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the processors and nodes are consecutively indexed, that is, V = {n1 , n2 , , n|V| } and Pc0, = {P1 , P2 , , P|Pc0, | } Algorithm 6 Optimal Scheduling G = (V, E, w) on Pc0, , with |Pc0, | |V| Insert {n V} in topological order into sequential list L for each ni in L do DRT 0 for each nj pred(ni ) do DRT max{DRT , tf (nj )} end for ts (ni ) DRT ; tf (ni ) ts (ni ) + w(ni ) proc(ni ) Pi , Pi Pc0, end for Theorem 44 (Optimal Scheduling on Unlimited Processors) Let G = (V, E, w) be a task graph and Pc0, a parallel system, with |Pc0, | |V| Algorithm 6 produces an optimal schedule Sopt of G on Pc0, Proof The produced schedule Sopt is feasible: (1) each node is scheduled on a distinct processor, hence the processor constraint (Condition 41) is always adhered to; and (2) the nodes comply with the precedence constraints (Condition 42), since the start time of each node is not earlier than its DRT The correct calculation of the DRTs is guaranteed by the topological order in which the nodes are processed The produced schedule Sopt is optimal, because each node starts execution at the earliest possible time, the time when its data is ready The DRT cannot be reduced, since it is not a function of the processors (De nition 416); that is, scheduling the involved nodes on other processors does not change the DRT Algorithm 6 has a time complexity of O(V + E): calculating a topological order is O(V + E) (Algorithm 4) To calculate the DRT of each node (O(V)), every predecessor is considered, which amortizes to O(E) predecessors for the entire graph Hence, this part is also O(V + E) While Algorithm 6 is simple, it does not utilize the processors in an ef cient way Usually, there will be a schedule S that uses less than |V| processors, but whose schedule length is also optimal, that is, sl(S) = sl(Sopt ) Yet again, nding the minimum number of processors for achieving the optimal schedule length is an NP-hard problem (Darte et al [52]) A simple heuristic to educe the number of processors, while maintaining the optimal schedule length, is linear clustering, which is presented in Section 532 Used Processors In Lemma 42 of Section 422 it was stated that the more processors are available for the task graph to be scheduled on, the smaller (or equal) the optimal schedule length The situation becomes less trivial and more interesting when one considers the used processors (De nition 411) in a schedule.

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is called The current version of is attached with updated members but the original timestamp The original timestamp takes care of the optimistic concurrency check If the apping is set up to skip the optimistic concurrency check, the current version is all that is called is needed for an update when In many multitier applications, the entire entity often is not sent across tiers for simplicity, interoperability, or privacy For example, a supplier may de ne a data contract for a web service that differs from the Order entity used on the middle tier Likewise, a web page may show only a subset of the members of an Employee entity Hence, the multitier support also accommodates such cases Only the members belonging to one or more of the following categories need to be transported between tiers and set before : calling Members that are part of the entity s identity Members that have been changed Members that participate in the optimistic concurrency check LINQ to SQL uses minimal updates It checks which columns are changed from the original to the current version and creates a SQL command to set just the corresponding columns So when you create an entity instance on the mid-tier for attaching, anything that is not changed and not required for the optimistic concurrency check can be skipped The remaining members can have default values in both the original and curuses a timestamp and only the rent versions For example, if an has been changed, you need to set only three properties when up an Order entity , , and the timestamp you property When attaching objects in original state, the entity graph is recursively explored for each attached entity until a known entity is reached By default, no further action is taken for attached entities For example, if you in original state, all related entities are also attach an s to effect an attached You are free to make changes to update, or you can even take action to have them inserted or deleted.

characteristics of the network in order to minimize overheads An intuitive approach for organizing into clusters is according to node mobility Clusters are not static but change as nodes move around As nodes move, they may drop out from their existing cluster and join a different cluster that is closer to them Clusters in some cases may join together or a single cluster may split into two or more clusters Typically a node might be elected in the cluster as a leader and such a node is typically called the clusterhead There are several algorithms that have been proposed in the literature for organizing nodes in clusters based on a variety of criteria, including how the clusters adapt to connectivity changes due to mobility, and how clusterheads for each cluster are elected (although certain cluster-based algorithms do not require the election of clusterheads) We will now discuss some representative clustering algorithms and will then consider how these concepts can be applied to intrusion detection We consider clustering both with and without the use of clusterheads 54311 Clustering with Clusterheads One of the heuristics for selecting clusterheads that has been proposed in [87] is the highest-degree algorithm In this algorithm, the nodes that have the largest number of direct neighbors become the clusterheads In this case, neighbors are those nodes that are within each other s transmission ranges The clusterhead selection can be implemented as follows Each node discovers through periodic hello messages its direct neighbors and the number of neighbors that each of them have Based on that information, a node selects itself as the clusterhead if it has more neighbors than its neighbors This is a very simple algorithm that has a ow rate of clusterhead change Another potential algorithm is the Graphcluster algorithm [88] The rst part of this algorithm involves forming a spanning tree by using the breadth- rst-search (BFS) tree algorithm In the BFS algorithm, a node initiates the algorithm and becomes the root If multiple nodes initiate the algorithm, the one with the lowest ID wins The root starts transmitting periodic tree discovery beacons and ooding them in the network As each node receives a beacon it transmits its own beacon which includes its id, the root s id, the id of its parent (its neighbor that is closest to the root), and its distance from the root Nodes that receive such beacons can discover the shortest path to the root and they send updated tree building beacons Eventually the tree forms One such possible tree is shown with bold lines in Figure 57 In the Graphcluster algorithm, once the tree is formed through the BFS algorithm, clusters are then formed as follows Each node counts the number of its children (it can discover those through the BFS advertisements) This information is then included in the subsequent tree building beacons These cluster members then start to be aggregated from the leaf nodes towards the root Once a node discovers that it has more than k children it forms either one cluster (if it has ,2k children) or multiple clusters by partitioning its children s subtrees into clusters Figure 58 shows one example for the formation of clusters (assuming k 3) Once nodes become part of a cluster they are not counted by their parents in the cluster formation algorithm The algorithm terminates once the cluster formation is stable After that the algorithm tries to maintain the clusters as they exist by making small changes for nodes that join or leave clusters due to mobility 54312 Clustering Without Clusterheads Algorithms have also been proposed for creating clusters that do not involve the election of an actual leader in the cluster The advantage of these approaches is that they do not require that a node take on extra.

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