Linked List Ranking

1. Linked List Ranking Parallel Algorithms

2. Work Analysis – Pointer Jumping • Number of Steps: Tp = O(Log N) • Number of Processors: N • Work = O(N log N) • T1 = O(N) • Work Optimal??

3. Common List Ranking Strategy • Reduce Rank Expand Strategy • Similar to Odd-Even Prefix Sums • Algorithms vary in how items are selected for deletion in the Reduce Step • Some variation in the Expand Step.

4. Reduce Rank Expand • Reduction Step - Delete non-adjacent elements in parallel until the list is of size O(p) (algorithms vary is this step) • Rank Step – Pointer jumping to rank the p remaining elements • Expand Step – Reinsert elements (in the reverse order), computing the rank as it is inserted

5. Asynchronous PRAM – APRAM • Variant of the PRAM model • Features of CRCW PRAM – Plus • Processors may have different clock speeds and work asynchronously • Each processor has own random number generator

6. Randomized Algorithm • Some portion of the algorithm’s outcome is non-deterministic • Performance stated in terms of expected order of time complexity – EO(f(n)) • EO(f(n)) – with high probability the results are within order f(n)

7. APRAM List Ranking by Martel & Subramonian • Select m=EO(n/log n) elements at random • Each generates a random number between 1-N, selects those with values 1 to N / log N • Perform log log n iterations of pointer jumping • Each points to selected, nil, or pointer length log n • Pack selected elements to smaller array • Scan: each selected points to next selected or nil • Compute ranks of selected list-pointer jumping • Copy back to original list; scan to complete ranks *EO(n log log n) using p = O(n/log n)

8. Demo: APRAM Randomized Algorithm

9. Modifications for EREW Model APRAM • Each PU operates asynchronously • Must synchronize each at step • Steps 2, 4, 6 allow for Concurrent Reads EREW • Default execution is synchronous • Partition the array to eliminate Concurrent Reads

10. Randomized EREW Algorithm (4.1) • Select O(p) elements at random • Each generates a random number between 1-N, selects those with values 1 to N / log N • Pack selected items into smaller array • Scan from selected item to next to set rank • Rank selected elements by pointer jumping • Write ranks back to original array • Scan from selected items to set remaining ranks

11. Demo: EREW Randomized Algorithm

12. Complexity Analysis • Selection - EO(n/p) • Compaction - prefix sums EO(n/p + log p) • Form LL - EO(n/p) • Pointer Jumping - EO ( n/p + log p) • Sequential Scan - EO(n/p) • OVERALL - EO ( n/p + log p) • Optimal speedup for p <= O (n / log n) • Which gives EO ( log n) time • See paper for proof.

13. What next?? • Can we accomplish an O(log n) deterministic algorithm? • What was randomized in other algorithms? • Can it be made deterministic?

14. Answer! • Yes & No, • Can Reduce deterministically! • Near Optimal • 2-Ruling Set (Cole & Vishkin) • A subset of the list such that no 2 elements are adjacent & there are exactly 1 or 2 non-ruling set elements between each ruling set element & its nearest successor

15. Deterministic EREW LLR Step 1 • Select a Ruling Set of O(p) elements, O(n/p) distance apart • Keep track of actual distance (for ranking) • Repeatedly apply 2-Ruling Set Algorithm to increase the distance between the elements • Remaining steps are same

16. Complexity Analysis • Select O(p) elements O(n/p) distance apart – O(n/p * log (n/p)) • Compact – O(n/p + log p) • Rank – O(n/p + log p) • Write back then Scan – O(n/p) O((log p) + (n/p) * log(n/p)) = O(n/p * log (n/p) ) for n>= p log p = O(log n log log n) for p =O(n/log n) Work O(n log log n)

18. Advantages of New Algorithms • Simple, use basic strategies • Small constants • Small space requirements • Optimal & Near Optimal