Leetcode 444. Sequence Reconstruction (topological sort)

444. Sequence Reconstruction

Topological sorting is an algorithm used on directed graphs where the nodes represent tasks or dependencies between elements. It arranges the nodes in a linear order such that for every directed edge from node A to node B, node A appears before node B in the ordering. This ordering is valid for acyclic directed graphs, known as DAGs (Directed Acyclic Graphs).

The algorithm typically involves:

1. **Finding a "start" node**: Identifying a node that has no incoming edges, often termed as a "source" or "root" node.
2. **Removing the node and its outgoing edges**: Deleting this node and its outgoing edges, updating the graph structure.
3. **Repeating the process**: Iterating through the graph by continually identifying and removing nodes with no incoming edges until no nodes remain.

The result is a linear ordering of nodes that satisfies the directed dependencies present in the graph.

In LeetCode, various problems employ topological sorting, especially in scenarios involving task scheduling, dependency resolution, or ordering problems. Some problems may not explicitly mention topological sort, but the underlying solution can be solved using this technique. Here are a few LeetCode problems that involve topological sorting:

1. **Course Schedule (LeetCode #207)**: Given the total number of courses and a list of prerequisite pairs, determine if it is possible to finish all courses.

2. **Alien Dictionary (LeetCode #269)**: Given a sorted dictionary of an alien language, find the order of characters in this language.

3. **Sequence Reconstruction (LeetCode #444)**: Given sequences, reconstruct the original sequence if it's unique and valid.

4. **Minimum Height Trees (LeetCode #310)**: Given an undirected graph where each edge is a tree edge, find all the minimum height trees.

5. **Reconstruct Itinerary (LeetCode #332)**: Given a list of airline tickets represented by pairs of departure and arrival airports, reconstruct the itinerary in order.

These problems often involve understanding dependencies or sequences and using topological sorting to determine a valid order or resolve dependencies efficiently.

The application of topological sorting extends beyond these specific problems, and it's a fundamental technique in graph theory, used in various scenarios requiring ordering or sequencing based on directed relationships.