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Graph Algorithms, Slides of Data Structures and Algorithms

George Mason University (GMU)Data Structures and Algorithms

An overview of various graph algorithms, including the shortest path problem, minimum spanning tree (mst) problem, and algorithms such as dijkstra's algorithm, bellman-ford algorithm, and prim's algorithm. It covers the definitions, motivations, and implementations of these algorithms, as well as related concepts like graph representation and traversal. The document also includes references to useful visualization tools and additional resources for further exploration of graph algorithms. This information could be valuable for students studying data structures and algorithms, particularly in the context of computer science courses covering graph theory and graph-based problem-solving.

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2020/2021

Uploaded on 05/03/2024

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Road Map

•Shortest path problem

•Find the shortest path for each vertex for a specified

starting vertex

•Unweighted graph: breadth-first traversal

•Non-negative-weighted graph: Dijkstra’s algorithm

•Negative-weighted graph: Bellman-Ford algorithm

•Minimum spanning tree ß

•Definition

•Prim’s algortihm

Graph Algorithms CS310 George Mason University 42

Discover Slides of Data Structures and Algorithms George Mason University (GMU)

Partial preview of the text

Download Graph Algorithms and more Slides Data Structures and Algorithms in PDF only on Docsity!

Road Map

Shortest path problem
- Find the shortest path for each vertex for a specified

starting vertex

Unweighted graph: breadth-first traversal
Non-negative-weighted graph: Dijkstra’s algorithm
Negative-weighted graph: Bellman-Ford algorithm
Minimum spanning tree ß
Definition
Prim’s algortihm

Warmup: Dijkstra’s Algorithm

Minimum Spanning Trees

• MST: a spanning tree with the minimum total edge

weight

• Example

• Applications?

Graph G

MST of G

Spanning Tree Motivation

Prim’s Algorithm

Idea: identify which edges should be included so that
- All nodes are connected
- No cycle is formed; total weight is minimum
Prim
- Assume undirected graphs
- Start with any vertex S and greedily grow a tree T from S
- For each step, pick the minimum edge that connects the

current T and (G-T) (node v in G-T)

Update T / G-T
Update the set of edges connecting these two sets (v and its nbrs)
Repeat until all vertices are connected in the tree

Prim Example

Initialize
Pick node with minimal cost to add to T
- Starting vertex 0 is the trivial winner

Node 0 1 2 3 4 5 6

Edge cost 0 ∞ ∞ ∞ ∞ ∞ ∞

Prev - 1 - - - - - -

T = {}

G-T = {0,1,2,3,4,5,6}

Prim Example

Only consider edges between T and (G-T)
Update cost if lowered
Next pick: 1 (assume lower ID preferred)

Node 0 1 2 3 4 5 6

Edge cost 0 1 1 3 ∞ ∞ ∞

Prev - 1 0 0 0 - - -

T = {0}

G-T = {1,2,3,4,5,6}

Prim Example

Only consider edges between T and (G-T)
Edges between {0, 1} and {2,3,4,5,6}
- Updating 1’s nbrs with a shorter edge: 4 and 6

Node 0 1 2 3 4 5 6

Edge cost 0 1 1 3 2 ∞ 7

Prev - 1 0 0 0 1 - 1

T = {0,1}

G-T = {2,3,4,5,6}

Prim Example

Only consider edges between T and (G-T)
Edges between {0,1,2} and {3,4,5,6}
- Pick node 3, update?

Node 0 1 2 3 4 5 6

Edge cost 0 1 1 3 1 2 ∞ 7

Prev - 1 0 0 0 2 1 - 1

T = {0,1,2}

G-T = {3,4,5,6}

Prim Example

Only consider edges between T and (G-T)
Edges between {0, 1,2,3} and {4,5,6}
- Pick node 4, update?

Node 0 1 2 3 4 5 6

Edge cost 0 1 1 1 2 1 ∞ 7

Prev - 1 0 0 2 1 3 - 1

T = {0,1,2,3}

G-T = {4,5,6}

Prim Example

Only consider edges between T and (G-T)
Edges between {0,1,2,3,4, 5} and {6}
- Pick node 6

Node 0 1 2 3 4 5 6

Edge cost 0 1 1 1 1 1 7 1

Prev - 1 0 0 2 3 4 1 5

T = {0,1,2,3,4,5}

G-T = {6}

Prim Example

Final result
- All nodes processed
- No cycle
- Minimum total weight (6)

Node 0 1 2 3 4 5 6

Edge cost 0 1 1 1 1 1 1

Prev - 1 0 0 2 3 4 5

T = {0,1,2,3,4,5, 6}

G-T = {}

Compare…

Dijkstra’s Shortest Path Algorithm:
- All nodes start at ∞ (except source)
- Take the node with minimum cost (path)
- Update connecting nodes (current cost + edge cost)
- Finds the SHORTEST PATH
Prim’s Minimum Spanning Tree Algorithm :
- All nodes start at ∞ (except source)
- Take the node with minimum cost (edge)
- Update connecting nodes (edge cost only )
- Finds the MINIMUM SPANNING TREE

Other MST Algorithms

Kruskal
- Sort edges based on their weights in ascending order
- For each step, pick the lowest one to add to T unless that

will create a cycle

Both Kruskal and Prim are greedy algorithms
Many other MST algorithms
http://en.wikipedia.org/wiki/Minimum_spanning_tree

Graph Algorithms, Slides of Data Structures and Algorithms

Related documents

Partial preview of the text

Download Graph Algorithms and more Slides Data Structures and Algorithms in PDF only on Docsity!

Road Map

starting vertex

Warmup: Dijkstra’s Algorithm

Minimum Spanning Trees

• MST: a spanning tree with the minimum total edge

weight

• Example

• Applications?

Spanning Tree Motivation

Prim’s Algorithm

current T and (G-T) (node v in G-T)

Prim Example

T = {}

G-T = {0,1,2,3,4,5,6}

Prim Example

T = {0}

G-T = {1,2,3,4,5,6}

Prim Example

T = {0,1}

G-T = {2,3,4,5,6}

Prim Example

T = {0,1,2}

G-T = {3,4,5,6}

Prim Example

T = {0,1,2,3}

G-T = {4,5,6}

Prim Example

T = {0,1,2,3,4,5}

G-T = {6}

Prim Example

T = {0,1,2,3,4,5, 6}

G-T = {}

Compare…

Other MST Algorithms

will create a cycle