Implementing the skip list data structure

The link list element structure used to implement a Skip List
- The link list element used to implement the skip list has 4 links (not including the data portion):

The Entry strcuture in a Skip List (the SkipListEntry class)

Skip List entry:

public class SkipListEntry { public String key; public Integer value; public SkipListEntry up; // up link public SkipListEntry down; // down link public SkipListEntry left; // left link public SkipListEntry right; // right link ... (methods) }

Note:

As you can see, my entry type is again very specific (no generic types):
When I write the demo program, I will do it using specific types (classes), not parameterized classes
I have showed you how to convert a specific class into a parameterized class, so you can write one if you want to

Reason for using specific classes:

My choice is didactic in nature; I don't want to spend time analyzing the overly complex syntax of parameterized classes
I want to spend my time teaching algorithms, not Java syntax

Making (and using) the special −∞ and +∞ elements

Representing the −∞ element and the +∞ element:

We can accommodate the −∞ element and the +∞ element by defining 2 special key value:

public class SkipListEntry { public String key; public Integer value; public SkipListEntry up, down, left, right; public static String negInf = "-oo"; // -inf key value public static String posInf = "+oo"; // +inf key value .... }

How to instantiate a Skip List entry containing +∞:

SkipListEntry x = new SkipListEntry( SkipListEntry.posInf, null );

How to check if an Skip List entry x contains +∞:

key == SkipListEntry.posInf

OK, now we move on to the Skip list itself....

Structure (class) to represent a Skip List

Remember that a Skip List is a very complicated list

But.... It is never the less a list

To represent a list, we only use a pointer (that points to the first element)
Often, we use more pointers for improve efficiency (such as a tail pointer)

Variables in the SkipList class:

Note:

The Random object r is used to determine the height of a newly added entry
(We use r to simulate a coin toss experiment)

Example illustrating how the variables are used:

Note:

Since the logical top level does not contain any entries:
The variables head and tail provide quick access to the end elements of the real top layer
Usage of head and tail:

Constructing a Skip List object

The constructor will construct an empty Skip List which looks like this:

Constructor code:

public SkipList() // Constructor... { SkipListEntry p1, p2; /* ----------------------------------- Create an -oo and an +oo object ----------------------------------- */ p1 = new SkipListEntry(SkipListEntry.negInf, null); p2 = new SkipListEntry(SkipListEntry.posInf, null); /* -------------------------------------- Link the -oo and +oo object together --------------------------------------- */ p1.right = p2; p2.left = p1; /* -------------------------------------- Initialize "head" and "tail" --------------------------------------- */ head = p1; tail = p2; /* -------------------------------------- Other initializations --------------------------------------- */ n = 0; // No entries in Skip List h = 0; // Height is 0 r = new Random(); // Make random object to simulate coin toss }

The SkipList class so far:

public class SkipList { public SkipListEntry head; // First element of the top level public SkipListEntry tail; // Last element of the top level public int n; // number of entries in the Skip List public int h; // Height public Random r; // Coin toss public SkipList() // Constructor... { SkipListEntry p1, p2; p1 = new SkipListEntry(SkipListEntry.negInf, null); p2 = new SkipListEntry(SkipListEntry.posInf, null); head = p1; tail = p2; p1.right = p2; p2.left = p1; n = 0; h = 0; r = new Random(); } ... }

Implementing the basic Map operations
- Basic Map operations:
  Notice that each basic operation must first find (search) the appropriate entry (using a key) before the operation can be completed.
  So we must learn how to search a Skip List for a given key first....

