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In breadth-first and depth-first search, an undiscovered node is marked discovered when it is first encountered, and marked processed when it has been completely searched. At any given moment, several nodes might be simultaneously in the discovered state. (a) Describe a graph on n vertices and a particular starting vertex v such that Θ(n) nodes are simultaneously in the discovered state during a breadth-first search starting from v. (b) Describe a graph on n vertices and a particular starting vertex v such that Θ(n) nodes are simultaneously in the discovered state during a depth-first search starting from

In breadth-first and depth-first search, an undiscovered node is marked discovered when it is first encountered, and marked processed when it has been completely searched. At any given moment, several nodes might be simultaneously in the discovered state. (a) Describe a graph on n vertices and a particular starting vertex v such that Θ(n) nodes are simultaneously in the discovered state during a breadth-first search starting from v. (b) Describe a graph on n vertices and a particular starting vertex v such that Θ(n) nodes are simultaneously in the discovered state during a depth-first search starting from

Database System Concepts
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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In breadth-first and depth-first search, an undiscovered node is marked discovered when it is first encountered, and marked processed when it has been completely searched. At any given moment, several nodes might be simultaneously in the discovered state.

(a) Describe a graph on n vertices and a particular starting vertex v such that Θ(n) nodes are simultaneously in the discovered state during a breadth-first search starting from v.


(b) Describe a graph on n vertices and a particular starting vertex v such that Θ(n) nodes are simultaneously in the discovered state during a depth-first search starting from v.

# Expression Tree and DAG Representation The image illustrates two different representations of the mathematical expression \(2 + 3 \times 4 + (3 \times 4) / 5\): as a tree and as a Directed Acyclic Graph (DAG). ## Tree Representation (Left Diagram) - **Root Node (Top Level):** The root node is a plus sign (+), which indicates the main operation of addition. - **Second Level:** - The left child of the root node is a division sign (/), representing the operation \((3 \times 4) / 5\). - The right child is another plus sign (+), representing the addition of \(2 + 3 \times 4\). - **Third Level:** - Under the division node (/), there are two children: - The left child is a multiplication sign (*), representing \(3 \times 4\). - The right child is the number 5. - Under the second plus node (+): - The left child is the number 2. - The right child is a multiplication sign (*), denoting \(3 \times 4\). - **Fourth Level:** - Under each multiplication node (*), there are two numbers: 3 and 4. These represent the operands of the multiplication operation. ## DAG Representation (Right Diagram) - **Root Node (Top Level):** The root node is a plus sign (+), representing the entire expression. - **Second Level:** - The left child of the root node is another plus sign (+), for \(2 + 3 \times 4\). - The right child is a division sign (/), for \((3 \times 4) / 5\). - **Third Level (Multiple Parents):** - From the plus node on the left side: - The left child is the number 2. - The right child is a multiplication sign (*), which has two parent nodes (the plus node on the left and the division node on the right). - From the division node on the right: - The left child is the shared multiplication node (*). - The right child is the number 5. - **Fourth Level:** - The shared multiplication node (*) has two children: the numbers 3 and 4, representing the operands for \(3 \times 4\). By using a DAG,
Transcribed Image Text:# Expression Tree and DAG Representation The image illustrates two different representations of the mathematical expression \(2 + 3 \times 4 + (3 \times 4) / 5\): as a tree and as a Directed Acyclic Graph (DAG). ## Tree Representation (Left Diagram) - **Root Node (Top Level):** The root node is a plus sign (+), which indicates the main operation of addition. - **Second Level:** - The left child of the root node is a division sign (/), representing the operation \((3 \times 4) / 5\). - The right child is another plus sign (+), representing the addition of \(2 + 3 \times 4\). - **Third Level:** - Under the division node (/), there are two children: - The left child is a multiplication sign (*), representing \(3 \times 4\). - The right child is the number 5. - Under the second plus node (+): - The left child is the number 2. - The right child is a multiplication sign (*), denoting \(3 \times 4\). - **Fourth Level:** - Under each multiplication node (*), there are two numbers: 3 and 4. These represent the operands of the multiplication operation. ## DAG Representation (Right Diagram) - **Root Node (Top Level):** The root node is a plus sign (+), representing the entire expression. - **Second Level:** - The left child of the root node is another plus sign (+), for \(2 + 3 \times 4\). - The right child is a division sign (/), for \((3 \times 4) / 5\). - **Third Level (Multiple Parents):** - From the plus node on the left side: - The left child is the number 2. - The right child is a multiplication sign (*), which has two parent nodes (the plus node on the left and the division node on the right). - From the division node on the right: - The left child is the shared multiplication node (*). - The right child is the number 5. - **Fourth Level:** - The shared multiplication node (*) has two children: the numbers 3 and 4, representing the operands for \(3 \times 4\). By using a DAG,
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