From fd3b1952b1f37423b483e0fbf2d69984f9454906 Mon Sep 17 00:00:00 2001 From: Prad Nukala Date: Tue, 22 Sep 2026 13:28:23 -0400 Subject: [PATCH] docs(docs): add Topological Sort documentation page with example code and images --- apps/docs/content/topological-sort/index.mdx | 69 +++++++++++++++++++ apps/docs/public/topo_intro_1.png | Bin 0 -> 65516 bytes apps/docs/public/topo_intro_2.png | Bin 0 -> 88739 bytes 3 files changed, 69 insertions(+) create mode 100644 apps/docs/content/topological-sort/index.mdx create mode 100644 apps/docs/public/topo_intro_1.png create mode 100644 apps/docs/public/topo_intro_2.png diff --git a/apps/docs/content/topological-sort/index.mdx b/apps/docs/content/topological-sort/index.mdx new file mode 100644 index 0000000..d4ef87a --- /dev/null +++ b/apps/docs/content/topological-sort/index.mdx @@ -0,0 +1,69 @@ +--- +title: Topological Sort +description: 'Topological sort or topological ordering of a directed graph is an ordering of nodes such that every node appears in the ordering before all the nodes it points to. ' +hidden: true +--- + +## Directed Graph + +Before we get started on topological order, let's talk about directed graphs first. A graph is directed when its edges have directions (these edges are also called arcs). Suppose that v, w are vertices in a directed graph. Let's get familiar with a few concepts. + +- in-edge - an edge pointing to or coming into the node +- out-edge - an edge pointing away or going out from the node (we will refer those nodes on the other end of the out-edges of v to v's neighbors) +- in-degree - the number of in-edges coming into the node +- out-degree - the number of out-edges going out from the node +- dependency - "v depends on w" / "v is dependent on w" means that there is a directed path from w to v where v, w are vertices in a directed graph. With these new terms under our belt, it will make more sense when we talk about directed graphs. + +## What is a topological order? + +Topological sort is not unique. For example, for a graph like this: + +![Topological sort example](../../public/topo_intro_1.png) + +Both `[4, 5, 2]` and `[5, 4, 2]` are valid topological orders. + +![Topological sort example](../../public/topo_intro_2.png) + +Task 3 is completely independent of task 2 and 4, and it can be anywhere in the order as long as it is before task 1 which depends on it. All of the following ordering are valid topological orderings. + +`[4, 2, 3, 1], [4, 3, 2, 1], [3, 4, 2, 1]` + + +:::warning[Graphs with Cycles Do Not Have Topological Ordering] +It should be obvious that if a graph with a cycle does not have a topological ordering. In the above example, 2 has to come before 5 which has come before 4 which has to come before 2 which has to come before 5... which is impossible. +::: + + +## Example Code + +:::note +Notice the topological sort algorithm is very similar to BFS. The main difference is that we only push nodes with 0 in-degree into the queue in topological sort whereas in BFS we push all the neighboring nodes into the queue. +::: + +```py +from collections import deque + +def find_indegree(graph): + indegree = { node: 0 for node in graph } # dict + for node in graph: + for neighbor in graph[node]: + indegree[neighbor] += 1 + return indegree + + +def topo_sort(graph): + res = [] + q = deque() + indegree = find_indegree(graph) + for node in indegree: + if indegree[node] == 0: + q.append(node) + while len(q) > 0: + node = q.popleft() + res.append(node) + for neighbor in graph[node]: + indegree[neighbor] -= 1 + if indegree[neighbor] == 0: + q.append(neighbor) + return res if len(graph) == len(res) else None +``` diff --git a/apps/docs/public/topo_intro_1.png b/apps/docs/public/topo_intro_1.png new file mode 100644 index 0000000000000000000000000000000000000000..3e246ab07e3ae212d99083ddacee17f0db977901 GIT binary patch literal 65516 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