4.3: SkiplistList - An Efficient Random-Access List

A SkiplistList implements the List interface using a skiplist structure. In a SkiplistList, \(L_0\) contains the elements of the list in the order in which they appear in the list. As in a SkiplistSSet, elements can be added, removed, and accessed in \(O(\log \mathtt{n})\) time.

For this to be possible, we need a way to follow the search path for the \(\mathtt{i}\)th element in \(L_0\). The easiest way to do this is to define the notion of the length of an edge in some list, \(L_{\mathtt{r}}\). We define the length of every edge in \(L_{0}\) as 1. The length of an edge, \(\mathtt{e}\), in \(L_{\mathtt{r}}\), \(\mathtt{r}>0\), is defined as the sum of the lengths of the edges below \(\mathtt{e}\) in \(L_{\mathtt{r}-1}\). Equivalently, the length of \(\mathtt{e}\) is the number of edges in \(L_0\) below \(\mathtt{e}\). See Figure \(\PageIndex{1}\) for an example of a skiplist with the lengths of its edges shown. Since the edges of skiplists are stored in arrays, the lengths can be stored the same way:

Figure \(\PageIndex{1}\): The lengths of the edges in a skiplist.

    class Node {
        T x;
        Node[] next;
        int[] length;
        @SuppressWarnings("unchecked")
        Node(T ix, int h) {
            x = ix;
            next = (Node[])Array.newInstance(Node.class, h+1);
            length = new int[h+1];
        }
        int height() {
            return next.length - 1;
        }
    }

The useful property of this definition of length is that, if we are currently at a node that is at position \(\mathtt{j}\) in \(L_0\) and we follow an edge of length \(\ell\), then we move to a node whose position, in \(L_0\), is \(\mathtt{j}+\ell\). In this way, while following a search path, we can keep track of the position, \(\mathtt{j}\), of the current node in \(L_0\). When at a node, \(\mathtt{u}\), in \(L_{\mathtt{r}}\), we go right if \(\mathtt{j}\) plus the length of the edge \(\texttt{u.next[r]}\) is less than \(\mathtt{i}\). Otherwise, we go down into \(L_{\mathtt{r}-1}\).

    Node findPred(int i) {
        Node u = sentinel;
        int r = h;
        int j = -1;   // index of the current node in list 0
        while (r >= 0) {
            while (u.next[r] != null && j + u.length[r] < i) {
                j += u.length[r];
                u = u.next[r];
            }
            r--;
        }
        return u;
    }

    T get(int i) {
        if (i < 0 || i > n-1) throw new IndexOutOfBoundsException();
        return findPred(i).next[0].x;
    }
    T set(int i, T x) {
        if (i < 0 || i > n-1) throw new IndexOutOfBoundsException();
        Node u = findPred(i).next[0];
        T y = u.x;
        u.x = x;
        return y;
    }

Since the hardest part of the operations \(\mathtt{get(i)}\) and \(\mathtt{set(i,x)}\) is finding the \(\mathtt{i}\)th node in \(L_0\), these operations run in \(O(\log \mathtt{n})\) time.

Adding an element to a SkiplistList at a position, \(\mathtt{i}\), is fairly simple. Unlike in a SkiplistSSet, we are sure that a new node will actually be added, so we can do the addition at the same time as we search for the new node's location. We first pick the height, \(\mathtt{k}\), of the newly inserted node, \(\mathtt{w}\), and then follow the search path for \(\mathtt{i}\). Any time the search path moves down from \(L_{\mathtt{r}}\) with \(\mathtt{r}\le \mathtt{k}\), we splice \(\mathtt{w}\) into \(L_{\mathtt{r}}\). The only extra care needed is to ensure that the lengths of edges are updated properly. See Figure \(\PageIndex{2}\).

Figure \(\PageIndex{2}\): Adding an element to a SkiplistList.

Note that, each time the search path goes down at a node, \(\mathtt{u}\), in \(L_{\mathtt{r}}\), the length of the edge \(\texttt{u.next[r]}\) increases by one, since we are adding an element below that edge at position \(\mathtt{i}\). Splicing the node \(\mathtt{w}\) between two nodes, \(\mathtt{u}\) and \(\mathtt{z}\), works as shown in Figure \(\PageIndex{3}\). While following the search path we are already keeping track of the position, \(\mathtt{j}\), of \(\mathtt{u}\) in \(L_0\). Therefore, we know that the length of the edge from \(\mathtt{u}\) to \(\mathtt{w}\) is \(\mathtt{i}-\mathtt{j}\). We can also deduce the length of the edge from \(\mathtt{w}\) to \(\mathtt{z}\) from the length, \(\ell\), of the edge from \(\mathtt{u}\) to \(\mathtt{z}\). Therefore, we can splice in \(\mathtt{w}\) and update the lengths of the edges in constant time.

Figure \(\PageIndex{3}\): Updating the lengths of edges while splicing a node \(\mathtt{w}\) into a skiplist.

This sounds more complicated than it is, for the code is actually quite simple:

    void add(int i, T x) {
        if (i < 0 || i > n) throw new IndexOutOfBoundsException();
        Node w = new Node(x, pickHeight());
        if (w.height() > h) 
            h = w.height();
        add(i, w);
    }

    Node add(int i, Node w) {
        Node u = sentinel;
        int k = w.height();
        int r = h;
        int j = -1; // index of u
        while (r >= 0) {
            while (u.next[r] != null && j+u.length[r] < i) {
                j += u.length[r];
                u = u.next[r];
            }
            u.length[r]++; // accounts for new node in list 0
            if (r <= k) {
                w.next[r] = u.next[r];
                u.next[r] = w;
                w.length[r] = u.length[r] - (i - j);
                u.length[r] = i - j;
            }
            r--;
        }
        n++;
        return u;
    }

By now, the implementation of the \(\mathtt{remove(i)}\) operation in a SkiplistList should be obvious. We follow the search path for the node at position \(\mathtt{i}\). Each time the search path takes a step down from a node, \(\mathtt{u}\), at level \(\mathtt{r}\) we decrement the length of the edge leaving \(\mathtt{u}\) at that level. We also check if \(\texttt{u.next[r]}\) is the element of rank \(\mathtt{i}\) and, if so, splice it out of the list at that level. An example is shown in Figure \(\PageIndex{4}\).

Figure \(\PageIndex{4}\): Removing an element from a SkiplistList.

    T remove(int i) {
        if (i < 0 || i > n-1) throw new IndexOutOfBoundsException();
        T x = null;
        Node u = sentinel;
        int r = h;
        int j = -1; // index of node u
        while (r >= 0) {
            while (u.next[r] != null && j+u.length[r] < i) {
                j += u.length[r];
                u = u.next[r];
            }
            u.length[r]--;  // for the node we are removing
            if (j + u.length[r] + 1 == i && u.next[r] != null) {
                x = u.next[r].x;
                u.length[r] += u.next[r].length[r];
                u.next[r] = u.next[r].next[r];
                if (u == sentinel && u.next[r] == null)
                    h--;
            }
            r--;
        }
        n--;
        return x;
    }

\(\PageIndex{1}\) Summary

The following theorem summarizes the performance of the SkiplistList data structure:

Theorem \(\PageIndex{1}\).

A SkiplistList implements the List interface. A SkiplistList supports the operations \(\mathtt{get(i)}\), \(\mathtt{set(i,x)}\), \(\mathtt{add(i,x)}\), and \(\mathtt{remove(i)}\) in \(O(\log \mathtt{n})\) expected time per operation.