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Melding Priority Queues
Ran Mendelson Robert E. Tarjan Mikkel Thorup Uri Zwick SWAT 2004
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Non-meldable Priority Queue Meldable Priority Queue
Improved analysis of transformation Non-meldable Priority Queue Meldable Priority Queue pq(n)+α(n) time per operation pq(n) time per operation or pq(n)α(n,n/pq(n)) time per operation
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Second transformation
Meldable Priority Queue pq(n) time per operation pq(N) time per operation n – number of elements in priority queue Keys are is {1,2,…,N}
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Meldable Priority Queues Insert Delete Find-Min O(1) O(log n) Dec-Key
10 25 4 7 13 2 17 1 Dec-Key O(1) 5 38 Meld Amortized [Fredman-Tarjan ’87] Worst case [Brodal ’96] Best possible comparison based results
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using our transformation
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) 010010 001001 011010 Dec-Key using our transformation Meld O(1) NO [Thorup ’03]
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At most O(log2n) elements!
Atomic heaps Insert Delete Find-Min O(1) 011010 000010 010011 At most O(log2n) elements! Meld NO [Fredman-Willard ’94]
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Union Find makeset union find delete O(1) O(α(m,n)) Amortized
b c d e Amortized [Tarjan ’75 , Tarjan & van Leeuven ’84 ]
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α(m,n) = min{ k : Ak(m/n) ≥ n }
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 extremely slow
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Union Find Represent each set as a rooted tree Union by rank
Path compression
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Union by rank r+1 r2 r r r1
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Path Compression
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Non-meldable priority queue + Union Find
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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
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Handling deletions using path compression
The amortized delete cost is O(pq(n)α(n)) [MTZ’04] [van Emde Boaz, Kaas, Zijlstra ’77 ]
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Flavor of improved analysis
rank ≥ k At most n/2k nodes size ≥ 2k rank < k size < 2k Choose k=2loglog n. If f>n/log n, we are done.
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More flavor of improved analysis
rank ≥ k size ≥ 2k rank < k size ≥ 2k rank < k size < 2k
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Conclusion Sorting Worst-case non-meldable priority queues
Amortized meldable priority queues
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