Overview
In this article, we cover results on persistent data structures, which are data structures where we keep all information about past states. Persistent data structures are part of the larger class of temporal data structures. The other kind of temporal data structures, retroactive data structures, are the topic of MIT lecture 8.
Usually we deal with data structure updates by mutating something in the existing data structure: either its data or the pointers that organize it. In the process we lose information previous data structures states. Persistent data structures do not lose any information.
For several cases of data structures and definitions of persistence it is possible to transform a plain data structure into a persistent one with asymptotically minimal extra work or space overhead.
A recurring theme in this area is that the model is crucial to the results.
Partial and full persistence correspond to time travel with a branching universe model such as the one in Terminator, and Deja Vu parts 1 and 2
Model and definitions
The pointer machine model of data structures
In this model we think of data structures as collections of nodes of a bounded size with entries for data. Each piece of data in the node can be either actual data, or a pointer to a node.
The primitive operations allowed in this model are:

x = new Node()

x = y.field

x.field = y

x = y + z, etc (i.e. data operations)

destroy(x) (if no other pointers to x)
Where x, y, z are names of nodes or fields in them.
Data structures implementable with these shape constraints and these operations includes linked lists and binary search trees, and in general corresponds to struct’s in C or objects in Java. An example of a data structure not in this group would be a structure of variable size such as an array.
Definitions of persistence
We have vaguely referred to persistence as the ability to answer queries about the past states of the structure. Here we give several definitions of what we might mean by persistence.

Partial Persistence – In this persistence model we may query any previous version of the data structure, but we may only update the latest version. We have operations read(var, version) and newversion = write(var, val). This definition implies a linear ordering on the versions like patial persistence in version diagrams.

Full Persistence – In this model, both updates and queries are allowed on any version of the data structure. We have operations read(var, version) and newversion = write(var, version, val). The versions form a branching tree as in full persistence in version diagrams.

Confluent Persistence – In this model, in addition to the previous operation, we allow combination operations to combine input of more than one previous versions to output a new single version. We have operations read(var, version), newversion = write(var, version, val) and newversion = combine(var, val, version1, version2). Rather than a branching tree, combinations of versions induce a DAG (direct acyclic graph) structure on the version graph, shown in confluent persistence in version diagrams.

Functional Persistence – This model takes its name from functional programming where objects are immutable. The nodes in this model are likewise immutable: revisions do not alter the existing nodes in the data structure but create new ones instead. Okasaki discusses these as well as other functional data structures in his book @okasaki.
The difference between functional persistence and the rest is we have to keep all the structures related to previous versions intact: the only allowed internal operation is to add new nodes. In the previous three cases we were allowed anything as long as we were able to implement the interface.
Each of the succeeding levels of persistence is stronger than the preceding ones. Functional implies confluent, confluent implies full, and full implies partial.
Functional implies confluent because we are simply restricting ways on how we implement persistence. Confluent persistence becomes full persistence if we restrict ourselves to not use combinators. And full persistence becomes partial when we restrict ourselves to only write to the latest version.
The diagrams in the following figures show what the version ‘genealogies’ can look like for each definition.
Version Diagrams
Gray means version is read only and blue means version is readwrite.
Patial Persistence
Full Persistence
Confluent/Functional Persistence
References
[1] Gerth Stolting Brodal: Partially Persistent Data Structures of Bounded
Degree with Constant Update Time. Nord. J. Comput. 3(3): 238255 (1996)
[2] Erik D. Demaine, John Iacono, Stefan Langerman: Retroactive data
structures. SODA 2004: 281290
[3] Erik D. Demaine, Stefan Langerman, and Eric Price: Confluently
Persistent Tries for Efficient Version Control. Algorithmica (2008).
[4] Paul F. Dietz, Daniel Dominic Sleator: Two Algorithms for Maintaining
Order in a List STOC 1987: 365372
[5] Paul F. Dietz: Fully Persistent Arrays (Extended Array). WADS 1989:
6774
[6] James R. Driscoll, Neil Sarnak, Daniel Dominic Sleator, Robert Endre
Tarjan: Making Data Structures Persistent. J. Comput. Syst. Sci.
38(1): 86124 (1989)
[7] Yoav Giora, Haim Kaplan: Optimal dynamic vertical ray shooting in
rectilinear planar subdivisions ACM Transactions on Algorithms 5(3)
(2009)
[8] Haim Kaplan, Chris Okasaki, Robert Endre Tarjan: Simple Confluently
Persistent Catenable Lists. SIAM J. Comput. 30(3): 965977 (2000)
[9] Amos Fiat, Haim Kaplan: Making data structures confluently persistent.
J. Algorithms 48(1): 1658 (2003)
[10] Chris Okasaki: Purely Functional Data Structures. New York: Cambridge
University Press, 2003.
[11] Sebastien Collette, John Iacono, and Stefan Langerman. Confluent
Persistence Revisited. In Symposium on Discrete Algorithms (SODA),
pages 593601, 2012.
[12] T. H. Cormen and C. E. Leiserson and R. L. Rivest and C. Stein,
Introduction to Algorithms. 3rd. Edition. The MIT Press, 2009.
[13] Nicholas Pippenger. Pure Versus Impure Lisp. ACM Transactions on
Programming Languages and Systems, Vol. 19, No. 2. (March 1997), pp.
223238.
[14] Gerth Stolting Brodal, Christos Makris, Kostas Tsichlas: Purely
Functional Worst Case Constant Time Catenable Sorted Lists. ESA 2006:
172183