![Shortest paths in DAGs : data structures
• Formally, for the vth element of arrays:
• dist[v] stores the weight of a shortest path from s to v.
• pred[v] stores the predecessor of v on the so-far identified
shortest path.
Figure: Finding single source shortest path
(DAA, Dr. Ashish Gupta) ULC403 : Unit-I/II 96 / 101](https://image.slidesharecdn.com/306shortestpathsindags-250419171705-664b2f9a/85/306_Shortest_Paths_in_DAGs_design_andanalysisofalgorithms-pdf-5-320.jpg)
306_Shortest_Paths_in_DAGs_design_andanalysisofalgorithms.pdf
- 1. DIRECTED ACYCLIC GRAPHS Shortest Paths (DAA, Dr. Ashish Gupta) ULC403 : Unit-I/II 92 / 101
- 2. Shortest paths in DAGs General Problem. Given a digraph G = (V, E), edge lengths le ≥ 0, source s ∈ V, and destination t ∈ V, find the shortest directed path from s to t. Single-source shortest paths problem in DAG Input Given a DAG, G = (V, E) and a source vertex s in V. Output Find the shortest path from s to every other vertex in G, i.e, both the sequence of nodes on the shortest path and its length. Figure: Example Problem (DAA, Dr. Ashish Gupta) ULC403 : Unit-I/II 93 / 101
- 3. How to proceed ? ■ STEP-1 OF 2 : linearize DAG, i.e, arrange nodes of G on a line so that all edges go from left to right. ■ Find the topological sort of the following graphs: (DAA, Dr. Ashish Gupta) ULC403 : Unit-I/II 94 / 101
- 4. Match your solution (DAA, Dr. Ashish Gupta) ULC403 : Unit-I/II 95 / 101
- 5. Shortest paths in DAGs : data structures • Formally, for the vth element of arrays: • dist[v] stores the weight of a shortest path from s to v. • pred[v] stores the predecessor of v on the so-far identified shortest path. Figure: Finding single source shortest path (DAA, Dr. Ashish Gupta) ULC403 : Unit-I/II 96 / 101
- 6. Shortest paths in DAGs: Step-2 • STEP-2 OF 2 • RELAX sub-procedure. (DAA, Dr. Ashish Gupta) ULC403 : Unit-I/II 97 / 101
- 7. Algorithm for shortest paths in DAG (DAA, Dr. Ashish Gupta) ULC403 : Unit-I/II 98 / 101
- 8. Example (DAA, Dr. Ashish Gupta) ULC403 : Unit-I/II 99 / 101
- 9. Running time for shortest paths in DAG Time complexity (DAA, Dr. Ashish Gupta) ULC403 : Unit-I/II 100 / 101
- 10. (DAA, Dr. Ashish Gupta) ULC403 : Unit-I/II 101 / 101