Understanding the Problem
We are given a connected, weighted, undirected graph. Our task is to find a Minimum Spanning Tree (MST), which is a subset of edges that connects all vertices in the graph with the minimum possible total edge weight, and without forming any cycles.
Prim’s Algorithm helps us solve this problem using a greedy approach. At every step, it picks the smallest weight edge that connects a node already in the MST to a node outside of it. This ensures we keep the tree connected and avoid cycles.
Step-by-Step Solution with Example
Step 1: Choose a starting point
We start from an arbitrary node. Let’s say we begin from node 0
.
Step 2: Add all edges from the start node to a Min Heap
This allows us to always access the smallest available edge. Each heap entry stores a tuple like (weight, from_node, to_node)
.
Step 3: Pop the smallest edge
We take the edge with the least weight from the heap. If the destination node is not yet in the MST, we add it to the MST and include this edge in our result.
Step 4: Add new edges from the newly included node
From the new node, push all edges to unvisited neighbors into the heap. This helps explore other possible minimum connections.
Step 5: Repeat until all nodes are in the MST
We continue this process of picking the smallest edge and expanding the MST until all vertices are included.
Step 6: Example
Let’s consider the following graph with 5 nodes and these edges:
Edges:
0 --(1)-- 1
0 --(3)-- 2
1 --(1)-- 2
1 --(6)-- 3
2 --(5)-- 3
3 --(2)-- 4
We start at node 0
:
- Add edges (0-1, weight 1) and (0-2, weight 3) to the heap.
- Pick edge (0-1). Add node 1 to MST. Total weight = 1
- Add edges (1-2, weight 1), (1-3, weight 6)
- Pick edge (1-2). Add node 2. Total weight = 2
- Add edge (2-3, weight 5)
- Pick edge (2-3). Add node 3. Total weight = 7
- Add edge (3-4, weight 2)
- Pick edge (3-4). Add node 4. Total weight = 9
Final MST edges: (0-1), (1-2), (2-3), (3-4)
Total weight of MST: 9
Edge Cases
- Single node graph: No edges, MST weight is 0.
- Disconnected graph: Prim’s Algorithm assumes the graph is connected. If not, we cannot form a valid MST.
- Multiple edges with same weight: The algorithm still works as it always chooses the minimum available weight. Tie-breaking is handled by heap order.
- Negative weight edges: Prim’s Algorithm can handle them safely, as it doesn’t depend on edge weight signs—only comparisons.
Finally
Prim’s Algorithm is an elegant and efficient way to find a Minimum Spanning Tree using a greedy approach. The use of a Min Heap ensures that we always pick the optimal next edge. For large graphs, the heap makes the algorithm run efficiently in O(E log V)
time.
Understanding each step and visualizing the MST growth helps beginners grasp how the algorithm avoids cycles and ensures minimum total weight.
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