Question

Sixteen stations, numbered I through 16, are contending for the use of a shared channel by using the adaptive tree-walk protocol. If all the stations whose addresses are prime numbers suddenly become ready at ance, bow many bit slots are needed to resolve the contention?

Short answer

3 bit slots are needed to resolve contention using a binary tree of depth 3.

Step-by-step solution

Identify Prime Numbered Stations

First, identify which of the stations, numbered from 1 to 16, have prime numbers as their identifiers. These are the stations that will contend for the channel. The prime numbers between 1 and 16 are 2, 3, 5, 7, 11, 13, and 17, making a total of 7 stations.

Understand Adaptive Tree-Walk Protocol

The adaptive tree-walk protocol is used to resolve contention by organizing the stations into a binary tree structure. Each level of the tree checks groups of stations to determine if there are any contenders.

Determine Tree Depth for 7 Contenders

To resolve contention for 7 stations using a binary tree, assess how many levels the tree needs. A complete binary tree with 8 leaves can accommodate up to 7 stations (since a complete tree of depth 3 has \(2^3 = 8\) leaves). Thus, the tree will have a depth of 3.

Calculate Bit Slots Required

The adaptive tree-walk protocol resolves contention by allocating one bit slot per level of the tree until a single contender remains at any branch. With a binary tree of depth 3, it will take 3 bit slots to descend the tree and resolve contention among the 7 stations.

Key concepts

Contention Resolution

When multiple stations vie for access to a shared communication channel, a plan must be used to determine who gets the chance to communicate. This scenario leads to what's known as *contention*. Successfully managing this contention is called *contention resolution*.
In the example we are exploring, 16 stations are competing, but only those identified by prime numbers are involved in the contention. Various protocols address contention to ensure fair access, with the adaptive tree-walk protocol being one.
This protocol helps to systematically reduce conflicts by organizing contenders into a tree structure. It ensures that every station eventually gets a fair opportunity to access the channel without overwhelming the system.

Prime Numbers

Prime numbers are those natural numbers greater than 1 that have no divisors other than 1 and themselves. They play a unique role in mathematics due to their indivisibility.
In the context of this exercise, the contenders are identified by their prime number designation. Among the 16 stations, we identify the prime-numbered ones as 2, 3, 5, 7, 11, 13, and 17. Although 17 exceeds our initial range of 16, it isn't used in a direct calculation here but serves to affirm which stations are contenders.
Prime numbers are crucial to this exercise because they define the stations that will participate in the contention resolution. Hence, knowing which stations are prime-numbered is essential before attempting to solve the contention problem.

Binary Tree

A binary tree is a data structure where each node has at most two children, called left and right children. This hierarchical structure is efficient for organizing data to facilitate operations such as searching, addition, and deletion.
In our scenario, the binary tree organizes the contending stations. Beginning at the root, each level of the tree divides the contenders, helping to pinpoint which group contains active stations. As a result, a binary tree aids in systematically resolving contention without checking every station individually.
The tree’s depth impacts how quickly contention is resolved. For 7 prime-numbered stations, our binary tree requires a depth of 3 because a tree of this depth provides up to 8 leaves, efficiently splitting the 7 contending stations.

Bit Slots

Bit slots refer to the time or allocation used in protocols to handle signals. Within this exercise, bit slots are utilized as part of the adaptive tree-walk protocol for contention resolution.
Each layer in the binary tree represents a decision point, and thus, each layer requires a bit slot. The number of bit slots needed directly corresponds to the tree’s depth. Since our binary tree has a depth of 3, we will need 3 bit slots.
These slots represent attempts to progress through the tree and identify or isolate stations engaged in contention. Therefore, they are a crucial part of ensuring that every active station gets recognized and the contention can be resolved efficiently.