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A group of 15 routers are interconnected in a centralized complete binary tree with a router at each tree node. Router icommunicates with router j by sending a message to the root of the tree. The root then sends the message back down torouter j. The mean number of hops per message, assuming all possible router pairs are equally likely is
- a)3
- b)4.26
- c)4.53
- d)5.26
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
A group of 15 routers are interconnected in a centralized complete bin...
The path length differs for nodes from each level. For a node in level 4, we have maximum no. of hops as follows
So, mean no. of hops for a node in level 4
So, mean no. of hops for a node in level 4
as we have 1, 2, 4 and 8 nodes respectively in levels 1, 2, 3 and 4 and we discard the source one
in level 4.
Similarly, from a level 3 node we get mean no. of hops,
Similarly, from a level 3 node we get mean no. of hops,
From level 2, we get mean no. of hops
And from level 1, we get, mean no. of hops
So, now we need to find the overall mean no. of hops which will be
Most Upvoted Answer
A group of 15 routers are interconnected in a centralized complete bin...
Mean number of hops per message in a centralized complete binary tree:
A centralized complete binary tree is a binary tree in which all nodes have exactly two children, except for the leaf nodes. In this tree structure, routers are interconnected, and each router communicates with another router by sending a message to the root of the tree. The root then sends the message back down to the destination router.
To find the mean number of hops per message, we need to calculate the average number of hops it takes for a message to travel from one router to another.
Let's break down the solution into steps:
Step 1: Number of routers
In the given problem, there are 15 routers interconnected in a centralized complete binary tree.
Step 2: Height of the tree
In a centralized complete binary tree, the height (h) is the maximum number of hops required to reach any leaf node from the root. In this case, since there are 15 routers, the tree height (h) can be calculated as:
h = log2(15) ≈ 3.91
Step 3: Number of levels in the tree
The number of levels in a tree is equal to the height of the tree plus 1. In this case, the number of levels in the tree is 4.
Step 4: Number of leaf nodes
In a complete binary tree, the number of leaf nodes is given by 2^h. In this case, the number of leaf nodes is 2^3 = 8.
Step 5: Number of router pairs
In a complete binary tree, there are n/2 leaf nodes, where n is the total number of nodes. In this case, there are 15/2 = 7.5 leaf nodes, which means there are 7 pairs of routers.
Step 6: Average number of hops per message
To calculate the average number of hops per message, we need to consider all possible router pairs. Each router pair will require a certain number of hops, depending on the distance between them in the tree.
In a complete binary tree, the distance between any two leaf nodes is equal to the sum of their heights. In this case, the sum of heights for all 7 router pairs is:
3 + 3 + 3 + 3 + 3 + 3 + 3 = 21
The average number of hops per message can be calculated as:
Average = Total distance / Number of router pairs = 21 / 7 = 3
Hence, the mean number of hops per message in this centralized complete binary tree is 3, which corresponds to option (a) in the given options.
A centralized complete binary tree is a binary tree in which all nodes have exactly two children, except for the leaf nodes. In this tree structure, routers are interconnected, and each router communicates with another router by sending a message to the root of the tree. The root then sends the message back down to the destination router.
To find the mean number of hops per message, we need to calculate the average number of hops it takes for a message to travel from one router to another.
Let's break down the solution into steps:
Step 1: Number of routers
In the given problem, there are 15 routers interconnected in a centralized complete binary tree.
Step 2: Height of the tree
In a centralized complete binary tree, the height (h) is the maximum number of hops required to reach any leaf node from the root. In this case, since there are 15 routers, the tree height (h) can be calculated as:
h = log2(15) ≈ 3.91
Step 3: Number of levels in the tree
The number of levels in a tree is equal to the height of the tree plus 1. In this case, the number of levels in the tree is 4.
Step 4: Number of leaf nodes
In a complete binary tree, the number of leaf nodes is given by 2^h. In this case, the number of leaf nodes is 2^3 = 8.
Step 5: Number of router pairs
In a complete binary tree, there are n/2 leaf nodes, where n is the total number of nodes. In this case, there are 15/2 = 7.5 leaf nodes, which means there are 7 pairs of routers.
Step 6: Average number of hops per message
To calculate the average number of hops per message, we need to consider all possible router pairs. Each router pair will require a certain number of hops, depending on the distance between them in the tree.
In a complete binary tree, the distance between any two leaf nodes is equal to the sum of their heights. In this case, the sum of heights for all 7 router pairs is:
3 + 3 + 3 + 3 + 3 + 3 + 3 = 21
The average number of hops per message can be calculated as:
Average = Total distance / Number of router pairs = 21 / 7 = 3
Hence, the mean number of hops per message in this centralized complete binary tree is 3, which corresponds to option (a) in the given options.
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A group of 15 routers are interconnected in a centralized complete binary tree with a router at each tree node. Router icommunicates with router j by sending a message to the root of the tree. The root then sends the message back down torouter j. The mean number of hops per message, assuming all possible router pairs are equally likely isa)3b)4.26c)4.53d)5.26Correct answer is option 'C'. Can you explain this answer?
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A group of 15 routers are interconnected in a centralized complete binary tree with a router at each tree node. Router icommunicates with router j by sending a message to the root of the tree. The root then sends the message back down torouter j. The mean number of hops per message, assuming all possible router pairs are equally likely isa)3b)4.26c)4.53d)5.26Correct answer is option 'C'. Can you explain this answer? for Computer Science Engineering (CSE) 2025 is part of Computer Science Engineering (CSE) preparation. The Question and answers have been prepared according to the Computer Science Engineering (CSE) exam syllabus. Information about A group of 15 routers are interconnected in a centralized complete binary tree with a router at each tree node. Router icommunicates with router j by sending a message to the root of the tree. The root then sends the message back down torouter j. The mean number of hops per message, assuming all possible router pairs are equally likely isa)3b)4.26c)4.53d)5.26Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Computer Science Engineering (CSE) 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A group of 15 routers are interconnected in a centralized complete binary tree with a router at each tree node. Router icommunicates with router j by sending a message to the root of the tree. The root then sends the message back down torouter j. The mean number of hops per message, assuming all possible router pairs are equally likely isa)3b)4.26c)4.53d)5.26Correct answer is option 'C'. Can you explain this answer?.
A group of 15 routers are interconnected in a centralized complete binary tree with a router at each tree node. Router icommunicates with router j by sending a message to the root of the tree. The root then sends the message back down torouter j. The mean number of hops per message, assuming all possible router pairs are equally likely isa)3b)4.26c)4.53d)5.26Correct answer is option 'C'. Can you explain this answer? for Computer Science Engineering (CSE) 2025 is part of Computer Science Engineering (CSE) preparation. The Question and answers have been prepared according to the Computer Science Engineering (CSE) exam syllabus. Information about A group of 15 routers are interconnected in a centralized complete binary tree with a router at each tree node. Router icommunicates with router j by sending a message to the root of the tree. The root then sends the message back down torouter j. The mean number of hops per message, assuming all possible router pairs are equally likely isa)3b)4.26c)4.53d)5.26Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Computer Science Engineering (CSE) 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A group of 15 routers are interconnected in a centralized complete binary tree with a router at each tree node. Router icommunicates with router j by sending a message to the root of the tree. The root then sends the message back down torouter j. The mean number of hops per message, assuming all possible router pairs are equally likely isa)3b)4.26c)4.53d)5.26Correct answer is option 'C'. Can you explain this answer?.
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