Partitioning & Divide and Conquer Strategies Prof. Wenwen Li School of Geographical Sciences and Urban Planning 5644 Coor Hall

Outline 1. Partitioning Strategies 2. Divide and Conquer Strategies 3. Examples 4. Applications 5. Summary

Partitioning Strategies Partitioning divides a complex problem into parts. Data division: data partitioning or domain decomposition Program division: functional decomposition

Partitioning Strategies Types of partitioning* Single-level partitioning: Range partitioning: maps data to partitions based on ranges of values of the partitioning key that you establish for each partition. Hash partitioning: maps data to partitions based on a hashing algorithm. List partitioning: enables explicitly controlling how rows map to partitions by specifying a list of discrete values for the partitioning key in the description for each partition. *

Partitioning Strategies Types of partitioning* Single-level partitioning: *

Partitioning Strategies Types of partitioning* Composite partitioning: Composite Range-Range Partitioning Composite Range-Hash Partitioning Composite Range-List Partitioning Composite List-Range Partitioning Composite List-Hash Partitioning Composite List-List Partitioning *

Partitioning Strategies Types of partitioning* Composite partitioning: *

Divide and Conquer Strategies Characterized by dividing problem into sub-problems of same. Further divisions into still smaller sub-problems, usually done by recursion. Recursive divide and conquer amenable to parallelization because separate processes can be used for divided parts. Also usually data is naturally localized.

Examples Operations on sequences of number such as simply adding them together Several sorting algorithms can often be partitioned or constructed in a recursive fashion Numerical integration N-body problem

Applications Partitioning a sequence of numbers into parts and adding the parts

Applications Tree construction

Applications Dividing a list into parts

Applications Partial summarization

Applications Quadtree

Applications Numerical integration Computing the “area under the curve.” Parallelized by divided area into smaller areas (partitioning). Can also apply divide and conquer -- repeatedly divide the areas recursively.

Applications Numerical integration Numerical integration by rectangles:

Applications Numerical integration Numerical integration by trapezoidal method:

Applications Adaptive quadrature Based on the fixed number of intervals, we can obtain more accurate result through increasing the number of intervals between two rounds.

Summary Numerical integration Computing the “area under the curve.” Parallelized by divided area into smaller areas (partitioning). Can also apply divide and conquer -- repeatedly divide the areas recursively.