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Transitive Closure: Roy-Floyd-Warshall Algorithm Overview

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Transitive Closure: Roy-Floyd-Warshall Algorithm Overview

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Questions and Answers

What is the main focus of the Roy-Floyd-Warshall algorithm?

Implementing adjacency functions for immediate predecessors

Bypassing matrices and arrays in favor of a direct mathematical model (correct)

Using matrices and arrays for efficient computation

Iterating over elements to create a sparse relation

In the transitive closure algorithm, what does step 0 do?

Applies the adjacency functions

Connects all paths over intermediate elements

Starts with the full transitive closure

Begins with the original relation (correct)

Who is the earliest known describer of the algorithm often named after Floyd and Warshall?

Warshall

Floyd

Roy-Floyd

B.Roy (correct)

How is the transitive closure algorithm implemented in functional programming languages?

Maintaining a cumulative approximation of the result Signup and view all the answers

What aspect makes the Roy-Floyd-Warshall algorithm particularly suitable for sparse relations?

Bounded closure property Signup and view all the answers

In which alternative formulation does the Roy-Floyd-Warshall algorithm not use matrices and arrays?

Second formulation Signup and view all the answers

What is the purpose of the theorem transitive-closure-correctness?

To demonstrate the correctness of an alternative formulation of a transitive closure algorithm Signup and view all the answers

What does the inductive definition in the alternative formulation represent?

A logic-program imitation Signup and view all the answers

Which statement is true about the relation between Steps and steps in the alternative formulation?

Steps and steps have a direct equivalence in the algorithm Signup and view all the answers

What is the main focus of the core algorithm expressed in a declarative manner in the text?

Describing the algorithm using inductive definitions Signup and view all the answers

What is the base case for the inductive-set Clos?

(x, y) ∈ Clos rel n Signup and view all the answers

In the proof of Clos-closure, what does the theorem state about the relation Clos rel n?

Clos rel n captures the bounded closure. Signup and view all the answers

What does the lemma Clos-Suc state about the relation Clos rel n?

It gets extended to include more elements. Signup and view all the answers

What is the significance of the step case in the inductive-set Clos?

It allows elements to be added to the closure set iteratively. Signup and view all the answers

How is the theorem Clos-closure related to the bounded closure concept?

It ensures a bound on closure size is maintained. Signup and view all the answers

What role does the lemma Clos-Suc play in relation to the growth of closures?

It helps in incrementally expanding the closure set. Signup and view all the answers