Understanding Skip Lists

In the world of computer science and data management, there are numerous ways to organize and access data efficiently. One such innovative data structure is the skip list. This blog aims to provide a comprehensive yet engaging primer on skip lists, covering their structure, operations, and analysis. By the end of this read, you'll have a solid understanding of skip lists and their significance in computer science.

What is a Skip List?

A skip list is an advanced data structure that allows fast search, insertion, and deletion operations. It is essentially a hierarchy of linked lists where each level is a subset of the level below it. This hierarchical organization enables higher-level linked lists to "skip" many elements, making search operations more efficient.

Structure of a Skip List

Levels and Sentinels: Each skip list contains multiple levels, starting from the bottom level (L0) that includes all elements. Each higher level contains a subset of elements from the level below. Notably, each level contains two sentinel nodes: $-\infty $ and $+\infty $, which represent the negative and positive infinity bounds.
Nodes and Towers: Each non-sentinel element in a skip list is a node. Nodes can be part of multiple levels, forming a structure known as a tower. Every node points to the node below it and the node after it.

Operations on a Skip List

Skip lists support three primary operations: search, insertion, and deletion. Let's delve into each of these operations with a step-by-step breakdown.

1. Search

The search operation in a skip list begins at the topmost level and progresses downwards until the desired element is found or confirmed absent.

Algorithm:

skipList::search(int k) {
    // Get the list of predecessors of key k
    std::stack<Node*> P = getPredecessors(k);

    // Get the predecessor of k in the bottom-level list (L0)
    Node* p0 = P.top();

    // Check if the key after p0 is k
    if (p0->after->key == k) {
        // If found, return the key-value pair at p0->after
        return p0->after->kvp;
    } else {
        // If not found, return "not found"
        return "not found";
    }
}

Step 1: Start at the highest level and move right until the key is found or the next key is larger than the target.
Step 2: Move down one level and repeat the process.
Step 3: Continue until reaching the bottom level (L0). If the key is found, return it; otherwise, indicate its absence.

Get Predecessors:

Get predecessors takes in a node with key k, and returns a stack containing every predecessor of the given node on each list of the skip list.

std::stack<Node*> skipList::getPredecessors(int k) {
    // Start at the root node
    Node* p = root;

    // Initialize a stack of nodes, initially containing p
    std::stack<Node*> P;
    P.push(p);

    // Traverse downwards until reaching the bottom level
    while (p->below != NULL) {
        // Move to the node below
        p = p->below;
        // Traverse rightwards until the key after p is not less than k
        while (p->after->key < k) {
            p = p->after;
        }
        // Push the current node onto the stack
        P.push(p);
    }

    // Return the stack of predecessors
    return P;
}

2. Insertion

Inserting an element involves determining a random height for the node and then placing it appropriately across different levels.

Algorithm:

void skipList::insert(int k, int v) {
    // Determine the random tower height
    int i;
    for (i = 0; random(2) == 1; i++);

    // Ensure the skip list has enough levels
    int h = 0;
    Node* p = root.below;
    while (p != NULL) {
        // Check if we need to increase the skip list height
        while (i >= h) {
            // Step 4: Create a new sentinel-only list and link it in below the topmost list
            createNewSentinelList();
            h++;
        }
        // Move to the node below
        p = p->below;
        h++;
    }

    // Get the list of predecessors for key k
    std::stack<Node*> P = getPredecessors(k);

    // Insert (k, v) in the bottom-level list (L0)
    Node* p = P.top();
    P.pop();
    Node* zbelow = new Node(k, v);
    zbelow->after = p->after;
    p->after = zbelow;

    // Insert k in levels L1 to Li
    while (i > 0) {
        p = P.top();
        P.pop();
        Node* z = new Node(k);
        z->after = p->after;
        p->after = z;
        z->below = zbelow;
        zbelow = z;
        i--;
    }
}

Step 1: Determine a random height for the new node.
Step 2: If necessary, create new levels to accommodate the height of the new node.
Step 3: Identify the predecessors for the new node across all relevant levels.
Step 4: Insert the new node at each level, linking it appropriately to maintain the skip list structure.

3. Deletion

Deletion involves finding all instances of the node across different levels and removing them.

Algorithm:

skipList::delete(k)
1. P ← get-predecessors(k)
2. while P is non-empty
3. p ← P.pop() // predecessor of k in some list
4. if p.after.key = k
5. p.after ← p.after.after
6. else break // no more copies of k

Step 1: Identify all predecessors of the node to be deleted.
Step 2: Remove the node from each level where it appears.

Analysis of Skip Lists

Since a skiplist is randomly generated from the insert method, we analyze the expected space and height from the skip list (the worst case is that insert doesn't terminate due to creating a node with infinite height).

Expected Space Usage: $O (n)$ , where $n$ is the number of elements in the skip list. This space complexity ensures that skip lists are space-efficient.
Expected Height: $O (\log n)$ . The height of a skip list grows logarithmically with the number of elements due to the probabilistic nature of node heights.
Time Complexity: All operations (search, insert, delete) have an expected time complexity of $O (\log n)$ . This is because each level of the skip list contains fewer nodes, making it faster to traverse the list.