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Tuesday, August 14, 2012

Skip Lists







Skip Lists

Thomas A. Anastasio
22 November 1999

http://www.csee.umbc.edu/courses/undergraduate/341/fall01/Lectures/SkipLists/skip_lists/skip_lists.html

Introduction

Skip Lists were developed around 1989 by William Pugh1 of the University of Maryland. Professor Pugh sees Skip Lists as a viable alternative to balanced trees such as AVL trees or to self-adjusting trees such as splay trees.
The find, insert, and remove operations on ordinary binary search trees are efficient, $O(\lg n)$, when the input data is random; but less efficient, $O(n)$, when the input data are ordered. Skip List performance for these same operations and for any data set is about as good as that of randomly-built binary search trees - namely $O(\lg n)$
In an ordinary sorted linked list, find, insert, and remove are in $O(n)$ because the list must be scanned node-by-node from the head to find the relevant node. If somehow we could scan down the list in bigger steps (``skip'' down, as it were), we would reduce the cost of scanning. This is the fundamental idea behind Skip Lists.
In simple terms, Skip Lists are sorted linked lists with two differences:
  1. the nodes in an ordinary list have one 'next' reference. The nodes in a Skip List have many 'next' references (called forward references).
  2. the number of forward references for a given node is determined probabilistically.
We speak of a Skip List node having levels, one level per forward reference. The number of levels in a node is called the size of the node.
In an ordinary sorted list, insert, remove, and find operations require sequential traversal of the list. This results in $O(n)$ performance per operation. Skip Lists allow intermediate nodes in the list to be ``skipped'' during a traversal - resulting in an expected performance of $O(\lg n)$ per operation.

The Basic Idea

To introduce the Skip List data structure, let's look at three list data structures that allow skipping, but are not truly Skip Lists. The first of these, shown in Figure 1 allows every other node to be skipped in a traversal. The second, shown in Figure 2 additionally allows every fourth node to be skipped. The third, shown in Figure 3 additionally allows every eighth node to be skipped, but suggests further development of the idea to skipping every $2^i$-th node.
For all three pseudo-skip-lists, there is a header node and the nodes do not all have the same number of forward references. Every node has a reference to the next node, but some have additional references to nodes further along the list.
Here is pseudo-code for the find operation on each list. This same algorithm will be used with real Skip Lists later. 

 
   Comparable & find(const Comparable & X)
    {
     node = header node
     for (reference level of node from (nodesize - 1) down to 0)
       while (the node referred to is less than X)
           node = node referred to
     if (node referred to has value X)
       return node's value
     else
      return item_not_found
    }
We start at the highest level of the header node and follow the references along this level until the value at the node referred to is equal to or larger than X. At this point, we switch to the next lower level and continue the search. Eventually, we shall be dealing with a reference at the lowest level of a node. If the next node has the value X, then we return that value; otherwise we return a value that signals unsuccessful search.

Skip Every Second Node

Figure 1 shows a 16-node list in which every second node has a reference two nodes ahead. The value stored at each node is shown below it (and corresponds in this example to the position of the node in the list). The header node has two levels; it's no smaller than largest node in the list. Node 2 has a reference to node 4, two nodes ahead. Similarly for nodes 4, 6, 8, etc - every second node has a reference two nodes ahead.

Figure 1: Skip Every 2nd Node
\begin{figure}\centering\leavevmode {\framebox [\textwidth] {\psfig{figure=Figures/skip_lists_intro_2node.eps,width=0.9\textwidth}}} \end{figure}
It's clear that the find operation does not need to visit each node. It can skip over every other node, then do a final visit at the end. The number of nodes visited is therefore no more than $\lceil\frac{n}{2}\rceil + 1$. For example, the nodes visited in scanning for the node with value  15 would be 2, 4, 6, 8, 10, 12, 14, 16, 15, a total of $(\lceil\frac{16}{2}\rceil + 1) = 9$.
Follow the algorithm for find on page [*] for this simple example to be sure you understand it thoroughly.

Skip Every Second and Fourth Node

The second example is a list in which every second node has a reference two nodes ahead and additionally every fourth node has a reference four nodes ahead. Such a list is shown in Figure 2. The header is still no smaller than the largest node in the list.

