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sters, and the cyclic indices indicate different node positions in a uses cubical indices to represent a, and uses cyclic indices to represent . Consistent hash function, denoted by H, is used to generate the hash value of attributes. Locality preserving hashing function , denoted by H, is used to generate the hash value of attribute value. Thus, the ID of a resource rescID=(H,H).A node reports its available resources to the system periodically via interface Insert(rescID,rescInfo).Therefore, the information of the same attribute will be mapped to the same cluster. We call the node or the ID of the node as the root of the rescID or rescInfo. Within each cluster, each node is responsible for the information of a resource whose cyclic index falls into the ID space sector it supervises.Proposition : In LORM, given a range query [, ]for a resource where , a node that contains attribute value within [, ] must have an ID that satisfies root(H ,) ID root(H ,).Proof: In LORM with n = d nodes, a node reports its resource information using the Cycloid interface Insert((H,Ha),rescInfo). Attribute a with value will be stored in root(H,Ha) whose ID is the closest to (H,Ha). According to the locality preserving hashing, because the resource information of value v will be stored in node i that satisfies to the condition root(H ()) i root(H ()). A node uses Lookup(rescID) to query for resources,and the query is routed to the directory node for the desired resource. A multiattribute query is posed of a set of subqueries on each attribute, which are processed in parallel. For example, when a node k needs a multipleattribute resource, say CPU and 2GB memory, it sends requests Lookup(H,Hcpu,cpu, ,ip_addr(k)) and Lookup(H,mem,ip_addr(k)), which will be resolved in parallel. The queries will arrive at node a and node e, which reply to the requester node k with the requested resource information mem, ip_addr(i) where= 2 and CPU, , ip_addr(j) where =. The requester node then concatenates the results in a databaselike “join” operation based on ip addr. The results are the nodes that have desired resource by the requester. For range queries such as“”and“Freememory2GB”, in addition to responding with satisfied resource information in their own directories, node a and e forward the resource queries to their immediate successors in their own clusters. The successors repeat the same process. This process is repeated until a successor has no satisfied resource information. If the requested resource range is less than a value, then nodes forward queries to their predecessors. If the queries have lower and upper bounds such as “” and“1GBmemory2GB”, the queries will be forwarded in both directions. Cycloid has a selforganization mechanism to maintain its structure and stored objects, which helps LORM to handle dynamism with node joins and departures.IV. COMPARATIVE STUDY AND ANALYSISWe use Mercury , SWORD , MAAN as representatives of multipleDHTbased, singleDHTbased centralized and singleDHTbased decentralized classes, and analyze LORM in parison with the approaches. LORM maps resource attribute and value or string description to two levels of a hierarchical Cycloid DHT. Mercury uses multiple DHTs with one DHT responsible for each attribute and maps resource value to each DHT. SWORD maps resource information including both attribute and value in a flat DHT,and MAAN maps attribute and value separately to a flat DHT. To be parable, we use Chord for attribute hubs in Mercury, and we replace Bamboo DHT with Chord in SWORD.In Mercury, for higher efficiency of resource query, a node within one of the hubs can hold the data record while the other hubs can hold a pointer to the node. This strategy can also be applied to other methods. To make the different methods be parable, we don’t consider this strategy in the parative study. We analyze their performance in terms of structure maintenance overhead, resource information maintenance overhead, and the efficiency of resource discovery. In the analysis, we use “attribute value” to represent the locality preserving hash value of both attribute value and attribute string description. We use directory size to represent the number of resource information pieces in a directory.A . Maintenance OverheadTheorem : In a grid system with n nodes and m resource attributes, with high probability, LORM can improve the structure maintenance overhead of multipleDHTbased methods (. Mercury) by no less than m times.Proof: LORM is based on Cycloid, in which each node is responsible for maintaining d ≤ log(n) neighbors. In multipleDHTbased methods such as Mercury, each node is responsible for maintain。
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