Single-Source Shortest Paths Introduction Negative Weights and - - PowerPoint PPT Presentation

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Apr 01, 2023 44 likes •155 views

Single-Source Shortest Paths Introduction Negative Weights and Cycles Initialize-Single-Source and Relax Bellman-Ford Algorithm Dijkstras Algorithm CS 3343 Analysis of Algorithms Single-Source Shortest Paths 1 Shortest Path Definitions

SLIDE 1

CS 3343 Analysis of Algorithms Single-Source Shortest Paths – 1

Single-Source Shortest Paths

Introduction Negative Weights and Cycles Initialize-Single-Source and Relax Bellman-Ford Algorithm Dijkstra’s Algorithm

SLIDE 2

Shortest Path Definitions

⊲ Definition

Subpaths Examples Basic Methods Bellman-Ford 1 Bellman-Ford 2 Bellman-Ford 3 Dijkstra 1 Dijkstra 2 Dijkstra 3

CS 3343 Analysis of Algorithms Single-Source Shortest Paths – 2

SLIDE 3

Subpath Property

Definition

⊲ Subpaths

Examples Basic Methods Bellman-Ford 1 Bellman-Ford 2 Bellman-Ford 3 Dijkstra 1 Dijkstra 2 Dijkstra 3

CS 3343 Analysis of Algorithms Single-Source Shortest Paths – 3

SLIDE 4

Examples

Definition Subpaths

⊲ Examples

Basic Methods Bellman-Ford 1 Bellman-Ford 2 Bellman-Ford 3 Dijkstra 1 Dijkstra 2 Dijkstra 3

CS 3343 Analysis of Algorithms Single-Source Shortest Paths – 4

SLIDE 5

Initialize-Single-Source and Relax

Definition Subpaths Examples

⊲ Basic Methods

Bellman-Ford 1 Bellman-Ford 2 Bellman-Ford 3 Dijkstra 1 Dijkstra 2 Dijkstra 3

CS 3343 Analysis of Algorithms Single-Source Shortest Paths – 5

SLIDE 6

Bellman-Ford Algorithm

Definition Subpaths Examples Basic Methods

⊲ Bellman-Ford 1

Bellman-Ford 2 Bellman-Ford 3 Dijkstra 1 Dijkstra 2 Dijkstra 3

CS 3343 Analysis of Algorithms Single-Source Shortest Paths – 6

SLIDE 7

Bellman-Ford Illustration

Definition Subpaths Examples Basic Methods Bellman-Ford 1

⊲ Bellman-Ford 2

Bellman-Ford 3 Dijkstra 1 Dijkstra 2 Dijkstra 3

CS 3343 Analysis of Algorithms Single-Source Shortest Paths – 7

SLIDE 8

Bellman-Ford Correctness

Definition Subpaths Examples Basic Methods Bellman-Ford 1 Bellman-Ford 2

⊲ Bellman-Ford 3

Dijkstra 1 Dijkstra 2 Dijkstra 3

CS 3343 Analysis of Algorithms Single-Source Shortest Paths – 8

SLIDE 9

Dijkstra’s Algorithm

Definition Subpaths Examples Basic Methods Bellman-Ford 1 Bellman-Ford 2 Bellman-Ford 3

⊲ Dijkstra 1

Dijkstra 2 Dijkstra 3

CS 3343 Analysis of Algorithms Single-Source Shortest Paths – 9

SLIDE 10

Dijkstra Illstration

Definition Subpaths Examples Basic Methods Bellman-Ford 1 Bellman-Ford 2 Bellman-Ford 3 Dijkstra 1

⊲ Dijkstra 2

Dijkstra 3

CS 3343 Analysis of Algorithms Single-Source Shortest Paths – 10

SLIDE 11

Dijkstra Running Time

Definition Subpaths Examples Basic Methods Bellman-Ford 1 Bellman-Ford 2 Bellman-Ford 3 Dijkstra 1 Dijkstra 2

⊲ Dijkstra 3

CS 3343 Analysis of Algorithms Single-Source Shortest Paths – 11