Introduction

Suffix trees are versatile data structures used to solve a variety of string processing problems. A tree representation of all possible suffixes of a string, they can be constructed in linear time and allow operations such as efficient substring searching, pattern matching, and longest common substring calculations. One of the key components of a suffix tree is the concept of suffix links. These links play a crucial role in ensuring the efficiency of certain operations and are especially important in algorithms like Ukkonen's for constructing suffix trees.

What is a Suffix Link?

A suffix link is a pointer from one node in a suffix tree to another. If a node represents the string `xα`, where `x` is a single character and `α` is a string, then the suffix link will point to another node representing the string `α`. These links make navigating the tree more efficient because they help in jumping to a smaller suffix of the string without starting from the root again.

When to Create a Suffix Link?

Suffix links are typically created during the construction of the suffix tree. They are crucial in algorithms like Ukkonen's algorithm, which builds the tree in an online fashion. Here's a brief overview of when to create suffix links:

1. During Node Creation: Whenever a new internal node is created in the tree, a suffix link from the new node to another node (which might correspond to a smaller length suffix) is added. The exact node that this link points to depends on various factors such as the context in which the node was created.
2. While Splitting an Edge: When an edge is split to accommodate a new suffix during tree construction, the newly created internal node should have a suffix link added to it.
3. Specific to Suffix Ends: When ends of suffixes are explicitly represented (as they are in some forms of tree representations), suffix links might sometimes need to be added or updated.

How to Create a Suffix Link?

The creation of a suffix link depends on understanding the algorithmic context. Below we'll discuss this within the realm of Ukkonen's algorithm, which is one of the efficient ways to build suffix trees.

Ukkonen's Algorithm

Ukkonen's algorithm is a linear-time algorithm for constructing suffix trees. It incrementally builds the tree one character at a time and is efficient due to its use of suffix links. Here's how it incorporates suffix links:

1. Phases and Extensions: The algorithm operates in phases, and for each phase, it performs multiple extensions. In each extension, it adds a new suffix to the tree. If a rule requires a new internal node, a suffix link is created from the new node to the appropriate existing node.
2. Creating Suffix Links During Rule 2:
   • When an edge is split to create a new internal node, if the previous processing resulted in an internal node that was created or visited during the last extension, a suffix link should point from that node to the newly created node. This is due to the phase-based operations where each suffix builds upon the previous.
3. Handling Existing Links: If a suffix link already exists, they should be navigated to efficiently append new suffixes as needed.
4. Root Node: There is no suffix link from the root as it doesn't represent any meaningful suffix transformation.

Example

Consider the string "bananas". As Ukkonen's algorithm processes this string, constructing respective suffix tree steps involves creating suffix links:

For "bananas", consider initially only "b":

• A root node and a child node for "b" are created.
• As the next character "a" is processed, the node "b" has a suffix link pointing to the node created for "a".

For deeper insights into this step-by-step process, here's a simplified table structure:

Summary Table

Conclusion

Suffix links are an integral part of the suffix tree that boosts the efficiency of operations like incremental suffix additions. Whether constructing the tree or utilizing it for queries, understanding when and how to create suffix links can have a substantial impact on performance. While subtle to grasp, mastering suffix links significantly contributes to mastery over computational string problems.