# Build your own Skip List

The skip list is one of my favorite data structures.

• It can be used to implement ordered lists or sets.
• It is easy to understand.
• It doesn’t require any complex re-balancing like some of the other ordered-list structures.
• It’s fast. And all of the operations - insert, search, delete - are O(log n) on average.

Here’s my ASCII-art schematic for what a skip list might look like for a sorted list of numbers from 0 through 9. Let’s dig into how we get here.

-------------------------->
|                             |
------>---------------->------>->
|         |                   |         |    |
(S)->(0)->(1)->(2)->(3)->(4)->(5)->(6)->(7)->(8)->(9)->(E)


A skip list starts as a plain old linked list. To add a value to a list you start at the head and traverse node by node until you run into the first value greater than the one you’re inserting; that’s where you stick the new value. But the difference between a linked list and a skip list becomes obvious immediately. Whenever you add a new value to the linked list, there is a probability (often 0.5 or 0.25) that you “project” a node into a faster “lane”. (Time to start looking at that figure above again.) If this is the first node in the faster lane then you also create a head node at the beginning of the list that points to this fast lane node. As soon as that node is generated, there is again a probability (the same probability as before) that another new node will be projected into an even faster lane. After that node is generated, there is a probability that yet another node will be generated - and so on.

The end result after several list insertions is something like you see in the image above. When it’s time to insert a value the top “lane” allows for lookups to quickly skip over large chunks of the list until we pass the position of the value. At this point we can back up to the last node that preceded the value, jump down to a slower lane and search again for the point at which we just pass the position of the value. We repeat this until we are in the slowest lane - the bottom linked list - and then we can insert the value like we normally would as a linked list.

## So let’s build one!

I’ve always wanted to build my own skip list from scratch and I finally had a chance to take a stab at it today. The code below is not completely general (I’m building this so that I can use it in a data sketch that I blog about here) but it does showcase how the data structure works.

import random

class Node():
def __init__(self, value = None):
self.left = None
self.right = None
self.down = None
self.up = None
self.value = value

def __repr__(self):
return '({})'.format(repr(self.value))

def __lt__(self, value):
if self.down:
return self.down < value
else:
return self.value < value

class StartNode(Node):
def __lt__(self, value):
return True

def __repr__(self):
return '(S)'

class EndNode(Node):
def __lt__(self, value):
return False

def __repr__(self):
return '(E)'

class SkipList():
def __init__(self, up_prob=0.5, verbose=False):
"""Skip list implementation

up_prob = probability that we extend upwards
"""
self.up_prob = up_prob
self.verbose = verbose

end = EndNode()

self.index = 1

if self.verbose:
messages = []
# find node to the left of value
skip_nodes = []  # the list of skip nodes above the final node
while True:
if node.right and node.right < value:
if self.verbose:
messages.append('\ttraverse right')
node = node.right
elif node.down:
if self.verbose:
messages.append('\ttraverse down')
skip_nodes.append(node)
node = node.down
else:
if self.verbose:
messages.append(
'\tadding {} after {}'.format(
value, node
)
)
break

# insert new node
new_node = Node(value)

right = node.right
new_node.right = right
if right:
right.left = new_node

left = node
new_node.left = left
left.right = new_node

# project upward
low_node = new_node
while True:
if random.random() > self.up_prob:
# extend upwards
projection_node = Node('s' + str(self.index))
self.index += 1
low_node.up = projection_node
projection_node.down = low_node

if skip_nodes:
left_node = skip_nodes.pop()
right_node = left_node.right
if self.verbose:
messages.append(
'\tplacing projection_node {} between left_node {} and right_node {}'.format(
projection_node, left_node, right_node or 'NONE'
)
)
left_node.right = projection_node
projection_node.left = left_node
projection_node.right = right_node
if right_node:
right_node.left = projection_node
else:
# we are projecting higher than the head node, so make a new head and a new lane
new_head = Node('s' + str(self.index))
self.index += 1
if self.verbose:
messages.append(
'\tpointing new_head {} to new projection_node {}'.format(
)
)
break
low_node = projection_node
else:
break

if self.verbose:
messages.append('')
print('\n'.join(messages))

return new_node

def pop(self, node=None):
if not node:
while node.down:
node = node.down
node = node.right
original_node = node

if self.verbose:
messages = []
messages.append('popping node {}'.format(node))
while True:
left = node.left
right = node.right
if self.verbose:
messages.append(
'\tconnecting {} to {}'.format(
left, right
)
)

if left:
left.right = right
if right:
right.left = left

node.left = None
node.right = None
node.down = None
if node.up:
old_node = node
node = node.up
old_node.up = None
else:
break

while True:
if left is self.head and left.right is None:
if self.verbose:
messages.append(
'\tremoving head {} so that {} is the new head'.format(
)
)
left = self.head = left.down
else:
break

if self.verbose:
print('\n'.join(messages))

return original_node

def __iter__(self):
while node.down:
node = node.down

node = node.right  # skip start node
while node.right:
yield node
node = node.right


