solve the following; no need for code; just written solution & answers with clarification

For illustration purposes. we use the following network:

$$edges $=[$

$(10,11),(11,12),(12,13),(13,14),(14,10),$

$(10,9),(9,7),(7,8),(8,4),(4,3),(3,1),(1,5),(5,6),(6,10),$

$(6,9),(6,7),(6,2),(6,\text{'}$s$\text{'}),$

$(\text{'}$s$\text{'},1),(\text{'}$s$\text{'},2),(\text{'}$s$\text{'},3),(\text{'}$s$\text{'},4),(\text{'}$s$\text{'},5),(\text{'}$s$\text{'},6),$

$(2,4),(2,6),(2,7),(2,8),(2,\text{'}$s$\text{'})$

$]$

Centralized approaches:

In the data Warehouse approach all sensor readings are first sent individually to the sink. Then, the latter performs data analytics on the received readings. In the in$-$network processing, rather sending all the readings to the sink, intermediary nodes in the spanning tree can perform data processing locally$($i$.$e$.,$selecting the max, min, etc.$)$before forwarding the result upper node. In both approaches, it is necessary to construct a spanning tree.

$1.$ Construct one possible spanning tree to be used for this example$($for consider Node S as the sink$).$

$2.$Calculate the corresponding cost in term of number of communications

$3.$ We consider the following request: select min$($temp$)$from sensors

$3\mathrm{.}1.$Give the average cost in term of number of hops for the evaluation of this request using warehouse approach

$3\mathrm{.}2$ Same evaluation using in$-$network processing approach

$4.$What are the main drawbacks of centralized approaches?

Serial approach:

In the serial approaches, the processing usually starts from one node, called starting node$($i$.$e$.,$Node$\backslash ($S$\backslash )$in our example$).$Each node receiving the message will, at its turn, process the received message and select the next hop. This is why we called them serial approaches.

GBT Algorithm if composed of tow rules, greedy Rule$($GR$)$and boundary Traversal Rule$($BTR$),$used by each node holding the query message. The query starts from the sink.

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Greedy Rule $($GR$)$: This rule is applied first by the current node and states that among unvisited neighbors, the next hop is the closest one to the SINK. In other words, based on the coordinates of the SINK, available in the message, the current node computes the distances between the SINK and all its unvisited neighbors and chooses the one with the smallest distance. It sends then the message next hop. When a node is not able to apply the greedy rule $($i$.$e$.,$ all its neighbors have been it applies the second rule, known as the Boundary Traversal Rule $($BTR$).$

Boundary Traversal Rule $($BTR$)$:The objective behind this rule is to allow the message to travel along of the visted region to nodes or regions of the network that have not yet been visited. iggers the boundary traversal process initiates the curved stick $($outside until a neighbor is hit

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This neighbor is then selected as the next hop. Before sending to it the message, the current node updates the flag RF in order to indicate that the current rule is the boundary traversal rule. This kind of message, which does not change the current query value, is known as the Boundary Traversal Rule $($BTR$)$ message. The next hop will apply the same rule whenever the greedy rule is not possible.

$1.$ Provide the sequence of visited nodes by GBT on the example as well as their nature $($GR or BTR$).$

$2.$ How can GBT stops?

$3.$Provide an evaluation of GBT in term of number of communications on the example above.

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