COMPSCI220 Chap.6 Weighted Graphs, Shortest Paths and Spanning Trees

Weighted Graphs, Shortest Paths and Spanning Trees

Define edge relaxation

The course material gives this chapter a concrete anchor: Weeks 10-12 in the schedule cover weighted graphs, single-source and all-pairs shortest paths and spanning trees, so their objectives and assumptions must be compared explicitly.

That edge relaxation anchor controls how shortest path is explained and how minimum spanning tree is tested in changed practice.

Weighted Graphs, Shortest Paths and Spanning Trees turns edge relaxation, shortest path and minimum spanning tree into executable reasoning.

The chapter's practical target is to execute relaxation and distinguish a shortest-path tree from a minimum spanning tree, so every explanation should connect syntax to program state, control flow and observable output.

Treat edge relaxation as a precise program object, not a loose label. Identify the value or responsibility of edge relaxation before execution, then trace what can read it, change it or depend on it.

This makes state changes visible before they become debugging guesses.

Use shortest path to explain the program's next move. Work through one representative shortest path input by hand and name the branch, iteration or call that follows.

If the shortest path trace cannot be stated, the code may run by accident rather than by understood design.

Trace shortest path

Bring in minimum spanning tree as the test of structure.

Compare normal, boundary and invalid inputs for minimum spanning tree; state the expected behaviour first; then use the mismatch between expectation and result to localise the defect.

For the application — execute relaxation and distinguish a shortest-path tree from a minimum spanning tree — write the smallest complete example that exposes the rule.

Explain why the minimum spanning tree result works, what would break it and how the program should signal or recover from that failure.

Before running an edge relaxation example, make a trace table with the important state before and after each operation. Include the value associated with edge relaxation, the control decision governed by shortest path and the output or object affected by minimum spanning tree.

The edge relaxation table turns an unexplained result into a sequence that can be tested one transition at a time.

Test three inputs: an ordinary case, a boundary case and an invalid case. State the expected minimum spanning tree result for each before execution, then compare it with what the program actually does.

A useful test of shortest path isolates one rule; changing several conditions at once cannot reveal which condition caused the failure.

Test with minimum spanning tree

Practise explaining the solution without reading the code.

For COMPSCI220, name the data representation, the control flow, the responsibility of each function or class and the reason the chosen design supports execute relaxation and distinguish a shortest-path tree from a minimum spanning tree.

This minimum spanning tree rehearsal matters when a written test or interview asks why the program works rather than whether it produces one correct output.

A complete response should make the task visible before the detail: identify what must be decided, define the relevant terms, connect the evidence to shortest path, and use minimum spanning tree to test the result.

The final sentence about minimum spanning tree should answer the question actually asked rather than merely repeat the topic.

The controlling limit is specific: Algorithm correctness depends on edge-weight and graph assumptions, and a minimum spanning tree does not minimise the path between every pair.

Keep that minimum spanning tree limit beside the worked example, because it separates a careful COMPSCI220 answer from one that sounds confident but claims more than the task or evidence supports.

For revision, retrieve edge relaxation, shortest path and minimum spanning tree without notes, explain their relationship aloud, then complete a changed version of the application: execute relaxation and distinguish a shortest-path tree from a minimum spanning tree.

Record the first failed shortest path reasoning move and repair it before attempting another case.

What this chapter covers

• 01

  edge relaxation

• 02

  shortest path

• 03

  minimum spanning tree

• 04

  Applying edge relaxation

• 05

  Limits of shortest path and minimum spanning tree

AskSia practice: apply Weighted Graphs, Shortest Paths and Spanning Trees

Key terms

edge relaxation

An update improving a tentative distance when a route through an examined edge is cheaper. Use this definition when the task is to execute relaxation and distinguish a shortest-path tree from a minimum spanning tree.

shortest path

A path with minimum total edge weight among those allowed by the model and algorithm assumptions. Use this definition when the task is to execute relaxation and distinguish a shortest-path tree from a minimum spanning tree.

minimum spanning tree

A minimum-total-weight acyclic edge set connecting every vertex in a connected undirected graph. Use this definition when the task is to execute relaxation and distinguish a shortest-path tree from a minimum spanning tree.

Weighted Graphs, Shortest Paths and Spanning Trees FAQ

Exam move

Reconstruct the relationship among edge relaxation, shortest path and minimum spanning tree; complete the chapter application without notes; then test the result against this limit: Algorithm correctness depends on edge-weight and graph assumptions, and a minimum spanning tree does not minimise the path between every pair.

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