexts.size());
        auto curr = head_;
        for (int i = head_->nexts.size() - 1; i >= 0; --i) {
            while (curr->nexts[i] && curr->nexts[i]->num < num) {
                curr = curr->nexts[i];
            }
            prevs[i] = curr;
        }
        return prevs;
    }
    
    int random_level() {
        static const int P_NUMERATOR = 1;
        static const int P_DENOMINATOR = 2;  // P = 1/4 in redis implementation
        static const int MAX_LEVEL = 32;  // enough for 2^32 elements
        int level = 1;
        while (uniform_int_distribution<int>{1, P_DENOMINATOR}(gen_) <= P_NUMERATOR &&
               level < MAX_LEVEL) {
            ++level;
        }
        return level;
    }
    
    void print_list() const {
        for (int i = head_->nexts.size() - 1; i >= 0; --i) {
            auto curr = head_->nexts[i];
            cout << curr->num;
            curr = curr->nexts[i];
            while (curr) {
                cout << "->" << curr->num;
                curr = curr->nexts[i];
            }
            cout << endl;
        }
    }

    default_random_engine gen_;
    int len_;
    shared_ptr<SkipNode> head_;
};

Beginner Explanation

What is Design Skiplist?

Design Skiplist (LeetCode #1206) is a Hard problem that primarily trains design.

How to think about it

Restate the goal in your own words before coding.
Work a tiny example by hand so the invariant becomes obvious.
Identify the pattern — this problem aligns with general problem-solving.
Only then translate the idea into code.

Why this problem matters

Hard problems force you to combine patterns and prove complexity carefully — interview gold.

AlgoForge explanations are original teaching notes. Always open the official problem statement on LeetCode for constraints and examples.

Interview Walkthrough

Interview approach for Design Skiplist

Opening (30–60 seconds)

Clarify inputs/outputs and edge cases (empty input, single element, duplicates, overflow).
State a brute force so the interviewer knows you can solve it naively.
Propose the optimal direction tied to general problem-solving.

Core solution narrative

Define the state you track (pointers, DP cell, set membership, stack top, etc.).
Explain the transition when you process the next element.
Call out time (O(logn), on average) and space (O(n)) before coding.
Code cleanly; narrate variable names.

What interviewers listen for

Correctness on edge cases
Complexity honesty
Ability to discuss trade-offs (e.g., hash map space vs. sort + two pointers)

Follow-up questions they may ask

Can you solve it with less memory?
What if the input stream is infinite / doesn't fit in RAM?
How would tests look for adversarial inputs?

Optimized Approach

Optimized solution notes

The reference solutions on AlgoForge target O(logn), on average time and O(n) space.

Pattern focus: general problem-solving

Use the pattern as a checklist:

Identify the dominant pattern and stick to one clear invariant

Multiple methods appear in the source solutions — compare them and explain when each is preferable.

Implementation tips

Prefer readable names over micro-optimizations in interviews.
Extract helpers only when they clarify (e.g., expand-around-center, DFS visit).
After AC-level logic, re-scan for off-by-one and null checks.

Complexity Analysis

Complexity

Measure	Bound
Time	O(logn), on average
Space	O(n)

How to justify this in an interview

Time: count loops, map/set operations, and recursive branching; state average vs worst case if relevant.
Space: include hash maps, recursion stack, and output allocation when the problem asks for it.

If your implementation differs from the reference, re-derive big-O from your code — never memorize a complexity you cannot defend.

Common Mistakes

Common mistakes on Design Skiplist

Skipping edge cases — empty collections, single-element inputs, max constraints.
Wrong invariant for general problem-solving — updating state too early or too late.
Mutating input unexpectedly when the problem forbids it.
Off-by-one in windows, ranges, or binary search bounds.
Ignoring overflow / precision for integer arithmetic problems.
Overengineering — jumping to an advanced structure when a simpler approach works.

Alternative Approaches

Alternatives

The source file includes more than one method. Compare:

Primary optimized path — best complexity for typical interviews.
Secondary approach — often brute force, sorting-based, or space-optimized variant.

Practice articulating when you would pick each (constraints, readability, follow-ups).

Edge Cases

Edge cases checklist

Minimum input size
Maximum input size / time limits
Duplicates and already-sorted input
Negative numbers / zeros (if applicable)
Disconnected structures (graphs/trees)
Single path vs branching recursion depth

Pattern Recognition

Spotting this pattern

Signal phrases that point to general problem-solving:

Sorted input or ability to sort without changing the answer class
Need for contiguous subarray / substring → consider sliding window
Need for O(1) membership → hash set/map
Optimal substructure + overlapping subproblems → DP
Connectivity / components → graph DFS/BFS or Union-Find

Primary topics: design.

Follow-up Interview Questions

Follow-ups

How does the solution change if the input is a stream?
Can you solve it in-place?
What if duplicates must be handled differently?
How would you parallelize the approach?
Design tests that would break a buggy implementation.

Practice Recommendations

What to practice next

Re-solve Design Skiplist in a second language (cpp, python).
Drill 3–5 more problems tagged design.
Teach the solution out loud in under 5 minutes.
Add this problem to your revision calendar in 3 days and 14 days.

Visualization

Conceptual diagram for Design Skiplist: show input structure (design), highlight the moving parts of the general problem-solving approach, and annotate each step with the maintained invariant and complexity.

Study checklist

Read the official problem statement on LeetCode
Solve on paper / whiteboard first
Implement the general problem-solving approach
Verify edge cases from the checklist
State time and space complexity aloud
Compare with the AlgoForge reference solution
Schedule a revision session

Revision notes

Design Skiplist (#1206) — Hard. Pattern: general problem-solving. Complexity: O(logn), on average time / O(n) space. Re-derive the invariant before coding.

FAQs

What is the time complexity of Design Skiplist?+

The reference solutions aim for O(logn), on average time and O(n) space. Always re-derive complexity from the code you write in the interview.

What pattern does Design Skiplist use?+

It primarily maps to general problem-solving, within the broader topic of design.

Is Design Skiplist good for interviews?+

Yes — as a Hard problem it is a solid practice target. Pair it with related problems in the same pattern family for spaced repetition.

Where can I read the official statement?+

Open the official LeetCode page for constraints and examples: https://leetcode.com/problems/design-skiplist/

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