## Heuristic methods

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The aim of this thesis is to deepen our understand of how
IDA* heuristics influence the number of nodes expanded during
search. To this end, we develop Korf's formula for the number
of expanded nodes into a heuristic quality η which
expresses the quality of a heuristic function as a constant factor
on the number of expanded nodes, independent of a particular problem
instance.
We proceed to show how to compute η for some common kinds of
heuristics and how to estimate η by means of a random sample for
arbitrary heuristics. Using the value of η for some concrete
examples, we then inspect for which parts of the search space the
values of h(v) are particularly critical to the performance of the
heuristic, allowing us to build better heuristics for future problems.
This report originally appeared as a master thesis at Humboldt
University of Berlin.

This report aims to explain and extend pattern databases as a tool in finding optimal solutions to combination puzzles. Using sliding tile puzzles as an example, an overview over techniques to construct and use pattern databases is given, analysed, and extended. As novel results, zero aware additive pattern databases and PDB catalogues are introduced. This report originally appeared as a bachelor thesis at Humboldt University of Berlin under the title Notes on the Construction of Tablebases.

The performance of heuristic search algorithms depends crucially on the effectiveness of the heuristic. A pattern database (PDB) is a powerful heuristic in the form of a pre-computed lookup table. Larger PDBs provide better bounds and thus allow more cut-offs in the search process. Today, the largest PDB for the 24-puzzle is a 6-6-6-6 PDB with a size of 486 MB.
We created 8-8-8, 9-8-7 and 9-9-6 PDBs that are three orders of magnitude larger (up to 1.4 TB) than the 6-6-6-6 PDB. We show how to compute such large PDBs and we present statistical and empirical data on their efficiency. The largest single PDB gives on average an 8-fold improvement over the 6-6-6-6 PDB. Combining several large
PDBs gives on average an 12-fold improvement.