Purely Functional Data Structures

This repository contains selected examples and solutions to exercises from Purely Functional Data Structures.

Data Structures

Queues

instance	persistence	snoc	head	tail
Batched Queue	ephemeral	O(n) / O(1)*	O(1)	O(n) / O(1)*
Banker's Queue	persistent	O(n) / O(1)*	O(1)	O(n) / O(1)*
Physicist's Queue	persistent	O(n) / O(1)*	O(1)	O(n) / O(1)*
Real-Time Queue	persistent	O(1)	O(1)	O(1)
Hood-Melville Queue	persistent	O(1)	O(1)	O(1)
Bootstrapped Queue	persistent	O(log*(n))**	O(1)	O(log*(n))**
Implicit Queue	persistent	O(log(n)) / O(1)*	O(1)	O(log(n)) / O(1)*

* amortized time

** amortized time, log*(n) is constant in practice

Deques

Output-Restricted Queues

instance	persistence	cons	head	tail	snoc
Output-Restricted Banker's Deque	persistent	O(n) / O(1)*	O(1)	O(n) / O(1)*	O(n) / O(1)*
Output-Restricted Real-Time Deque	persistent	O(1)	O(1)	O(1)	O(1)
Output-Restricted Hood-Melville Deque	persistent	O(1)**	O(1)	O(1)	O(1)

Double-Ended Queues

instance	persistence	cons	head	tail	snoc	last	init
Banker's Deque	persistent	O(n) / O(1)*	O(1)	O(n) / O(1)*	O(n) / O(1)*	O(1)	O(n) / O(1)*
Real-Time Deque	persistent	O(1)	O(1)	O(1)	O(1)	O(1)	O(1)

* amortized time

** via the ConsQueue wrapper

Catenable Lists

instance	persistence	cons	snoc	++	head	tail
Catenable List	persistent	O(1)**	O(1)**	O(1)*	O(1)	O(n) / O(1)*

* amortized time

** depends on the complexity of snoc of the underlying queue

Catenable Deques

instance	persistence	cons	snoc	++	head	tail	last	init
Simple Catenable Deque	persistent	O(1)**	O(1)**	O(log(n))*	O(1)	O(1)*	O(1)	O(1)*
Implicit Catenable Deque	persistent	O(1)**	O(1)**	O(1)*	O(1)	O(1)*	O(1)	O(1)*

* amortized time

** amortized or worst-case time (depends on the underlying deque instance)

Random Access List

instance	cons	head	tail	lookup	update
Binary Random Access List	O(log(n))	O(log(n)) / O(1)**	O(log(n))	O(log(n))	O(log(n))
Alternative Binary Random Access List	O(log(n))	O(log(n)) / O(1)**	O(log(n))	O(log(n))	O(log(n))
Zeroless Binary Random Access List	O(log(n))	O(1)	O(log(n))	O(log(i))	O(log(i))
Zeroless Redundant Bin. Random Access List	O(log(n)) / O(1)*	O(1)	O(log(n)) / O(1)*	O(log(i))	O(log(i))
Skew Binary Random Access List	O(1)	O(1)	O(1)	O(min(i, log(n)))	O(min(i, log(n)))

where n is the list size and i is the index parameter of the lookup/update.

* amortized time

** with explicit reference to the head element

Heaps

instance	persistence	insert	merge	findMin	deleteMin
Leftist Heap	ephemeral	O(log(n))	O(log(n))	O(1)	O(log(n))
Binomial Heap	persistent	O(log(n)) / O(1)*	O(log(n))	O(log(n)) / O(1)**	O(log(n))
Scheduled Binomial Heap	persistent	O(1)	O(log(n))	O(log(n)) / O(1)**	O(log(n))
Skew Binomial Heap	persistent	O(1)	O(log(n))	O(log(n)) / O(1)**	O(log(n))
Splay Heap	ephemeral	O(n) / O(log(n))*	O(n) / O(log(n))*	O(n) / O(log(n))* / O(1)**	O(n) / O(log(n))*
Pairing Heap	ephemeral	O(1)	O(1)	O(1)	O(n) / O(log(n))*
Lazy Pairing Heap	persistent	O(1)	O(1)*	O(1)	O(log(n))*
Bootstrapped Heap	persistent	O(1)'	O(1)'	O(1)	O(log(n))'

* amortized time

** with explicit reference to the minimum element

' either worst-case or amortized depending on the underlying heap

Sortable Collections

instance	persistence	add	sort
Merge Sort	persistent	O(log(n))*	O(n)*
Scheduled Merge Sort	persistent	O(log(n))	O(n)

* amortized time

Sets

instance	persistence	member	insert
Binary Search Tree	ephemeral	O(n)	O(n)
Red-Black Tree	ephemeral	O(log(n))	O(log(n))

Finite Maps

instance	lookup	bind
Trie	O(qm)	O(qm)
Trie of Trees	O(qm)	O(qm)

where

• O(q) is the query complexity (e.g. size of an aggregated key [k] or Tree k)
• O(m) is the operation complexity of the underlying map m k

Terminology

See terminology for brief description and definition of concepts and techniques from the book that is used to describe data structure instances in the IHaskell notebooks.

Notebooks

Haskell code can be found in IHaskell notebooks under the notebooks directory. In order to view and edit them one can start a Docker container running Jupyter with IHaskell kernel via

make ihaskell

or just make.

Disclaimer

Note that this is a personal learning place that was created while reading the book. I do encourage others to buy and read it!