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**Sorting an Intransitive Total Ordered Set**

Sequential and Parallel Implementation for COMP 5704 Parallel Algorithms by Johannes Singler, Carleton University

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**Presentation Outline Introduction to the Problem Prerequisites**

Sequential Algorithm Parallelizing for a PRAM Questions?!

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**Introduction Definition of Transitivity:**

Traditional sorting relies upon (Transitive) Total Ordered Set (e. g. natural numbers and ≥). New Problem: Sorting of Intransitive Total Ordered Set (ITOS) Relying on intransitive relation

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**Intransitive Sorted Sequence**

An intransitive sorted sequence has to satisfy: Theorem: There exists always a solution (proof later if time remaining). The solution is not unique in general. Real-world example for an ITOS: tournament in sports: Every player wins or loses against each other. Therefore ITOS also referred to as a Tournament.

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**Representation as a Graph**

Tournament can be represented as a complete directed graph. Every two nodes are connected by an edge, direction denotes relation (who wins). A intransitive sorted sequence then corresponds to a Hamiltonian Path. Definition of Hamiltonian Path: A path in a graph that visits every node exactly once.

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**Sorting Prerequisites**

Sorting using Semi-Heap according to Wu [2] Definition of On 3 elements. Set in general (not unique any more). If one player beats both the others, he is the only maximum element, otherwise all three are maximal. Definition of a Semi-Heap Binary tree (complete in the beginning). Similar to a heap. Semi-Heap condition: 4 valid possibilities (and 4 invalid):

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**Actual Sorting Similar to Heap-Sort**

Build up heap Iteratively remove root, replace by rightmost leaf of deepest level before pushing this down the heap. Intransitive Sorting using Semi-Heap Build up semi-heap Iteratively remove root, update semi-heap top-down. Tree may not be complete any more after first removal.

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**Pseudo-Code Pseudo-Code:**

semi-heap-sort build-semi-heap While nodes left Remove root and append to sorted sequence Recursively replace by winning child top-down build-semi-heap (Q(n)) for i:=heapsize/2 downto 1 semi-heapify(i) semi-heapify(i) if(i not max) exchange with max child semi-heapify(former child position) recursively

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**Sequential Complexity**

Demo Sequential Complexity: O(n log n)

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**Parallelizing the Algorithm for a PRAM**

Parallelizing for fine-grained multiprocessing using Pipelining. One processor per two consecutive tree levels (overlapping): Q(log n) processors needed. Two phases: Either processors with even numbers or processor with odd numbers are active. While updating the tree top-down, the work is handed from one processor to the next. Demo

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**Complexity Issues Parallel Complexity:**

Q(n) on an EREW PRAM using Q(log n) processors, therefore cost-optimal. Building the initial semi-heap takes Q(n) sequentially and thus needs no parallelization. No concurrent access needed because of phases. Can be modified to use less processors according to Brent’s Law.

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Outlook Extension: A Hamiltonian Cycle exists in every strong connected Tournament. Definition of strong connected: Every node is reachable from each other Hamiltonian Cycle: Every node is visited exactly once and there is an edge from the last node to the first one:

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Questions? Do you have any questions left?

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Questions! Which of the following subtrees satisfy the semi-heap condition? What does Pipelining mean? How many possible tournaments of size n are there?

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Conclusion Presentation and demo program available at my website: References [1] Danny Soroker. Fast parallel algorithms for finding Hamiltonian paths and cycles in a tournament. Journal of Algorithms, 9(2):276–286, June [2] J. Wu. On sorting an intransitive total ordered set using semi-heap. In International Parallel and Distributed Processing Symposium, pages 257–262, 2000.

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**Appendix: Proof of Theorem**

By induction according to Soroker [1] Result clear for n=2 n → n+1 Let v be vertex of V(T), then V(T)-{v} has Hamiltonian Path v1,…,vn If v → v1, then v,v1,…,vn is a valid result Otherwise let I be the largest index such that vi → v, then v1,…,vi,v,vi+1,…,vn or v1,…,vn,v respectively (i=n) is a valid result.

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