1 / 17

Melding Priority Queues

Melding Priority Queues. Ran Mendelson Robert E. Tarjan Mikkel Thorup Uri Zwick. SWAT 2004. Improved analysis of transformation. Meldable Priority Queue. Non-meldable Priority Queue. pq(n)+ α (n) time per operation. pq(n) time per operation. or.

hubert
Download Presentation

Melding Priority Queues

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Melding Priority Queues Ran Mendelson Robert E. Tarjan Mikkel Thorup Uri Zwick SWAT 2004

  2. Improved analysisof transformation MeldablePriority Queue Non-meldablePriority Queue pq(n)+α(n) timeper operation pq(n) timeper operation or pq(n)α(n,n/pq(n)) timeper operation

  3. Second transformation MeldablePriority Queue pq(n) timeper operation pq(N) timeper operation n – number of elements in priority queue Keys are is {1,2,…,N}

  4. Meldable Priority Queues Insert Delete Find-Min O(1) O(log n) O(1) O(1) 10 25 4 7 2 13 17 1 O(1) Dec-Key 5 38 Meld Amortized[Fredman-Tarjan ’87] Worst case[Brodal ’96] Best possible comparison based results

  5. RAM Priority Queues Keys are integers that fit into a single machine word.Standard arithmetical and logical operations take constant time Insert Delete Find-Min O(1) O(log log n) O(1) O(1) 010010 001001 011010 Dec-Key using our transformation Meld O(1) NO [Thorup ’03]

  6. Atomic heaps Insert Delete Find-Min O(1) O(1) O(1) 011010 000010 010011 At most O(log2n) elements! Meld NO [Fredman-Willard ’94]

  7. Union Find makeset union find delete O(1) O(1) O(α(m,n)) O(1) a b c d e Amortized [Tarjan ’75 , Tarjan & van Leeuven ’84 ]

  8. Ackermann’s function A0(j) = j+1 Ai(j) = Ai-1(j+1)(j) Grows extremely FAST α(n) = min{ k : Ak(1) ≥ n } α(m,n) = min{ k : Ak(m/n) ≥ n } Grows extremelyslow

  9. Union Find Represent each set as a rooted tree Union by rank Path compression

  10. Union by rank r+1 r r2 r r1 0

  11. Path Compression

  12. Non-meldable priority queue+Union Find Meldable priority queue

  13. Use the union-find data stricture to maintain the sets Place a non-meldable priority queue at each node of a union-find tree holding the minimal element in each one of its subtrees 9 1 5 1 2 4 5 3 19 2 7 4 8 6 19 2 4 8 6

  14. Handling deletions using path compression The amortized delete cost is O(pq(n)α(n)) [MTZ’04] [van Emde Boaz, Kaas, Zijlstra ’77 ]

  15. Flavor of improved analysis rank ≥ k At mostn/2k nodes size ≥ 2k rank < k size < 2k Choose k=2loglog n. If f>n/log n, we are done.

  16. More flavor of improved analysis rank ≥ k size ≥ 2k rank < k size ≥ 2k rank < k size < 2k

  17. Conclusion Sorting Worst-case non-meldable priority queues Amortized meldablepriority queues

More Related