SLIDE 1
Pairing Heaps
CS 261 Oregon State University
Heaps
- Heap is a great data structure
- Any chance to make it even better?
Pairing Heaps CS 261 Oregon State University Heaps Heap is a - - PowerPoint PPT Presentation
Pairing Heaps CS 261 Oregon State University Heaps Heap is a - - PowerPoint PPT Presentation
Pairing Heaps CS 261 Oregon State University Heaps Heap is a great data structure Any chance to make it even better? Seeking an O(1) addition How do we obtain O(1) addition? Answer: only change the location of the root How to
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SLIDE 3
Seeking an O(1) addition
- How do we obtain O(1) addition?
- Answer: only change the location of the root
- How to maintain heap structure?
- Heap is partial order!
SLIDE 4
Robert Tarjan
- Professor of Princeton
- Probably the most influential DS researcher in the 1980s
- Many algorithms/advanced DS
- Splay trees
- Pairing heaps
- Fibonacci heaps
- Goldberg-Tarjan push-relabel max-flow
- Hopcroft-Tarjan Planarity-testing
- Turing Award in 1986
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Pairing heap (Fredman, Sedgewick, Sleator, Tarjan 1986)
- Maintain root and a list of subheaps
Instead of: Maintain
2 List of subheaps Root 3
9 12 14 10 16 11 5 7 8
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Pairing heap: O(1) insertion
- Merge 2 heaps operation:
2 List of subheaps 6 Insert 2 Root Root List of subheaps 3
9 12 14 10 16 11 5 7 8
3
9 12 14 10 16 11 5 7 8
6
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Pairing heap: O(1) insertion
2 List of subheaps 1 Insert List of subheaps Root Root 1 2 3
9 12 14 10 16 11 5 7 8
3
9 12 14 10 16 11 5 7 8
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Pairing heap: O(1) merging
2 List of subheaps 6
17 10 13 15 9
List of subheaps
Merge
2
17 10 13 15 9
6
Subheap pointer
3
9 12 14 10 16 11 5 7 8
3
9 12 14 10 16 11 5 7 8
SLIDE 9
Pairing heap: Deletion of root
- Am I cheating?
- What if I just inserted many elements?
- Deletion is going to be very difficult!
- No and yes
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Pairing heaps: deletion (step 1)
2
17 10 13 15 9
6 3
9 12 14 10 16 11 5 7 8
17 10 13 15 9
6 3
9 12 14 10 16 11 5 7 8
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Pairing heaps: deletion (step 2)
17 10 13 15 9
6 3
9 12 14 10 16 11 5 7 8
17 10 13 15 9
6 3
9 12 14 10 16 11 5 7 8
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Pairing heaps: deletion (step 3)
17 10 13 15 9
6 3
9 12 14 10 16 11 5 7 8
17 10 13 15 9
6 3
9 12 14 10 16 11 5 7 8
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Pairing heaps: formal deletion algorithm
SLIDE 14
Pairing heaps: merging sequence
SLIDE 15
Pairing heaps: deletion time
- Amortized log-n time:
- O((log n)+)
- Analysis too complicated here
- Basic point is that each deletion makes the heap more “binary” which makes
subsequent ones faster
- Why would pairing heaps work?
- Utilize multi-way trees
- Data structures can get more complicated than class!
SLIDE 16
Many more heaps
- https://en.wikipedia.org/wiki/Fibonacci_heap
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Practical Performance
- A Back-to-Basics Empirical Study of Priority Queues
- http://arxiv.org/pdf/1403.0252.pdf
- Pairing heap is the most efficient in practice!