CIS 121—Data Structures and Algorithms—Fall 2022
Topological Sort / Strongly Connected Components—Monday, October 24 / Tuesday, October 25
Readings
•Lecture Notes Chapter 18: DAGs and Topological Sort
•Lecture Notes Chapter 19: Strongly Connected Components
Review: Topological Sort
A topological sort of a directed acyclic graph (DAG) G= (V, E) is an ordering of the vertices such that for
each directed edge (u, v)∈E,uappears before vin the ordering. As described in the below algorithms,
topologically sorting a DAG only takes O(m+n) time, so given a DAG, it is helpful to topologically
sort it, since most graph algorithms take Ω(m+n) time anyway. In other words, topologically sorting a
DAG is usually a free step, and if not necessary for your algorithm, it can make reasoning/thinking about
the problem easier since it gives you a visual. Below are two algorithms to find a topological sort:
Kahn’s Algorithm
Every DAG has a source node, or a node with no incoming edges. Kahn’s algorithm relies on this intuition —
at a high-level, the algorithm operates by repeatedly finding a source node, putting it next in the topological
sort, removing the node and all of the edges incident on it from the graph, and repeating this process.
Kahn’s algorithm runs in O(m+n) time. As seen in the pseudocode, the first step to compute the in-degree
of each node takes O(m+n) time since for each node, we scan through its neighbors; the second step to
populate the queue takes O(n) time since we iterate through all of the vertices. In our while loop, note that
we enqueue each node exactly once and scan through each of its neighbors, performing constant work for
each, which takes O(m+n) time.
Tarjan’s Algorithm
Tarjan’s algorithm leverages the finishing times of DFS as shown in the pseudocode by just running DFS
and then returning the nodes in decreasing order of finishing times. Thus, it also runs in O(m+n) time.
This algorithm provides key insight into the following algorithm to find the strongly connected components
of a directed graph.
Review: Kosaraju’s Algorithm
Given a directed graph G= (V, E), a strongly connected comp onent (SCC) is a maximal set S⊆V
such that for all u, v ∈S, there exists a path u⇝vand a path v⇝u. Thus, we can decompose a directed
graph Ginto its SCCs, yielding GSCC or our kernel graph. Formally, GSCC = (VS CC , E SCC ). Each vertex
viin GSCC represents a single SCC Ciin G, and an edge (vi, vj) exists in GSC C if Gcontains the directed
edge (x, y) where xis in SCC Ciand yis in SCC Cj. Observe that GSC C is a DAG, meaning that we can
topologically sort it to make the problem easier to think about.
Kosaraju’s algorithm is an algorithm that we can use to compute the SCCs of a graph, and by extension, to
obtain GSCC . It operates by running two DFS traversals, one on Gand another on GT, the transposed graph
obtained by reversing the direction of edges in G; in the latter, we consider vertices in order of decreasing
finishing times.
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