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Jack has a complete binary tree with depth DD and N=2D+1-1N=2D+1-1 nodes (numbered 11 through NN). Node 11 is the root and each edge is either white or black. The shape of the tree is as follows: 2 5 7 Let's number the levels of the tree 00 through DD from top to bottom. Each node at an odd level is connected to its parent with a white edge (8,g, edges 1-21-2 and 1-31-3), while each node at an even level is connected to its parent with a black edge (e.g. edges 2-42-4, 2-52-5, 3–63-6, 3-73-7). Jack's friend Alice wants to pick two distinct nodes Uy and Vy and a colour CC (white or black) uniformly randomly and add an edge between the nodes Uy and Vy with colour CC. It is allowed for two parallel edges to exist between nodes Uy and Vx, after this operation. A strip is an alternating cycle of white and black edges. Each vertex may appear any number of times in a strip. Find the probability that a strip is created when Alice adds an edge to the graph. It can be proved that this probability can be expressed as a fraction PQPQ, where PP and QQ are positive integers and QQ is coprime with 1,000,000,0071,000,000,007. You should compute P-Q-1P-Q-1 modulo 1,000,000,0071,000,000,007, where Q-1Q-1 denotes the multiplicative inverse of QQ modulo 1,000,000,0071,000,000,007.

Jack has a complete binary tree with depth DD and N=2D+1-1N=2D+1-1 nodes (numbered 11 through NN). Node 11 is the root and each edge is either white or black. The shape of the tree is as follows: 2 5 7 Let's number the levels of the tree 00 through DD from top to bottom. Each node at an odd level is connected to its parent with a white edge (8,g, edges 1-21-2 and 1-31-3), while each node at an even level is connected to its parent with a black edge (e.g. edges 2-42-4, 2-52-5, 3–63-6, 3-73-7). Jack's friend Alice wants to pick two distinct nodes Uy and Vy and a colour CC (white or black) uniformly randomly and add an edge between the nodes Uy and Vy with colour CC. It is allowed for two parallel edges to exist between nodes Uy and Vx, after this operation. A strip is an alternating cycle of white and black edges. Each vertex may appear any number of times in a strip. Find the probability that a strip is created when Alice adds an edge to the graph. It can be proved that this probability can be expressed as a fraction PQPQ, where PP and QQ are positive integers and QQ is coprime with 1,000,000,0071,000,000,007. You should compute P-Q-1P-Q-1 modulo 1,000,000,0071,000,000,007, where Q-1Q-1 denotes the multiplicative inverse of QQ modulo 1,000,000,0071,000,000,007.

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
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Question 15: Code the given problem using python Jack has a complete binary tree with depth DD and N=2D+1-1N=2D+1-1 nodes (numbered 11 through NN). Node 11 is the root and each edge is either white or black. The shape of the tree is as follows: 5 6 7 Let's number the levels of the tree 00 through DD from top to bottom. Each node at an odd level is connected to its parent with a white edge (eg, edges 1-21-2 and 1-31-3), while each node at an even level is connected to its parent with a black edge (e.g. edges 2-42-4, 2-52-5, 3-63-6, 3-73-7). Jack's friend Alice wants to pick two distinct nodes Uy and Vy and a colour Cc (white or black) uniformly randomly and add an edge between the nodes Uy and Vx with colour CC. It is allowed for two parallel edges to exist between nodes Uy and V after this operation. A strip is an alternating cycle of white and black edges. Each vertex may appear any number of times in a strip. Find the probability that a strip is created when Alice adds an edge to the graph. It can be proved that this probability can be expressed as a fraction PQPQ, where PP and QQ are positive integers and QQ is coprime with 1,000,000,0071,000,000,007. You should compute P-Q-1P-Q-1 modulo 1,000,000,0071,000,000,007, where Q-1Q-1 denotes the multiplicative inverse of QQ modulo 1,000,000,0071,000,000,007. Input: 1 2 Output: 142857144
Transcribed Image Text:Question 15: Code the given problem using python Jack has a complete binary tree with depth DD and N=2D+1-1N=2D+1-1 nodes (numbered 11 through NN). Node 11 is the root and each edge is either white or black. The shape of the tree is as follows: 5 6 7 Let's number the levels of the tree 00 through DD from top to bottom. Each node at an odd level is connected to its parent with a white edge (eg, edges 1-21-2 and 1-31-3), while each node at an even level is connected to its parent with a black edge (e.g. edges 2-42-4, 2-52-5, 3-63-6, 3-73-7). Jack's friend Alice wants to pick two distinct nodes Uy and Vy and a colour Cc (white or black) uniformly randomly and add an edge between the nodes Uy and Vx with colour CC. It is allowed for two parallel edges to exist between nodes Uy and V after this operation. A strip is an alternating cycle of white and black edges. Each vertex may appear any number of times in a strip. Find the probability that a strip is created when Alice adds an edge to the graph. It can be proved that this probability can be expressed as a fraction PQPQ, where PP and QQ are positive integers and QQ is coprime with 1,000,000,0071,000,000,007. You should compute P-Q-1P-Q-1 modulo 1,000,000,0071,000,000,007, where Q-1Q-1 denotes the multiplicative inverse of QQ modulo 1,000,000,0071,000,000,007. Input: 1 2 Output: 142857144
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