title: Will paths with the power of two return the same shortest path using Dijkstra's?
body: <p>I have a graph, G = (V, E) with two weight functions w1(e) and w2(e), where
  w1(e) = (w2(e))^2. All edge weights are unique and positive.</p>
  
  <blockquote>
    <p>Under both weight functions, Kruskal’s algorithm will return the same
    minimum spanning tree.</p>
  </blockquote>
  
  <p>I know that kruskal is greedy and will choose the shortest/lowest cost path. Since they're positive, we will end up choosing the same MST as long as there are no paths costing 1.5 or such. </p>
  
  <blockquote>
    <p>Under both weight functions, Dijkstra’s algorithm will return the same
    shortest path.</p>
  </blockquote>
  
  <p>I'm unsure about this one. I think it will also be true, but I feel as though if we get large enough numbers, one path might actually end up larger. Can anyone confirm that if we were to exponentiate the path lengths?</p>