Ontological Layer: Network Relationships

Graphs

Connections, Traversals, and Optimization

The Freedom of Networks

Trees restrict us to strict hierarchies. Graphs shatter those restrictions, allowing any data point to connect to any other. By defining Nodes (Vertices) and Relationships (Edges), we can model maps, social networks, and internet routing. By assigning Weights to these edges, data structures transcend mere storage and become dynamic arenas for powerful optimization algorithms like Dijkstra's Shortest Path and Kruskal's Minimum Spanning Tree.

VERTICES & EDGES → ADJACENCY DATA → NETWORK STRUCTURE → QUEUE (BFS) vs STACK (DFS) → PATH WEIGHTS → RELAXATION → OPTIMIZATION

The 4 Network Nodes

Ontological Mapping: Representation to Optimization

GROUP 01

Graph Representation

A → BC

B → AC

C → AB

Vertices (V) and Edges (E)
Directed vs Undirected edges
Adjacency Matrix (O(V²) space)
Adjacency List (O(V+E) space)

GROUP 02

BFS & DFS Traversal

Breadth-First: Level-by-level (Queue)
Depth-First: Branch-by-branch (Stack)
Visited tracking to prevent infinite loops
Foundation for finding connectivity

GROUP 03

Shortest Path (Dijkstra)

Weighted edges & path cost
Edge Relaxation (updating best known distance)
Priority Queue for vertex selection
Finds optimal path from a single source

GROUP 04

Minimum Spanning Tree

Connect all vertices with minimum total weight
Must contain NO cycles (it is a Tree)
Prim's Algorithm (Vertex/Tree growth)
Kruskal's Algorithm (Edge selection)

FINAL PROTOCOL

05 — Network Algorithms Laboratory

1. RELAX EDGE(u, v):

if dist[v] > dist[u] + weight(u,v)

dist[v] = dist[u] + weight(u,v)

[1] BUILD ADJACENCY DATA [2] ENQUEUE/PUSH NODES [3] DETECT CYCLES [4] RELAX DISTANCES [5] ANALYZE COMPLEXITY

Graphs represent the pinnacle of structural relationships. By combining the linear access disciplines of Stacks and Queues with the hierarchical logic of Trees, we can traverse networks and mathematically guarantee optimal routes through chaotic, interconnected data.

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