Understanding Graph Weight Modifications

The question asks us to analyze the impact of a specific change to edge weights on two fundamental graph concepts: Minimum Spanning Trees (MSTs) and Shortest Paths (SPs).

We start with a weighted undirected graph $G$ with positive edge weights. A positive constant value, denoted by $\alpha$, is added to the weight of every edge in the graph. We need to determine if the MSTs and SPs computed on the original graph remain the same or if they might change after this weight update.

Analysis of Changes After Weight Update

Impact on Minimum Spanning Trees (MSTs)

Let's consider how adding $\alpha$ to every edge weight affects the properties crucial for finding MSTs.

• Relative Edge Weights: MST algorithms, such as Kruskal's algorithm or Prim's algorithm, primarily rely on the relative ordering of edge weights. For any two edges $e_1$ and $e_2$ in the original graph $G$, let their weights be $w(e_1)$ and $w(e_2)$.
• Weight Update Rule: The new weight for any edge $e$ is $w'(e) = w(e) + \alpha$, where $\alpha > 0$.
• Order Preservation: If we compare two edges $e_1$ and $e_2$ in the original graph, and assuming $w(e_1) < w(e_2)$, then after adding $\alpha$, their new weights will be $w'(e_1) = w(e_1) + \alpha$ and $w'(e_2) = w(e_2) + \alpha$. Since $w(e_1) < w(e_2)$, it directly follows that $w(e_1) + \alpha < w(e_2) + \alpha$, meaning $w'(e_1) < w'(e_2)$.
• Conclusion for MSTs: The relative order of edge weights is completely preserved. MST algorithms select edges based on this order (e.g., Kruskal's sorts edges by weight and adds them if they don't form a cycle). Since the order remains unchanged, the set of edges that constitute a Minimum Spanning Tree will also remain the same. Therefore, every MST in the original graph remains an MST in the updated graph.

Impact on Shortest Paths (SPs)

Now, let's examine the effect of this constant addition on shortest paths between any two nodes in the graph.

• Path Length Calculation: The length of a path in a weighted graph is the sum of the weights of the edges constituting the path.
• Original Path Length: Consider a path $P$ consisting of $k$ edges. Let the original weights be $w(e_1), w(e_2), ..., w(e_k)$. The original length of path $P$ is $L(P) = \sum_{i=1}^{k} w(e_i)$.
• Updated Path Length: After adding $\alpha$ to each edge weight, the new weight of each edge is $w'(e_i) = w(e_i) + \alpha$. The new length of the same path $P$ becomes $L'(P) = \sum_{i=1}^{k} w'(e_i) = \sum_{i=1}^{k} (w(e_i) + \alpha)$.
• Mathematical Simplification: Expanding the sum, we get $L'(P) = (\sum_{i=1}^{k} w(e_i)) + (\sum_{i=1}^{k} \alpha) = L(P) + k \cdot \alpha$.
• Dependency on Number of Edges: The change in the path's total weight, $L'(P) - L(P)$, is $k \cdot \alpha$. This means the increase in path length is directly proportional to the number of edges ($k$) in the path.
• Potential Change in Shortest Path: Consider two different paths, $P_1$ and $P_2$, between the same pair of nodes. Suppose $P_1$ was the shortest path in the original graph ($L(P_1) < L(P_2)$), but $P_1$ has more edges than $P_2$ (i.e., $k_1 > k_2$). After adding $\alpha$, the new lengths are $L'(P_1) = L(P_1) + k_1 \cdot \alpha$ and $L'(P_2) = L(P_2) + k_2 \cdot \alpha$. It is possible that $L(P_1) + k_1 \cdot \alpha > L(P_2) + k_2 \cdot \alpha$, even if $L(P_1) < L(P_2)$. This happens because the term $k \cdot \alpha$ adds a larger value to paths with more edges. Consequently, a path that was previously the shortest might no longer be the shortest path after the weights are updated.
• Conclusion for SPs: Shortest paths are not guaranteed to remain the shortest paths after adding a constant $\alpha$ to all edge weights. The path with the overall minimum weight might change depending on the number of edges involved.

Overall Conclusion

Based on the analysis:

• Minimum Spanning Trees (MSTs) are preserved because the relative ordering of edge weights remains the same.
• Shortest Paths (SPs) are not necessarily preserved because the increase in path length depends on the number of edges, potentially altering the shortest path ranking.

Therefore, the statement "Every MST remains an MST, and SPs need not remain SPs" accurately describes the situation.