Figure 2: Skip Every 2nd and 4th Node
\begin{figure}\centering\leavevmode {\framebox [\textwidth] {\psfig{figure=Figures/skip_lists_intro_4node.eps,width=0.9\textwidth}}} \end{figure}
The find operation can now make bigger skips than those for the list in Figure 1. Every fourth node is skipped until the search is confined between two nodes of size 3. At this point, as many as three nodes may need to be scanned. It is also possible that some nodes will be visited more than once using the algorithm on page [*]. The number of nodes visited2 is no more than $\lceil\frac{n}{4}\rceil + 3$. As an example, look at the nodes visited in scanning for the node with value of 15. These are 4, 8, 12, 16, 14, 16, 15 for a total of $(\lceil\frac{16}{4}\rceil + 3) = 7$.

Skip Every $2^i$-th Node

This final example is for a list in which some skip lengths are even larger.

Figure 3: Skip Every $2^i$-th Node
\begin{figure}\centering\leavevmode {\framebox [\textwidth] {\psfig{figure=Figures/skip_lists_intro.eps,width=0.9\textwidth}}} \end{figure}
Every $2^i$-th node, $i = 1, \ldots \lceil \lg n \rceil$, has a forward reference $2^i$ nodes ahead. For example, every $2^{\rm nd}$ node has a reference 2 nodes ahead; every $8^{\rm th}$ node has a reference 8 nodes ahead, etc. Figure 3 shows a short list of this type. Once again, the header is no smaller than the largest node on the list. It is shown arbitrarily large in the Figure.
Suppose the Skip List in Figure 3 contained 32 nodes and consider a search in it. Working down from the highest level, we first encounter node 16 and have cut the search in half. We then search again, one level down in either the left or right half of the list, again cutting the remaining search in half. We continue in this manner till we find the sought-after node (or not). This is quite reminiscent of binary search in an array and is perhaps the best way to intuitively understand why the maximum number of nodes visited in this list is in $O(\lg n)$.
This data structure is looking pretty good, but there's a serious problem with it for the insert and remove operations. The work required to reorganize the list after an insertion or deletion is in $O(n)$. For example, suppose that the first element is removed in Figure 3. Since it is necessary to maintain the strict pattern of node sizes, values from 2 to the end must be moved toward the head and the end node must be removed. A similar situation occurs when a new element is added to the list.
This is where the probabilistic approach of a true Skip List comes into play. A Skip List is built with the same distribution of node sizes, but without the requirement for the rigid pattern of node sizes shown. It is no longer necessary to maintain the rigid pattern by moving values around after a remove or insert operation. Pugh shows that with high probability such a list still exhibits $O(\lg n)$ behavior. The probability that a given Skip List will perform badly is very small.

Skip Lists - the Probabilistic Approach

Figure 4 shows the list of Figure 3 with the nodes reorganized. The distribution of node sizes is exactly the same as that of Figure 3; the nodes just occur in a different pattern. In this example, the pattern would require that 50% of the nodes have just one reference, 25% have just two references, 12.5% have just three references, etc. The distribution of node sizes is maintained, but the strict order is not required.

Figure 4: A Probabilistic Skip List
\begin{figure}\centering\leavevmode {\framebox [\textwidth] {\psfig{figure=Figures/skip_example_1.eps,width=0.9\textwidth}}} \end{figure}
Figure 4 shows one way this might work out. Of course, this is probabilistic, so there are many other possible node sequences that would satisfy the required probability distribution.
When inserting new nodes, we choose the size of the new node probabilistically. Every Skip List has an associated (and fixed) probability $p$ that determines the distribution of nodes. A fraction $p$ of the nodes that have at least $r$ references also have $r + 1$ references. The Skip List does not have to be reorganized when a new element is inserted.


An Aside on Node Distribution

Suppose we have an infinitely-long Skip List with associated probability $p$. This means that a fraction, $p$, of the nodes with a forward reference at level $i$ also have a forward reference at level $i + 1$.
Let $L_k$ be the fraction of nodes having precisely $k$ forward references ( i.e., $L_k$ is the fraction of nodes of size $k$). Then, $L_k = p L_{k-1}$ and
\begin{displaymath}L_1 = 1 - \sum_{i=2}^\infty L_i \end{displaymath}

Since
\begin{displaymath}\sum_{i=2}^\infty L_i =L_2 + L_3 + \cdots =L_2 + pL_2 + p^2L_2 +\cdots \end{displaymath}

we can write $L_1$ in terms of $L_2$ as
\begin{displaymath}L_1 = 1 - L_2 \sum_{i=0}^\infty p^i\end{displaymath}

Recall that $\sum_{i=0}^\infty p^i$ is the sum of the geometric progression with first term $1$ and common ratio $p$. Thus,
\begin{displaymath}\sum_{i=0}^\infty p^i = \frac{1}{1 - p} \end{displaymath}

Therefore, $L_1 = 1 - \frac{L_2}{1-p}$. Since $L_2 = pL_1$, we can write
\begin{displaymath}L_1 = 1 - \frac{pL_1}{1 - p} \Rightarrow L_1 = 1 - p \end{displaymath}

Example: In the situation shown in Figure 4, $p = 0.5$. Therefore, $1 - \frac{1}{2} = \frac{1}{2}$ of the nodes with at least one reference have two references; one-half of those with at least two references have three references, etc.
You should work out the distribution for a SkipList with associated probability of $\frac{1}{4}$ to be sure you understand how distributions are computed.