Search operation in a skip list

Consider the links traversed to locate the key 50:

Psuedo code:

p = head; repeat { Move to the right until your right neighbor node contains a key that is greater than k if ( not lowest level ) Drop down one level else exit }

Search algorithm for Skip List:

/* ------------------------------------------------------ findEntry(k): find the largest key x <= k on the LOWEST level of the Skip List ------------------------------------------------------ */ public SkipListEntry findEntry(String k) { SkipListEntry p; /* ----------------- Start at "head" ----------------- */ p = head; while ( true ) { /* ------------------------------------------------ Search RIGHT until you find a LARGER entry E.g.: k = 34 10 ---> 20 ---> 30 ---> 40 ^ | p must stop here p.right.key = 40 ------------------------------------------------ */ while ( (p.right.key) != SkipListEntry.posInf && (p.right.key).compareTo(k) <= 0 ) { p = p.right; // Move to right } /* --------------------------------- Go down one level if you can... --------------------------------- */ if ( p.down != null ) { p = p.down; // Go downwards } else { break; // We reached the LOWEST level... Exit... } } return(p); // Note: p.key <= k }

Note:

If the key k is found in the Skip List, findEntry(k) will return the reference to the entry containg the key k
If the key k is not found in the Skip List, findEntry(k) will return the reference to the floorEntry(k) entry containg a key that is smaller than k
Example: findEntry(42) will return the reference to 39:

Implementing the "get(Key k)" method

get(k):

/** Returns the value associated with a key. */ public Integer get (String k) { SkipListEntry p; p = findEntry(k); if ( k.equals( p.key ) ) return(p.value); else return(null); }

Put(k,v): inserting into a Skip List

Pseudo code for put(k,v):

put(k, v) { SkipListEntry p; p = findEntry(k); if ( k.equals( p.key ) ) // Found ! { p.value = v; // Update the value return; // Done } /* ================================================== Not found. Then: p == floorEntry(k) !!! ================================================== */ (1) insert (k,v) AFTER p (2) make a column of (k,v) of RANDOM height }

Recall what happens when we insert a new entry:

Before insertion:

After inserting key 42:

Note:

As part of the insert operation, we will make a column (see figure above) for that key
The height of the column will be random...
(We have also seen how to use a random "trial" to generate a random height)

Step-by-step depictions of the steps necessary for insertion: put("42", ??)

Before the insertion:
Step 1: find the insert position
Step 2: insert q after p:

Now make a column of random height: repeat these steps a random number of times

Starting at p, (using p to) scan left and find the first entry that has an up-entry:
Make p point to the up-element:
Create a new entry with the same key (we are making the "tower"):
Insert the newly created entry: right of p and up from q:
Make q point to the newly inserted entry (to continue the iteration if necessay)

I will repeat the steps and show the effect of building a "tower":

Starting at p, scan left and find the first entry that has an up-element:
Create a new entry (we are making another level of the "tower"):
Insert the newly created entry: right of p and up from q:
Make q point to the newly inserted entry (to continue the iteration if necessay)
(And so on)

Note:

Adding a (empty) layer

Before we can do anything, We need to what are the changes in the Skip List when we add an empty layer to the Skip List:

Add layer algorithm:

SkipListEntry p1, p2; /* ----------------------------- Make the -oo and +oo entries ---------------------------- */ p1 = new SkipListEntry(SkipListEntry.negInf, null); p2 = new SkipListEntry(SkipListEntry.posInf, null); /* -------------------- Link them -------------------- */ p1.right = p2; p1.down = head; p2.left = p1; p2.down = tail; head.up = p1; tail.up = p2; /* -------------------- Update head and tail -------------------- */ head = p1; tail = p2; h = h + 1; // One more level...