I encourage you to copy paste the code into a jupyter notebook and play around with it. You’ll notice that I’ve added a lot of logging if you turn the verbose parameter on. Let’s give it a spin:

def skip_list_insert(n):
skip_list = SkipList(verbose=True)
for i in random.sample(range(n), n):
return skip_list

skip_list = skip_list_insert(10)


The code above creates a randomly sorted list of numbers from 0 to 9 and inserts them one at a time into an initially empty skip list. The verbose commentary is shown below. Note in the commentary that the skip nodes are labeled as things like ('s4') while the nodes that hold values are labled as things like (3). The skip list starts out as a simple linked list that holds two items, the start node (S) and the end node (E). The commentary for each value insertion follows this pattern:

• the value we are adding
• how we traverse the skip list to get to the node just before it
• adding in the value after that node
• optionally adding in one or more projection nodes into faster lanes
adding 9
adding 9 after (S)

adding 1 after (S)
pointing new_head ('s2') to new projection_node ('s1')

traverse right
traverse down
adding 4 after (1)

traverse right
traverse down
traverse right
adding 7 after (4)
placing projection_node ('s3') between left_node ('s1') and right_node NONE

traverse right
traverse down
traverse right
adding 6 after (4)

traverse right
traverse right
traverse down
adding 8 after (7)
placing projection_node ('s4') between left_node ('s3') and right_node NONE

traverse right
traverse down
adding 2 after (1)

traverse down
adding 0 after (S)

traverse right
traverse down
traverse right
adding 3 after (2)

traverse right
traverse down
traverse right
traverse right
traverse right
adding 5 after (4)
placing projection_node ('s5') between left_node ('s1') and right_node ('s3')
pointing new_head ('s7') to new projection_node ('s6')


The end result looks like this:

-------------------------->
|                             |
------>---------------->------>->
|         |                   |         |    |
(S)->(0)->(1)->(2)->(3)->(4)->(5)->(6)->(7)->(8)->(9)->(E)


To make sure that the results are as expected let’s print them out

>>> nodes = list(skip_list)
>>> print(nodes)
[(0), (1), (2), (3), (4), (5), (6), (7), (8), (9)]


Perfect!

My version of the skip list also allows for deletion by node. Note that this is different than typical skip list implementation where you delete by value. When deleting by value you have to search for the node that holds that value, pull out that node, and then splice together the left and right pieces of the skip list. When deleting by node we skip the search step. But here’s an example result:

>>> skip_list.pop(nodes)
popping node (5)
connecting (4) to (6)
connecting ('s1') to ('s3')
connecting ('s7') to None
removing head ('s7') so that ('s2') is the new head


Pretty self explanatory I think.

I’ve also added in functionality so that if you call pop with out a node argument it will just pop off the first item in the list. In this way a skip list can be used as a priority queue.

## Performance

OK, so I build it… but does it really perform? Well check this out. In the plots below I’m comparing the add time of the skip list with that of a normal python list (using bisect.insort_left). The x-axis represents the number of elements in the list.

In this plot we can see that list performance scales linearly (just as the Python docs said it would) and we see that the skip list performs much much better.

If we zoom in on the skip list performance…

…we see that, maybe ignoring some noise, the performance does appear to scale as O(log n). Great! So there it is folks… there’s a skip list and it’s awesome.

Well… just for the sake of full disclosure, let’s look at one more plot:

This is the exact same data as the first performance plot, but this time plotted in loglog. This tells just a little different story doesn’t it? You can see that for really large numbers the performance of the vanilla python list explodes. But how often do you think you’ll find yourself in a tight loop inserting items into a list that contains 80M items? What this plot tells us is that if your list has less than 50K elements you might as well just use the Python list implementation. As a matter of fact, if your list contains about 3000 elements, the skip list implementation is going to be around 20x slower!

## Conclusion

Despite the poor performance of my skip list for low cardinality lists, I still think this data structure is pretty amazing. I’m sure that I (or someone much smarter than me) could reimplement the skip list in c and get much better performance. I’m confident of this because otherwise the skip list wouldn’t have found such favor among data store builders. The skip list features prominently in many data stores that I’m sure you’ve heard of: Cassandra, Lucene (e.g. Solr and Elasticsearch), Redis, HBase, and leveldb just to name a few.