Shortest Path Properties

The shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized. The shortest path between any two nodes of the graph can be founded using many algorithms, such as Dijkstra’s algorithm, Bellman-Ford algorithm, Floyd Warshall algorithm. There are some properties of finding the shortest paths based on which the algorithm to find the shortest path works:

1. Optimal Substructure Property
   • All the sub-paths of the shortest path must also be the shortest paths.
   • If there exists the shortest path length between two nodes U and V, then greedily choosing the edge with the minimum length between V to S will give the shortest path length between U and S.
   • All the algorithms listed above work based on this property.
   • For example, let P₁ be a sub-path from (X → Y) of the shortest path (S →X →Y → V) of graph G. And let P₂ be any other path (X → Y) in graph G. Then, the cost of P₁ must be less than or equal to the cost of P₂. Otherwise, the path (S →X →Y → V) will not be the shortest path between nodes S and V.
   

   Graph G

2. Triangle Inequality
   • Let d(a, b) be the length of the shortest path from a to b in graph G₁. Then,
     • d(a, b) ≤ d(a, x) + d(x, b)
   

   Graph G₁