Choosing a Node Probabilistically

When a new node is needed for a insert operation, the size of the node is chosen by consulting a random number generator r. Here is one way to do this for a Skip List with associated probability p and maximum node level maxLevel 
   int generateNodeLevel(double p, int maxLevel)
   {
    int level = 1;
   
    while (drand48() < p)
      level++;
    return (level > maxLevel) ? maxLevel : level;
   }
Note that the level of the new node is independent of the number of nodes already in the Skip List. Each node is chosen only on the basis of the Skip List's associated probability.
When the associated probability is $p$, the average number of comparisons that must be done for find, is $1 + \frac{\log_\frac{1}{p} n}{p} + \frac{1}{1 - p}$. For example, for a list of size 65,536, the average number of nodes to be examined is 34.3 for $p = \frac{1}{4}$ and 35 for $p = \frac{1}{2}$. This is a tremendous improvement over an ordinary sorted list for which the average number of comparisons is $\frac{n}{2} = 32768$.

Determining the Header's Level

The level of the header node is the maximum allowed level in the SkipList and is chosen at construction. Pugh shows that the maximum level should be chosen as $\log_\frac{1}{p} n$. Thus, for $p = \frac{1}{2}$, the maximum level for a SkipList of up to 65,536 elements should be chosen no smaller than $\lg_2 65536 = 16$.

Performance Considerations

The expected time to find an element (and therefore to insert or delete an element) is in $O(\lg n)$. It is possible for the time to be substantially larger if the configuration of node levels is unfavorable for a particular operation. Since the node sizes are generated randomly, it is possible to get a ``bad'' run of sizes. For example, it is possible that each node will be generated at the same size, producing the equivalent of an ordinary sorted list. A bad run of sizes will result in longer-than expected search (and therefore insert or remove) times; the SkipList will simply not be as efficient as expected. Obviously, a bad run will be proportionataly less important in a long SkipList than in a short one. The probability of poor performance decreases rapidly with the number of nodes (i.e., list size). The probability that an operation will take longer than expected is a function of the probability associated with the list. For example, Pugh calculates that for a list with $p = 0.5$ and 4096 elements, the probability that the actual time will exceed the expected time by a factor of 3 is less than one in 200 million.
The relative time and space performance of a Skip List depends on the probability level associated with the list. Pugh suggests that a probability of 0.25 be used in most cases. If the variability of performance is important, he suggests using a probability of 0.5 (variability decreases with increasing probability). Interestingly, the average number of references per node is only 1.33 when a probability of 0.25 is used. A binary search tree, of course, has 2 references per node, so Skip Lists can be more space-efficient.

Implementation of Skip Lists

The implementation embeds the SkipListNode class as a private member of SkipList. 


template 
class SkipList
{
private:
  // embedded node
  class SkipListNode
  {
  public:
    void setDatum(const Comparable & datum);
    void setForward(int i, SkipListNode * f);
    void setSize(int sz);
    SkipListNode(); // 16 levels, Comparable() for datum
    SkipListNode(const Comparable & datum, int levels);
    SkipListNode(const SkipListNode &);
    ~SkipListNode();

    const Comparable & getDatum() const;
    int getSize() const;
    SkipListNode * getForward(int level);
  private:
    int _levels;  // number of levels in this node
    vector _forward;  // forward pointers
    Comparable _datum;  // datum stored at this node
  }; // SkipListNode
continued below

SkipList implementation continued 
public:
  SkipList(); // use max_node_size of 16, prob of 0.25
  SkipList(int max_node_size, double probab);
  SkipList(const SkipList &);
  ~SkipList();

  int getHighNodeSize() const; // size of the node with most levels
  int getMaxNodeSize() const; // max node size allowed in this list
  double getProbability() const; // prob for this list

  // insert item into this skiplist.
  void insert(const Comparable & item);

  // Search for item in skiplist.  If found, return the item with
  // success equal true.  Otherwise, return a dummy item with
  // success equal false.
  const Comparable & find(const Comparable & item, bool & success);

