Consider an undirected graph G=(V, E) with nonnegative edge...

Consider an undirected graph $G=(V, E)$ with nonnegative edge weights $w_{e} \geq 0 .$ Suppose that you have computed a minimum spanning tree of $G,$ and that you have also computed shortest paths to all nodes from a particular node $s \in V$ Now suppose each edge weight is increased by 1: the new weights are $w_{e}^{\prime}=w_{e}+1$. (a) Does the minimum spanning tree change? Give an example where it changes or prove it cannot change. (b) Do the shortest paths change? Give an example where they change or prove they cannot change.

Consider an undirected graph $G=(V, E)$ with nonnegative edge weights $w_{e} \geq 0 .$ Suppose that you have computed a minimum spanning tree of $G,$ and that you have also computed shortest paths to all nodes from a particular node $s \in V$ Now suppose each edge weight is increased by 1: the new weights are $w_{e}^{\prime}=w_{e}+1$.
(a) Does the minimum spanning tree change? Give an example where it changes or prove it cannot change.
(b) Do the shortest paths change? Give an example where they change or prove they cannot change.

Key Concepts

Graph Edge Weight Transformation

This concept refers to the modification of edge weights by applying a constant or function to each weight, thereby exploring how such transformations influence graph properties and algorithmic outcomes. In the context of MSTs and shortest paths, understanding the impact of adding a constant to all weights is crucial because it can preserve the ordering of relative differences in some cases while altering aggregate path sums in others.

Shortest Paths

Shortest path algorithms are designed to find the minimum weight routes from a given source vertex to all other vertices in a graph. Algorithms like Dijkstra's rely on the precise values of edge weights to determine the least costly path. Uniform additive changes to all edge weights can change the overall path costs and even switch the order in which paths become optimal, thereby affecting the resulting shortest paths.

Minimum Spanning Tree

A minimum spanning tree (MST) of a graph is a subset of the edges that connects all vertices together without any cycles and with the minimal possible total edge weight. It is typically computed using greedy algorithms such as Kruskal’s or Prim’s algorithm, and its structure is determined by the relative ordering of edge weights. Changes in edge weights, even uniform shifts, can potentially alter the MST if the order of edge addition or removal is affected.

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