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Topological Sort


Implement a function that performs a topological sort on a directed graph (in adjacency list format), where the keys represent nodes and values are an array of nodes that have an outgoing edge to the current node.


const graph1 = {
A: ['B', 'C'],
B: ['C', 'D', 'E'],
C: ['F'],
D: [],
E: ['F'],
F: [],
topologicalSort(graph1); // ['A', 'B', 'C', 'D', 'E', 'F']
const graph2 = {
A: ['B', 'C'],
B: ['C', 'D'],
C: ['D'],
D: ['E'],
E: ['F'],
F: [],
topologicalSort(graph2); // ['A', 'B', 'C', 'D', 'E', 'F']
const graph3 = {
A: [],
B: ['A'],
C: ['B'],
D: ['C'],
E: ['D'],
F: ['E'],
topologicalSort(graph3); // ['F', 'E', 'D', 'C', 'B', 'A']

Note that there can be multiple valid topological orderings for a directed graph, but for our test cases there is only one valid solution.

A Queue data structure is also provided for you at the bottom of the skeleton code.


Topological sort is an algorithm which is used to sort the elements of a directed acyclic graph (DAG) in a linear order such that the order respects the order defined by edges within the graph.

The easy way to understand this is to think of the nodes of a graph as tasks, and an edge from node A to node B represents that node B has a dependency on node A. What a topological sort does is to produce an array of nodes out of the graph such that the order of nodes in the array respects all the dependencies defined in the graph.

Why is it that a topological sort only works on DAGs and not any graph? This is because the graph has to have directed edges in order to represent dependencies, and acyclic (no cycles) because any cycles would represent an unresolvable dependency between nodes.

Topological sorting is often used in scheduling problems, such as scheduling tasks with dependencies, or in compilation, where the order of compilation is determined by the dependencies between modules.

There are many ways to implement topological sort, but one simple and intuitive way is using the Kahn's algorithm, which works as follows:

  1. Initialize a queue and a list to store the sorted nodes.
  2. For each node in the graph, if it has no incoming edges, add it to the queue.
  3. While the queue is not empty:
  4. Dequeue a node from the front of the queue.
  5. Add this node to the list of sorted nodes.
  6. For each child of this node, decrease its in-degree (the number of incoming edges) by 1.
  7. If a child's in-degree becomes 0, add it to the queue.
  8. If the length of the sorted list is less than the number of nodes in the graph, this means that there is a cycle in the graph, and no topological ordering is possible.


  • Strictly speaking, we can also use a set instead of a queue.