PairingHeap
PairingHeap is a pure Ruby priority queue implementation using a pairing heap as the underlying data structure. While a pairing heap is asymptotically less efficient than the Fibonacci heap, it is usually faster in practice. This makes it a popular choice for Prim's MST or Dijkstra's algorithm implementations.
PairingHeap is currently being used as the priority queue data structure in RGL.
Also implementation without priority change support is provided(SimplePairingHeap), while the asymptotical complexity of the methods stay the same, bookkeeping of elements is not needed making, the constant smaller.
Installation
Add this line to your application's Gemfile:
gem 'pairing_heap'
And then execute:
$ bundle install
Or install it yourself as:
$ gem install pairing_heap
Documentation
https://rubydoc.info/gems/pairing_heap
Usage
require 'pairing_heap'
# Simple PairingHeap
simple_heap = PairingHeap::SimplePairingHeap.new
simple_heap.push(:a, 1)
simple_heap.push(:b, 2)
simple_heap.push(:c, 3)
simple_heap.peek # => :a
simple_heap.peek_priority # => 1
simple_heap.pop_with_priority # => [:a, 1]
simple_heap.pop # => :b
# Min priority queue
best_defenses = PairingHeap::MinPriorityQueue.new
best_defenses.push('Chelsea', 24)
best_defenses.push('City', 30)
best_defenses.push('Tottenham', 25)
best_defenses.any? # => true
best_defenses.size # => 3
best_defenses.decrease_key('City', 15)
best_defenses.min # => 'City'
best_defenses.pop # => 'City'
best_defenses.extract_min # => 'Chelsea'
best_defenses.extract_min # => 'Tottenham'
best_defenses.any? # => false
# Max priority queue
best_teams = PairingHeap::MaxPriorityQueue.new
best_teams.push('City', 56)
best_teams.push('United', 46)
best_teams.push('Leicester', 46)
best_teams.increase_key('Leicester', 47)
best_teams.max # => 'City'
best_teams.pop # => 'City'
best_teams.extract_max # => 'Leicester'
# Custom comparator(it defaults to :<=.to_proc)
compare_by_length = PairingHeap::PairingHeap.new { |l, r| l.length <= r.length }
compare_by_length.push(:a, '11')
compare_by_length.push(:b, '1')
compare_by_length.push(:c, '11')
compare_by_length.change_priority(:c, '')
compare_by_length.peek # => :c
compare_by_length.pop # => :c
compare_by_length.pop # => :b
compare_by_length.pop # => :a
# SafeChangePriortyQueue
queue = PairingHeap::SafeChangePriorityQueue.new
queue.push(:a, 1)
queue.push(:b, 2)
queue.change_priority(:a, 3) # This works and does not throw an exception
queue.peek # => :b
See also test/performance_dijkstra.rb for a Dijkstra algorithm implementation.
Changes from lazy_priority_queue
This API is a drop-in replacement of lazy_priority_queue with the following differences:
- Custom comparator provided to constructur, compares weights, not internal nodes
change_priorityreturnsselfinstead of the first argumentenqueuereturnsselfinstead of the first argument- Queue classes are in the
PairingHeapnamespace, sorequire 'pairing_heapdoes not loadMinPriorityQueueto the global scope top_conditionconstructor argument is removed
Time Complexity
| Operation | Time complexity | Amortized time complexity |
|---|---|---|
| enqueue | O(1) | O(1) |
| peek | O(1) | O(1) |
| dequeue | O(n) | O(log n) |
| * change_priority | O(1) | o(log n) |
| * delete | O(n) | O(log n) |
| ^ merge | O(1) | O(1) |
* Not available in SimplePairingHeap
^ Only available in SimplePairingHeap
Benchmarks
I picked the three fastest pure Ruby priority queue implementations I was aware of for the comparison:
- lazy_priority_queue that uses a lazy binomial heap. This is probably the most popular option. It was used in RGL until PairingHeap replaced it.
- Pure Ruby implementation of Fibonacci Heap from priority-queue (link to source)
- rb_heap that uses a binary heap. Note however that this implementation does not support change_priority operation.
All tests except for the third one were executed by benchmark-ips with parameters time = 60 and warmup = 15, on an Intel(R) Core(TM) i7-10700K CPU @ 3.80GHz.