  // remove the first occurrence of the specified item
  void remove(const Comparable & item);  
private:
  // Search for item in this skiplist, starting at startnode
  // startnode will usually be the header.  If item is found, 
  // return the node at which it is found with success equal true.
  // Otherwise, return the preceding node with success equal false.
  // Precondition:  startnode is not NULL
  SkipListNode * find(const Comparable & item, 
        SkipListNode * startnode);

  // Accessor for _head
  SkipListNode * getHeader() const; // return the header node
continued below

SkipList implementation continued 
  // Finds the point at which item would be inserted using a node
  // of the given size.
  // Returns a "lookback" node that has each of its forward 
  // pointers set to the closest node (toward the header) at that level.
  SkipListNode * findInsertPoint(const Comparable & item, 
     int nodesize);

  // insert item into this skiplist, using a node of the 
  // specified size. success returned true if insertion succeeds
  void insert(const Comparable & item, int nodesize, bool & success);

  int _high_node_size;  // highest-level node now in this list
  int _max_node_size;  // highest-level node allowed in this list
  double _prob;  // probability associated with this list
  SkipListNode * _head;  // header node of this list
};  // SkipList

Overview of SkipList Methods

We will examine SkipList methods insert, and remove in some detail below. Pseudocode for find was given on page [*].
Insertion and deletion involve searching the SkipList to find the insertion or deletion point, then manipulating the references to make the relevant change.
When inserting a new element, we first generate a node that has had its level selected randomly. The SkipList has a maximum allowed level set at construction time. The number of levels in the header node is the maximum allowed. For convenience in searching, the SkipList keeps track of the maximum level actually in the list. There is no need to search levels above this actual maximum.

findInsertPoint Method

In an ordinary linked list, insertion and deletion require having a pointer to the previous node. Insertion is done after this previous node, deletion deletes the node following the previous node. We call the point in the list after the previous node, the insertion point, even if we are doing deletions. Finding the insertion point is the first step in doing an insertion or a deletion. In Skip Lists, we need pointers to all the see-able previous nodes between the insertion point and the header. Imagine standing at the insertion point, looking back toward the header. All the nodes you can see are the see-able nodes. Some nodes may not be see-able because they are blocked by higher nodes. Figure 5 shows an example.

Figure 5: Update Node Example
\begin{figure}\centering\leavevmode {\framebox [\textwidth] {\psfig{figure=Figures/skip_update.eps,width=0.9\textwidth}}} \end{figure}



In the figure, the insertion point is between nodes 6 and 7. Looking back toward the header, the nodes you can see at the various levels are 
  level  node seen
    0      6
    1      6
    2      4
    3      header


We construct a backLook node that has its forward pointers set to the relevant see-able nodes. This is the type of node returned by the findInsertPoint method.

insert Method

Once we have the backLook node returned by findInsertPoint and have constructed the new node to be inserted, doing an insertion is easy. Here's how. The public insert(const Comparable &) method decides on the new node's size by random choice, then calls the overloading private method
insert(const Comparable &,int,bool &) to do all the work. 



template
void 
SkipList::insert(const T & item)
{
  int nodesize = randomNodeSize(getProbability(), getMaxNodeSize());
  insert(item, nodesize);
}



This is the private insert method. 


template
void 
SkipList::insert(const T & item, int nodesize)
{
  SkipList::SkipListNode * backLook = findInsertPoint(item, nodesize);
  SkipList::SkipListNode * newnode = 
    new SkipList::SkipListNode(item, nodesize);

  // newnode may have more levels than are currently in list
  // If so, adjust high_node_size of list.
  if (nodesize > getHighNodeSize())
    setHighNodeSize(nodesize);

  //  Rearrange the relevant previous pointers using information
  //  in the backLook node.
  for (int i = 0; i < nodesize; i++)
    {
      SkipList::SkipListNode * prev = backLook->getForward(i);
      SkipList::SkipListNode * next = prev->getForward(i);
      newnode->setForward(i, next);
      prev->setForward(i, newnode);
    }

  incrementListSize();
}

remove Method

The remove method is entirely analogous to the insert method. It uses the backLook node returned by findInsertPoint to reassign the previous nodes' pointers around the node to be removed.

Footnotes

... Pugh1
Communications of the ACM, June 1990, pp 668-676
... visited2
This is not the same value given by Pugh who may not have counted multiple visits to a node. Pugh's value is $\lceil\frac{n}{4}\rceil + 2$. This difference in the additive constant does not change the conclusion of $O(\lg n)$ performance for Skip Lists.

Thomas Anastasio
2000-02-19
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