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Updated Mar 22, 2026

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3 pages

Mastering Graph Algorithms for Further Maths Decision

Amy

@amy_udqc

Graph algorithms are powerful tools that help solve real-world problems... Show more

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Minimum Spanning Trees

Ever wondered how engineers decide the cheapest way to connect multiple cities with roads? Minimum spanning trees (MST) solve exactly this problem by finding the lowest-cost way to connect all vertices in a graph.

Kruskal's algorithm works like a bargain hunter - it sorts all edges by weight and picks the cheapest ones first. Start by listing all edges from smallest to largest weight, then keep adding the cheapest edge that doesn't create a cycle. You'll stop when all vertices are connected, guaranteed to have the minimum total cost.

Prim's algorithm takes a different approach by growing the tree from a single starting point. Pick any vertex to begin, then repeatedly add the cheapest edge that connects your existing tree to a new vertex. This method feels more natural since you're building outward step by step.

Top Tip: Kruskal's focuses on edges globally, whilst Prim's grows locally from your current tree - both give the same minimum weight!

Distance Matrices and Shortest Paths

Working with distance matrices makes Prim's algorithm even more systematic, especially when dealing with dense graphs. You'll delete rows and number columns as you build your spanning tree, circling the smallest values to identify your next connection.

Dijkstra's algorithm is your go-to method for finding the shortest path between two specific points. Think of it like exploring a map systematically - you assign final labels to vertices in order of their distance from the start. Each vertex keeps working values that get updated as you discover potentially shorter routes.

The beauty of Dijkstra's lies in its guarantee: once a vertex receives its final label, you've definitely found the shortest path to it. No vertex gets labeled until you're absolutely certain of the optimal route.

Remember: Dijkstra's only works with non-negative weights - negative edges would break the algorithm's logic!

Finding All Shortest Paths

Floyd's algorithm tackles the ultimate challenge: finding the shortest path between every pair of vertices simultaneously. Rather than running Dijkstra's multiple times, this clever method updates all distances in systematic iterations.

You'll work with two tables - a distance table showing current shortest distances and a route table tracking the actual paths. The key insight is simple: if going through vertex A makes any journey shorter, update both tables accordingly.

Each iteration focuses on one vertex as a potential "stepping stone." Shade that vertex's row and column, then check if routing through it improves any existing paths. The rule is straightforward: if distance via A plus distance from A is less than the current direct distance, make the change.

Pro Strategy: Floyd's algorithm is perfect for small graphs where you need all shortest paths - it's more efficient than running Dijkstra's from every vertex!

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Further Maths

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176

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Updated Mar 22, 2026

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3 pages

Mastering Graph Algorithms for Further Maths Decision

Amy

@amy_udqc

Graph algorithms are powerful tools that help solve real-world problems like finding the cheapest way to connect cities or the shortest route between locations. You'll master three essential algorithms that appear frequently in exams and have practical applications in everything... Show more

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Minimum Spanning Trees

Top Tip: Kruskal's focuses on edges globally, whilst Prim's grows locally from your current tree - both give the same minimum weight!

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Distance Matrices and Shortest Paths

Remember: Dijkstra's only works with non-negative weights - negative edges would break the algorithm's logic!

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Finding All Shortest Paths

Pro Strategy: Floyd's algorithm is perfect for small graphs where you need all shortest paths - it's more efficient than running Dijkstra's from every vertex!