(a) Why does the greedy algorithm work for the spanning tree...

(a) Why does the greedy algorithm work for the spanning tree problem?
(b) Show by example that the greedy algorithm could fail to find the shortest path from $s$ to $t$, by starting with the shortest edge.

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Key Concepts

Limitations of Greedy Strategies in Shortest Path Problems

Unlike MSTs, the shortest path problem does not always satisfy the greedy-choice property; choosing the shortest edge at each step can lead to a local optimum that is not globally optimal. This highlights that greedy algorithms can fail in problems where the optimal substructure is not compatible with a purely local decision process.

Minimum Spanning Tree Algorithms

Algorithms such as Kruskal's and Prim's are classical examples of greedy algorithms applied to find MSTs. They work well because the problem guarantees that locally optimal decisions (selecting the minimum edge) will combine to form a globally optimal tree.

Greedy Choice Property in MST

In the context of minimum spanning trees (MST), the greedy choice property allows the algorithm to safely choose the smallest available edge that does not form a cycle. This is ensured by the cut property, which states that the cheapest edge across any partition of the graph is part of some MST.

Greedy Algorithms

A greedy algorithm builds a solution incrementally by always making a locally optimal choice at each step, with the hope that these local optima will lead to a global optimum. Its effectiveness depends on the problem having properties like the greedy-choice property and optimal substructure.

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