Question

Suppose host \(\mathrm{A}\) is sending to a multicast group; the recipients are leaf nodes of a tree rooted at A with depth \(N\) and with each nonleaf node having \(k\) children; there are thus \(k^{N}\) recipients. (a) How many individual link transmissions are involved if A sends a multicast message to all recipients? (b) How many individual link transmissions are involved if A sends unicast messages to each individual recipient? (c) Suppose A sends to all recipients, but some messages are lost and retransmission is necessary. Unicast retransmissions to what fraction of the recipients is equivalent, in terms of individual link transmissions, to a multicast retransmission to all recipients?

Short answer

Multicast = \(k^N\); Unicast = \(k^N \times N\); \(f = \frac{1}{N \times (k - 1)}.\)

Step-by-step solution

Understand the Tree Structure

Note that the tree is rooted at A with depth \(N\) and each nonleaf node has \(k\) children. Therefore, every nonroot node has \(k\) connections branching from it.

Calculate the Number of Links for Multicast

For a tree of depth \(N\), where each nonleaf node has \(k\) children, the number of individual link transmissions required for multicast equals the total number of links in the tree. For a complete \(k\)-ary tree of depth \(N\) rooted at A, there are \(1 + k + k^2 + ... + k^{N}\) nodes. Thus, the number of edges (links) is \((1 + k + k^2 + ... + k^{N-1})\). Therefore, the total number of link transmissions is \ \text{Total Multicast Transmissions} = \frac{k^N - 1}{k - 1}. For large N, it can be approximated by \(k^N\).

Calculate the Number of Links for Unicast

For unicast, node A sends an individual message to each of the \(k^N\) recipients. Each transmission from A traverses \(N\) links (depth of the tree). Therefore, the number of link transmissions required is \(k^N \times N\).

Unicast Retransmissions Equivalence

To find the fraction of recipients that equates unicast retransmissions to multicast retransmissions, we need to equate the number of link transmissions involved. Let \(f\) be the fraction of recipients that require unicast retransmissions. The link transmissions involved in unicast retransmissions are \(f \times k^N \times N\). Similarly, the multicast retransmissions lead to \ \text{Multicast Retransmissions} = \frac{k^N - 1}{k - 1}. Setting these equal,\ \ f \times k^N \times N = \frac{k^N - 1}{k - 1}. \ \ solve for f,\ \ f = \frac{1}{N \times (k - 1)}.

Key concepts

Multicast Transmission

In networking, multicast transmission is a method where a single data packet is sent from a source to multiple recipients. Unlike unicast transmission, where each recipient gets an individual copy of the packet, multicast transmission allows for one copy of the data packet to be distributed simultaneously to multiple recipients. This is highly efficient in terms of bandwidth usage because it drastically reduces the total number of packets sent across the network.
In the context of the given problem, the tree has a root at point A, and by using multicast, the message sent from A will traverse through the tree's links, ultimately reaching all leaf nodes (recipients). This is calculated using the formula for a complete k-ary tree of depth N:

• The number of link transmissions involved is the total number of edges in the tree.
• For a tree rooted at A with depth N and each node having k children, the total link transmissions can be derived using the geometric series sum.
• This gives us the formula: \[\text{Multicast Transmissions} = \frac{k^N - 1}{k - 1}\]. This considerably reduces the number of required link transmissions compared to unicast transmission.

Unicast Transmission

Unicast transmission involves sending a data packet individually to each recipient. In this method, if a node A needs to send a message to multiple recipients, it will send separate packets for each recipient. This method ensures that every recipient gets an exact copy of the message. In the problem scenario, considering the tree structure with depth N and k children per nonleaf node, if node A uses unicast to send messages, each of the \(k^N\) recipients will receive an individual message.

• The number of link transmissions for unicast is thus significantly higher.
• Each message to a recipient will traverse N links. Hence, for \(k^N\) recipients, the total number of link transmissions required is \[k^N \times N\].

Unicast transmission is less efficient than multicast, especially in scenarios involving a large number of recipients, as it results in a higher burden on the network's bandwidth.

Tree Structure in Network

Understanding the tree structure in a network is crucial for comprehending how data traverses from a root node to various leaf nodes (recipients). In network communication, a tree structure helps in effectively managing and optimizing the flow of data.

• A tree consists of nodes (vertices) connected by edges (links).
• The root node is the starting point, and from there, multiple branches lead to other nodes until the leaf nodes are reached.
• In the given problem, the tree has a root at A, the depth is N, and each nonleaf node has k children, forming a complete k-ary tree.

Each nonleaf node has connections equal to the number of its children (k). Calculating the number of link transmissions in such a tree involves summing up the links at each level of the tree. For multicast transmission, we consider the entire tree, while for unicast, we look at the path each message takes from A to a recipient.

• This structured approach helps in efficiently managing both multicast and unicast transmissions and is fundamental in network design and optimization.