By Pavle Jonoski

# Persistent Data Structures

Persistent data structures are data structures that preserve the previous versions of the structure on modification.

They are effectively immutable, and depending on the implementation offer a special kind of memory optimization.

Being immutable, the persistent data structures are useful when writing concurrent code and the need arises to use some kind of data structure like list, tree, map or other.

When a persistent structure is modified, the original structure remains unchanged (it is a previous version), and the modification method returns a new structure that contains the modification.

Consider the following list:

``````1
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a: 1 -> 2 -> 3

``````

if we add another element to it, the list itself is modified:

``````1
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a: 1 -> 2 -> 3 -> 4

``````

However if we use a persistent list, then the same operation would look like this:

``````1
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a: 1 -> 2 -> 3

print(a)

1 -> 2 -> 3

print(b)

1 -> 2 -> 3 -> 4

``````

The method that adds an element to the list, actually returns a new list, containing the element we added.

The original list remains unchanged.

There are multiple ways that this can be achieved:

• Doing copy-on-write - we copy the entire list in `b` then actually add the element, leaving the original list `a` unchanged.

This however is not very memory efficient, as every modification would allocate memory for all previous elements - `O(n^2)` space complexity.

• Using a linked list, but we only keep the pointer to the head and the tail of the list. This way we can share the unmodified elements and

optimize the memory usage.

Let’s consider the second way of implementing this data structure. We want to reuse the elements from `a` and just add the final element.

``````1
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\                   |

+-> (1) -> (2) -> (3)

/                    \

b: head b                +-> (4) tail b

``````

`a` and `b` share the first 3 elements, but only `b` can “see” the final added element.

When removing or updating an element, then we’re only copying the list up to the affected elements.

Let’s say we want to update the second element. Then we have to copy the head and the second element:

``````1
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\                   |

+-> (1) -> (2) -> (3)

/               /    \

b: head b         /      +-> (4) tail b and c

/

(1) -> (7) -+

|

``````

After changing the element `2` to `7`, we have to copy all affected elements, but the rest of the list

remains unchanged and shared between the previous versions. Note that all changes return new versions of the

list and we cannot directly modify previous versions - `a` and `b` remain unchanged after we do `set(b, 2, 7)`.

## An example of persistent list in Python

Let’s implement this kind of structure in Python. Every list would contain nodes that hold the data

and point to the next node. The list itself will contain the HEAD and TAIL nodes.

``````1
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class Node:

def __init__(self, value):

self.value = value

self.next = None

def __repr__(self):

return str(self.value)

class PList:

self.tail = tail

'''Adds to the end of the list.

'''

node = Node(value)

return PList(node, node)  # Add first element in an empty list

self.tail.next = node

# Return the new list

def pop(self):

'''Removes the last element from the list.

'''

raise Exception('Cannot pop from empty list')

while tail.next != self.tail:

tail = tail.next

def update(self, index, value):

'''Updates the element at index and returns the updated list.

'''

raise Exception('Empty list.')

lst = PList()

i = 0

while i < index and node != self.tail:

node = node.next

i += 1

if i != index:

raise Exception('Index out of bounds')

lst.tail.next = node.next

def remove(self, index):

'''Removes the element at index.

'''

raise Exception('Cannot remove from empty list')

if index == 0:

return PList()

else:

raise Exception('Index out of bounds')

lst = PList()

i = 0

while i < (index - 1) and node != self.tail:

node = node.next

i += 1

if i != (index - 1):

raise Exception('Index out of bounds')

if node.next == self.tail:

# Actually pop

return self.pop()

tail = lst.tail

tail.next = node.next

def iterate(self):

'''Returns an iterable (iterator) that goes through all elements of the list.

'''

return []

while node != self.tail:  # we must terminate at the tail of the list.

yield node.value

node = node.next

yield node

def __repr__(self):

return '[]'

return str([v for v in self.iterate()])

print('Lists a and b after adding element to a:')

print('a:', a)

print('b:', b)

value, c = b.remove(2)

print('\nLists after removing element from b:')

print('a:', a)

print('b:', b)

print('c:', c, '(removed ', value, ')')

print('\nLists after updating element in b:')

d = b.update(1, 7)

print('a:', a)

print('b:', b)

print('c:', c)

print('d:', d)

``````

After execution, it will print:

``````1
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Lists a and b after adding element to a:

a: [1, 2, 3]

b: [1, 2, 3, 4]

Lists after removing element from b:

a: [1, 2, 3]

b: [1, 2, 3, 4]

c: [1, 3, 4] (removed  2 )

Lists after updating element in b:

a: [1, 2, 3]

b: [1, 2, 3, 4]

c: [1, 3, 4]

d: [1, 7, 3, 4]

``````

which is exactly what we intended and wanted - every list is immutable and we can add/remove/update elements and

by doing so we get new lists.