parallel lecture 19

Introduction to Parallel Computing and Sorting Algorithms

  • Overview of today's lecture on sorting algorithms in parallel computing.

  • Recap of last class: Odd-Even Sort.

  • Upcoming Lab Assignment: Implement Odd-Even Sort.

  • Homework Assignment: Generalization of Odd-Even Sort.

Odd-Even Sort Recap

Phases of Odd-Even Sort

  • The Odd-Even Sort operates in two main phases:

    • Odd Phase (Phase 1): Perform comparisons and swaps on odd indexed pairs.

    • Even Phase (Phase 2): Perform comparisons and swaps on even indexed pairs.

  • The source code for odd phases is uniform (e.g., Phase 1, Phase 3, Phase 5, etc.) and does not require modification.

  • Similarly, the source code for even phases remains consistent (e.g., Phase 2, Phase 4, Phase 6, etc.).

Programming Requirements

  • Students will program:

    • Phase 1 for odd indices

    • Phase 2 for even indices

  • Each phase requires only one implementation.

Computational Complexity

  • The worst-case scenario for how many phases must be executed:

    • For n elements, the maximum number of phases needed is n - 1.

    • The largest element moving sequentially from the leftmost position to the rightmost requires the maximum phases.

Pseudo-Code for Odd-Even Sort

  • Pseudo-code will differentiate between odd and even phases, utilizing modular arithmetic to define which block of code executes.

  • Rank parity can be determined easily using rank mod 2.

Functionality and Generalization

Phase Implementation

  • Addressing the need to generalize the sorting for larger datasets, not just 8 elements.

  • Suggestion for a function that alternately calls Phase 1 and Phase 2 appropriately based on the input size.

Parallel Complexity of Odd-Even Sort

  • When distributed across cores, the complexity depends on both n and p (number of processors).

  • Each swap requires local resuming in each core to retain sorted order post-exchange.

Scalability of the Algorithm

  • Initial implementation is not scalable for very large datasets with single element cores due to resource limits.

  • Introducing distribution of elements across multiple cores will improve scalability and feasibility.

Quick Sort Introduction

Overview of Quick Sort

  • Quick Sort is a highly efficient sorting algorithm.

  • Average-case complexity of O(n log n), while worst-case complexity can reach O(n²) due to poor pivot selection.

Pivot Selection in Quick Sort

  • Quick Sort involves selecting a pivot to partition the array:

    • All elements smaller than the pivot are arranged to the left, and larger elements are to the right.

    • The pivot's correctness and subsequent splits lead to a logarithmic depth of recursive calls.

Complexity Analysis of Quick Sort

Average Case Complexity

  • Breakdown of recursive steps: Quick Sort maintains average-case efficiency by effectively selecting pivots and partitioning.

  • Concerns on how to structure code to minimize comparisons while ensuring stability in sort.

Worst Case Scenario

  • Understanding the conditions that lead to O(n²) performance:

    • Choosing the smallest or largest element as the pivot consistently results in poor partitioning.

    • The process can be likened to repeated linear scans rather than logarithmic, leading to quadratic time complexity.

Recursive Structure of Quick Sort

  • Recursive call structure leads to a binary tree-like division of elements:

    • Each call reduces the problem size logarithmically.

    • Each level of recursion retains a linear number of comparisons.

Conclusion

  • Wrap up of current topics and a preview of exploring more on these algorithms in parallel computing.

  • Reminder of lab expectations focusing on Odd-Even Sort, and preparing for discussions on Quick Sort's implementation.