precedence relations. One exam as vertices and edges represents I, then there is an edge from ver can easily be done by performing

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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II. Read each problem carefully and present an algorithm with the required running-time to solve each
problem.
1. In class we discussed that directed acyclic graphs (DAG) can be used to represent
dependency/precedence relations. One example is modeling task dependency where tasks are
represented as vertices and edges represents direct dependencies between tasks, e.g., if task T
requires task T, then there is an edge from vertex i to vertex j. Arranging tasks with respect to their
dependencies can easily be done by performing topological sort to the DAG.
a. Describe an algorithm that runs in O(n + m) time that given two tasks, T, and I₁, determines if
task T, can be performed without performing task T₁. Note that I, is not required if it is not a direct
or indirect dependency of T. Task I, is an indirect dependency of task T, if there is no edge from
T, to T, but I must appear before 7, in the topological order.
b. Describe an algorithm that runs in O(n + m) time that given a task T₁, outputs the minimum
possible position of tasks T, in any topological order.
Transcribed Image Text:II. Read each problem carefully and present an algorithm with the required running-time to solve each problem. 1. In class we discussed that directed acyclic graphs (DAG) can be used to represent dependency/precedence relations. One example is modeling task dependency where tasks are represented as vertices and edges represents direct dependencies between tasks, e.g., if task T requires task T, then there is an edge from vertex i to vertex j. Arranging tasks with respect to their dependencies can easily be done by performing topological sort to the DAG. a. Describe an algorithm that runs in O(n + m) time that given two tasks, T, and I₁, determines if task T, can be performed without performing task T₁. Note that I, is not required if it is not a direct or indirect dependency of T. Task I, is an indirect dependency of task T, if there is no edge from T, to T, but I must appear before 7, in the topological order. b. Describe an algorithm that runs in O(n + m) time that given a task T₁, outputs the minimum possible position of tasks T, in any topological order.
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