The put() method

put(k,v) psuedo code:

p = findEntry(k); // Find insert location if ( entry found ) { update the value in p; exit; } /* ---------------------------------- Insert a brand new entry (k,v) p put q here | | V V [ ] <------> [ ] ---------------------------------- */ q = new Entry(k,v); // Make new entry link q after p; /* ------------------------ Make a random tower... ------------------------ */ while ( random() < 0.5 /* coin toss */ ) { if ( height of tower >= h ) { create a new TOP layer (see: click here) } p = Find the first left element in the next level above; q = new Entry(k,v); link q after p; }

The put() method for Skip List in Java:

public Integer put (String k, Integer v) { SkipListEntry p, q; int i; p = findEntry(k); // Try find the entry /* ------------------------ Check if key is found ------------------------ */ if ( k.equals(p.key) ) // If key found, update the value and we are done... { Integer old = p.value; // Remember the old value p.value = v; // Update value return(old); // Return the old value } /* ------------------------------------------------------------- Key k is not found, then p = floorEntry(k) (See: click here) The rest of the code will insert a new entry (k,v) ------------------------------------------------------------- */ q = new SkipListEntry(k,v); // Create a new entry with k and v /* -------------------------------------------------------------- Insert q into the lowest level after SkipListEntry p: p put q here p q | | | | V V V V V Lower level: [ ] <------> [ ] ==> [ ] <--> [ ] <--> [ ] --------------------------------------------------------------- */ q.left = p; q.right = p.right; p.right.left = q; p.right = q; /* ----------------------------------------------------- Make a "tower" of the entry e or RANDOM height ----------------------------------------------------- */ i = 0; // Current level = 0 while ( r.nextDouble() < 0.5 /* Coin toss */ ) { // Coin toss success ! ---> build one more level !!! /* ------------------------------------------------------------------- Check if we need to increase the height of the -oo and +oo "pillars ------------------------------------------------------------------- */ if ( i >= h ) // We reached the top level !!! { Create a new empty TOP layer (see: click here) (Put the code from above here.... I left it out for brevity) } /* ------------------------------------ Find first element with an UP-link ------------------------------------ */ while ( p.up == null ) { p = p.left; } /* -------------------------------- Make p point to this UP element -------------------------------- */ p = p.up; /* --------------------------------------------------- Add one more (k,*) to the column Schema for making the linkage: p <--> e(k,*) <--> p.right ^ | v q ---------------------------------------------------- */ SkipListEntry e; e = new SkipListEntry(k, null); // Don't need the value... /* --------------------------------------- Initialize links of e --------------------------------------- */ e.left = p; e.right = p.right; e.down = q; /* --------------------------------------- Change the neighboring links.. --------------------------------------- */ p.right.left = e; p.right = e; q.up = e; q = e; // Set q up for next iteration (if there is one) // See here for more detail: click here i = i + 1; // Current level increases by one } n = n + 1; // One more entry in the Skip List return(null); // No old value }

Example Program: (Demo above code)

SkipListEntry.java Prog file: click here
SkipList.java Prog file: click here
Test program 1 (inserts 4 entries): click here
Test program 2 (inserts 40 entries): click here

Example output: (The keys are strings)

- - - - - - - - - - 10 13 15 15 2 21 25 31 31 31 33 33 33 33 33 33 33 33 33 33 36 38 39 39 39 39 39 41 41 41 42 42 42 42 5 5 5 54 54 57 59 59 59 59 59 59 59 60 60 63 63 65 69 7 71 71 71 71 71 72 77 77 81 82 86 88 90 92 92 99 + + + + + + + + + +

Deleting an entry from a Skip List

What you must do to the skip list to remove an entry:

Step-by-step to accomplish: remove(25)

Before the deletion:
Step 1: locate the desired element (at the lowest level of the skip list):

While p != null, repeat these steps to remove the column:

Unlink the element at p (by making the left neighbor and the right neighbor pointing to each other)
Move p upward (prepare for loop)

Result of removal:

The Removal Algorithm

Psuedo code:

p = findExntry(k); if (p.key != k) return(null); // Not found, don't remove /* ------------------------------------------------------------ We are at level 0 Travel up the tower and link the left and right neighbors ------------------------------------------------------------ */ while ( p != null ) { p.left.right = p.right; p.right.left = p.left; }

Removal in Java:
- It's gonna be a home work assignment