Representing: Shortest PathGiven a graph G = (V, E), we maintain for each vertex v ∈ V a predecessor π [v] that is either another vertex or NIL. During the execution of shortest paths algorithms, however, the π values need not indicate shortest paths. As in breadthfirst search, we shall be interested in the predecessor subgraph G_{n}= (V_{n},E_{n}) induced by the value π. Here again, we define the vertex set V_{π}, to be the set of vertices of G with non  NIL predecessors, plus the source s: V_{π}= {v ∈ V: π [v] ≠ NIL} ∪ {s} } The directed edge set E_{Π} is the set of edges induced by the Π values for vertices in V_{Π}: E_{Π}= {(Π[v], v) ∈ E: v ∈ V_{Π}  {s}} A shortest  paths tree rooted at s is a directed subgraph G = (V' E'), where V'∈ V andE'∈E, such that
Shortest paths are not naturally unique, and neither is shortest  paths trees. Properties of Shortest Path:1. Optimal substructure property: All subpaths of shortest paths are shortest paths. Let P_{1} be x  y sub path of shortest s  v path. Let P_{2} be any x  y path. Then cost of P_{1}≤ cost of P_{2},otherwise P not shortest s  v path. 2. Triangle inequality: Let d (v, w) be the length of shortest path from v to w. Then, 3. Upperbound property: We always have d[v] ≥ δ(s, v) for all vertices v ∈ V, and once d[v] conclude the value δ(s, v), it never changes. 4. Nopath property: If there is no path from s to v, then we regularly have d[v] = δ(s, v) = ∞. 5. Convergence property: If s>u>v is a shortest path in G for some u, v ∈ V, and if d[u] = δ(s, u) at any time prior to relaxing edge (u, v), then d[v] = δ(s, v) at all times thereafter.
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