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Exploring Shortest-Paths Trees: Efficient Algorithms and Their Applications

This document outlines the concept of Shortest-Paths Trees (SPT), which aims to identify the optimal set of edges connecting all nodes, minimizing the total edge lengths from a source node. It covers key algorithms such as Dijkstra's and Bellman-Ford, detailing methods to handle negative edges in directed graphs. By illustrating the process of edge relaxation and construction of SPTs, this resource serves as a practical guide for students and professionals in computer science, especially within the fields of graph theory and algorithm design.

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Exploring Shortest-Paths Trees: Efficient Algorithms and Their Applications

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  1. Shortest-Paths Trees Kun-Mao Chao (趙坤茂) Department of Computer Science and Information Engineering National Taiwan University, Taiwan E-mail: kmchao@csie.ntu.edu.tw WWW: http://www.csie.ntu.edu.tw/~kmchao

  2. Shortest-Paths Trees • The objective is to find the set of edges connecting all nodes such that the sum of the edge lengths from the source to each node is minimized. • In order to minimize the total path lengths, the path from the source to each node must be a shortest path connecting them.

  3. Shortest-Paths Trees(source: vertex a)

  4. Negative edges in an undirected graph

  5. Directed graphs

  6. Dijkstra's Algorithm

  7. Choose a

  8. Relax (a, b) and (a, g)

  9. Choose b; Add (a, b) to T

  10. Relax (b, c) and (b, d)

  11. Choose g; Add (a, g) to T

  12. Relax (g, e) and (g, h)

  13. Choose d; Add (b, d) to T

  14. Relax (d, e)

  15. Choose h; Add (g, h) to T

  16. Choose e; Add (d, e) to T

  17. Relax (e, f)

  18. Choose c; Add (b, c) to T

  19. Relax (c, h)

  20. Choose f; Add (e, f) to T

  21. Relax (f, d) and (f, h)

  22. The resulting SPT

  23. Negative edge

  24. Choose a

  25. Choose b; Add (a, b) to T

  26. Choose d; Add (b, d) to T

  27. Choose c; Add (b, c) to T

  28. Choose e; Add (d, e) to T

  29. Something went wrong

  30. A wrong SPT

  31. A correct SPT

  32. The Bellman-Ford Algorithm

  33. δ[b] and δ[g] modified

  34. δ[c], δ[d], δ[e] and δ[h] modified

  35. δ[f] modified

  36. δ[h] modified

  37. A correct SPT

  38. Try this in class

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