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Stefan Edelkamp (University of Dortmund, Germany) and Shahid Jabbar (University of Dortmund, Germany)

DOI: 10.4018/978-1-59904-849-9.ch076

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TopMost modern operating systems provide a general-purpose memory management scheme called *Virtual Memory* to compensate for the limited RAM. Unfortunately, such schemes pay off only when the algorithm’s memory accesses are local, i.e., it works on a particular memory address range for a while, before switching the attention to another range. Search algorithms, especially those that order the nodes on some particular node property, do not show such behaviour. They jump back and forth to pick the best node, in a spatially unrelated way for only marginal differences in the node property.

External memory algorithms are designed with a hierarchy of memories in mind. They are analyzed on an *external memory model* as opposed to the traditional von Neumann RAM model. We use the two-level memory model by Vitter and Shriver (1994) to describe the search algorithms. The model provides the necessary tools to analyze the asymptotic number of block accesses (I/O operations) as the input size grows. It consists of

*•**M*: Size of the internal memory in terms of the number of elements,*•**N*>>*M*: Size of the input in terms of the number of elements, and*•**B*: Size of the data block that can be transferred between the internal memory and the hard disk; transferring one such block is called as a single I/O operation.

The complexity of external memory algorithms is conveniently expressed in terms of predefined I/O operations, such as, *scan*(*N*) for scanning a file of size *N* with a complexity of Θ(*N*/*B*) I/Os, and *sort*(*N*) for external sorting a file of size *N* with a complexity of Θ(*N*/*B* log_{M}_{/}* _{B}* (

In the following, we assume a graph as a tuple (*V*, *E*, *c*), where *V* is the set of nodes, *E* the set of edges, and *c* the weight function that assigns a non-zero positive integer to each edge. If all edges have the same weight, the component *c* can be dropped and the graphs are called as *unweighted*. Given a start node s and a goal node g, we require the search algorithm to return an optimal path wrt. the weight function.

Memory Hierarchy: Modern hardware has a hierarchy of storage mediums: starting from the fast registers, the L1 and L2 caches, moving towards RAM and all the way to the slow hard disks and tapes. The latency timings on different levels differ considerably, e.g., registers: 2ns, cache: 20ns, hard disk: 10ms, tape: 1min.

Graph: A set of nodes connected through edges. The node at the head of an edge is called as the target and at the tail as the source. A graph can be undirected, i.e., it is always possible to return to the source through the same edge – the converse is a directed graph. If given beforehand in the form of adjacency lists (e.g., a road network), we call it an explicit graph. Implicit graphs – another name for ‘state spaces’ – are generated on-the-fly from a start node and a set of rules/actions to generate the new states (e.g., a checkers game).

Delayed Duplicate Detection: In difference to hash tables that eliminate duplicate states on-the-fly during the exploration, the process of duplicate detection can be delayed until a large set of states is available. It is very effective in external search, where it is efficiently achieved by external sorting and scanning.

Value Iteration: Procedure that computes a policy (mapping from states to action) for a probabilistic or non-deterministic search problem most frequently in form of a Markov Decision Problem (MDP).

Search Algorithm: An algorithm that when given two graph nodes, start and goal, returns a sequence of nodes that constitutes a path from start to the goal, if such a sequence exists. A search algorithm generates the successors of a node through an expansion process, after which, the node is termed as a closed node. The newly generated successors are checked for duplicates, and when found as unique, are added to the set of open nodes.

Model Checking: It is an automated process that when given a model of a system and a property specification, checks if the property is satisfied by the system or not. The properties requiring that ‘something bad will never happen’ are referred as safety properties, while the ones requiring that ‘something good will eventually happen’ are referred as liveness properties.

Heuristic Function: A function that assigns a node, an estimated distance to the goal node. For example, in route-planning, the Euclidean distance can be used as a heuristic function. A heuristic function is admissible, if it never overestimates the shortest path distance. It is also consistent, if it never decreases on any edge more than the edge weight, i.e., for a node n and its successor n’, h(n) – h(n’) = c(n,n’).

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