Understanding the Problem
We are given a network of servers (nodes) connected by undirected edges. Each connection is a two-way link. Our goal is to find all critical connections, also known as bridges.
A bridge is an edge that, if removed, would increase the number of connected components in the graph. In simple terms, removing that edge disconnects part of the network. This is a classic graph problem, and we can solve it efficiently using Tarjan’s Algorithm, which is based on Depth-First Search (DFS).
Step-by-Step Solution with Example
step 1: Represent the Graph
We first build an adjacency list from the input list of connections. This allows us to efficiently explore all neighbors of a node during DFS.
step 2: Setup Required Structures
We initialize:
visited[]
: to keep track of visited nodes
tin[]
: stores the discovery time (insertion time) for each node
low[]
: the lowest discovery time reachable from that node or its subtree
timer
: a global counter to assign discovery times
result[]
: a list to store all bridges found
step 3: Run DFS from Any Node
We begin DFS traversal from an unvisited node. During DFS:
- We mark the node as visited and assign its
tin
and low
values using the global timer.
- For each neighbor:
- If the neighbor is the parent (the node we came from), we skip it.
- If the neighbor is already visited, it means a back edge exists. We update the current node’s
low
value using the neighbor’s tin
.
- If the neighbor is unvisited, we recursively apply DFS on it. After returning, we update the current node’s
low
value based on the neighbor’s low
. If low[neighbor] > tin[current]
, then [current, neighbor]
is a bridge.
step 4: Example Walkthrough
Let's consider this input:
n = 5
connections = [[0,1],[1,2],[2,0],[1,3],[3,4]]
This forms the following graph:
- A cycle: 0–1–2–0
- Two edges: 1–3 and 3–4 (which are not part of any cycle)
Running Tarjan's algorithm:
- DFS visits nodes and assigns timestamps
- When we backtrack, we find that removing 1–3 or 3–4 would increase the number of components
Output: [[3, 4], [1, 3]]
Edge Cases
- Disconnected Graph: If the graph has multiple disconnected parts, run DFS for each unvisited node.
- No Bridges: If the graph is fully connected with cycles (like a complete graph), then no bridge will be found.
- Single Node: No connections to check, return an empty list.
- Tree Structure: Every edge in a tree is a bridge since removing any edge disconnects the graph.
Finally
Tarjan’s Algorithm is an elegant and efficient solution to detect bridges in a graph. By tracking the discovery time and the lowest reachable ancestor for each node, we can find edges that, if removed, disconnect the graph. The algorithm runs in O(V + E) time and is highly suitable for large graphs.
Understanding the idea of back edges and how they affect the low value is key to mastering this approach. With step-by-step DFS and careful updates, even beginners can grasp the intuition behind finding critical connections in a network.
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