Catálogo de cuentas

The flexible-DSDP is said to be correct, if no fault-free host Si ∈ S misdiagnoses the

correct state of any other host in the MANET, at the end of the diagnosis session. The following lemmas are defined and proved to support the correctness of the protocol.

Lemma 4.1. Let the undirected graph C(S, Lt_{) represents the topology of the MANET,}

where S is the set of hosts, and Lt _{is the set of logical links among them at time t.}

Assume that a ST covering all hosts is maintained. If Si ∈ S is disconnected from ST

then it is reconnected to the same during Maintenance phase.

Proof. Once Si finds that P arentSi is no longer in its neighborhood, it sends a

Reconnect _{message (line 4, Algorithm 4.1) to all its neighbors. Neighbors except} its own children confirm their wishes to adopt Si by sending AddParent messages

(line 25, Algorithm 4.3). Upon receiving these messages, Si adds the one having

lowest depth in ST, say Sj, as parent and sends an AddChild message to it (line 33,

Algorithm 4.3). Host Sj then includes Si in its children list (line 10, Algorithm 4.3).

Thus node Si is reconnected to the ST.

Lemma 4.2. Let S, the set of mobiles and Lt_{, the set of logical links at time t,}

collectively define an undirected graph C(S, Lt_{), that represents the communication}

graph of the MANET. At the end of the Comparison phase for round x, the actual fault status of each host except permanent hard faulty ones, is known to at least one fault-free host.

Proof. A fault-free host Si, irrespective of whether it is dynamic or stable, initiates

its diagnosis once it receives a message of Result type from any of its neighbors. At this point, it broadcasts a Result message after computing the common test task T . Upon receiving this message, its fault-free neighbor, say Sj, can deduce its correct

status immediately if Si has already received the same result from some other host

and not processed yet; otherwise, the actual status can be deduced after time-out occurs (line 17, Algorithm 4.4). Same reasoning also applies when Si is soft faulty

dynamic or stable host. In this case, Si sends the Result message with incorrect

result and the fault-free neighbor Sj after receiving it can deduce the status after

time-out. Note that the hard faulty hosts are detected after all local diagnostics have been sent to the initiator of the ST. The initiator then declares the undiagnosed hosts to be hard faulty (line 20, Algorithm 4.5).

Lemma 4.3. Let a MANET is represented by an undirected graph C(S, Lt_{). The set}

of hosts in MANET is denoted as S and the set of logical links among them at time t is denoted as Lt_{. After Comparison phase, if a node is found to be connected to a}

faulty parent in the ST then it is reconnected to a fault-free parent during Repairing phase.

Proof. If a node discovers its parent to be faulty then it triggers the Repairing phase by invoking Algorithm 4.1. The working of Repairing phase (Algorithm 4.3) is dis- cussed previously. More specifically, the second condition, i.e., (Sj ∈ F/ Si) in lines 9,

20, and 30 of this algorithm helps Si in choosing a fault-free parent.

Lemma 4.4. Let the topology of a MANET at an instance of time t is represented by an undirected graph C(S, Lt_{), where S and L}t _{respectively represents the set of mobiles}

and the set of logical links at time t. The local diagnostic view of any fault-free host Si ∈ S reaches the initiator correctly during Dissemination phase.

Proof. Any host Si can verify if it is a leaf node in the ST. If ChildrenSi is

NULL, then it starts disseminating its local view to its parent, say Sj. Node Sj

keeps track of the children who has already sent the local diagnostic through a list ChildrenSentLDSj. Here, three scenarios need to be clarified. First, if Si is a stable

neighbor of Sj then Si ∈ ChildrenSj, and Sj includes Si in ChildrenSentLDSj. Sec-

ond, if Si has sent its local view to Sj but has been migrated to some other parent

then Si ∈ ChildrenSentLDSj but Si ∈ Children/ Sj. Third, if Sihas sent its local view

to its old parent, but now has been migrated to Sj then Si ∈ ChildrenSentLDSj and

Si ∈ ChildrenSj. This is due to the fact that if host Si chooses a new parent Sj then

it transmits a boolean field SentLDSi in the AddChild message indicating if it has

already sent its local view to the old parent or not. If it has done so then Sj includes Si

in ChildrenSentLDSj (line 12, Algorithm 4.3). So (ChildrenSj∩ChildrenSentLDSj)

represents the set of children of Sj who have already sent their local diagnostics. Host

Sj collects local diagnostics from all its children and transmits a combined diagnostic

message to its parent. This process continues and the initiator collects all the local diagnostic views.

Theorem 4.1. (Proof of Correctness) Let the set of hosts in the MANET and the set of logical links between them are represented by S and Lt _{respectively. The topology}

of MANET at time t is defined by an undirected graph C(S, Lt_{). Every fault-free host}

in the ST correctly receives the global diagnostic information about the fault status of all other hosts in the MAENT.

Proof. Lemma 1 and 3 conveys that the ST is always maintained connected and each node has a fault-free parent in the tree. In Lemma 2, it is proved that each host except the hard faulty ones is correctly diagnosed by at least one fault-free host in the MANET. The initiator receives the local diagnostic messages from all fault-free mobiles correctly. This is proved in Lemma 4. If the initiator finds a node to be faulty in all r rounds, then it is categorized to be permanently faulty and is added to the global permanent faulty set (line 28, Algorithm 4.5). Similarly, if a node is found to be faulty in no round, then it is added to the global fault-free set. However, if the node is found to be faulty in some rounds and fault-free in others, then it is added to the global intermittent faulty set. These three sets constitute the global diagnostic view. The initiator then transmits the global view to its children. Upon receiving the global view each node relays it to the nodes in the next lower level in the ST, and sets

global diagnostic. This process continues until the global diagnostic message reaches the leaf nodes.

A special case must be clarified here. Consider a scenario where the parent of a node Si, say Sj, has not yet disseminated the global view. If Si finds that Sj is not

in the communication range, then it seeks for a new parent, say Sk. Now, if Sk has

already sent the global view, then Si will not receive the global view. To handle this

situation, the node Si intimates Sk if it has already received the global view or not.

This is accomplished by transmitting the boolean variable ReceivedGDSj through

AddChild _{message. Node Sk then retransmits the global view to Si. Thus, each} fault-free node receives the global diagnostic view correctly.