Stress test without changing priority test(N = 1000) source code
Original performance test from lazy_priority_queue
A stress test of 1,000,000 operations: starting with 1,000 pushes/0 pops, following 999 pushes/1 pop, and so on till 0 pushes/1000 pops.
ruby 3.3.0 (2023-12-25 revision 5124f9ac75) [x86_64-darwin23] Library Iterations Seconds Iterations per second pairing_heap (SimplePairingHeap) 26 62.249427 0.419 pairing_heap (PairingHeap) 17 61.624806 0.276(1.52x slower) rb_heap 16 63.656502 0.251(1.67x slower) lazy_priority_queue 7 63.339192 0.111(3.79x slower) Fibonacci 5 69.010737 0.073(5.77x slower) ruby 3.3.0 (2023-12-25 revision 5124f9ac75) +YJIT [x86_64-darwin23] Library Iterations Seconds Iterations per second pairing_heap (SimplePairingHeap) 39 60.725689 0.642 rb_heap 31 60.370348 0.514(1.25x slower) pairing_heap (PairingHeap) 25 62.269577 0.402(1.6x slower) lazy_priority_queue 9 62.144710 0.145(4.43x slower) Fibonacci 8 65.064385 0.123(5.22x slower) jruby 9.4.5.0 (3.1.4) 2023-11-02 1abae2700f OpenJDK 64-Bit Server VM 21+35-2513 on 21+35-2513 +indy +jit [x86_64-darwin] Library Iterations Seconds Iterations per second pairing_heap (SimplePairingHeap) 43 60.734661 0.709 rb_heap 34 61.677228 0.552(1.28x slower) pairing_heap (PairingHeap) 28 61.284382 0.458(1.55x slower) Fibonacci 22 61.396897 0.359(1.97x slower) lazy_priority_queue 20 61.841463 0.324(2.19x slower) truffleruby 23.1.2, like ruby 3.2.2, Oracle GraalVM JVM [x86_64-darwin] Library Iterations Seconds Iterations per second pairing_heap (SimplePairingHeap) 202 60.225639 3.388 rb_heap 140 60.190881 2.357(1.44x slower) pairing_heap (PairingHeap) 100 60.373610 1.692(2x slower) lazy_priority_queue 31 61.179244 0.510(6.65x slower) Fibonacci 11 64.413419 0.171(19.81x slower)
Stress test with changing priority(N = 1000) source code
A stress test of 1,501,500 operations: starting with 1,000 pushes/1000 change_priorities/0 pops, following 999 pushes/999 change_priorities/1 pop, and so on till 0 pushes/0 change_priorities/1000 pops.
| ruby 3.3.0 (2023-12-25 revision 5124f9ac75) [x86_64-darwin23] | |||
|---|---|---|---|
| Library | Iterations | Seconds | Iterations per second |
| pairing_heap | 15 | 60.817878 | 0.247 |
| lazy_priority_queue | 7 | 63.990376s | 0.109(2.26x slower) |
| Fibonacci | 5 | 70.148980s | 0.071(3.47x slower) |
| ruby 3.3.0 (2023-12-25 revision 5124f9ac75) +YJIT [x86_64-darwin23] | |||
| Library | Iterations | Seconds | Iterations per second |
| pairing_heap | 22 | 62.429264 | 0.353 |
| lazy_priority_queue | 9 | 63.464818 | 0.142(2.49x slower) |
| Fibonacci | 8 | 67.908812 | 0.118(2.99x slower) |
| jruby 9.4.5.0 (3.1.4) 2023-11-02 1abae2700f OpenJDK 64-Bit Server VM 21+35-2513 on 21+35-2513 +indy +jit [x86_64-darwin] | |||
| Library | Iterations | Seconds | Iterations per second |
| pairing_heap | 27 | 61.709517 | 0.438 |
| Fibonacci | 20 | 61.495636 | 0.326(1.34x slower) |
| lazy_priority_queue | 19 | 63.401601 | 0.309(1.46x slower) |
| truffleruby 23.1.2, like ruby 3.2.2, Oracle GraalVM JVM [x86_64-darwin] | |||
| Library | Iterations | Seconds | Iterations per second |
| pairing_heap | 93 | 60.125750 | 1.577 |
| lazy_priority_queue | 26 | 62.372660s | 0.419(3.77x slower) |
| Fibonacci | 11 | 62.620172s | 0.177(8.92x slower) |
Stress test with changing priority or push/pop(test ignored in summary) source code
Start with 500 pushes, then:
- If queue supports changing priority 500 change_priority calls, then 500 pops
- If does not support changing priority 500 push calls, then 1000 pops
ruby 3.3.0 (2023-12-25 revision 5124f9ac75) [x86_64-darwin23] Library Iterations per second pairing_heap (PairingHeap) 748.9 lazy_priority_queue 388.6(1.93x slower) pairing_heap (SimplePairingHeap) 336.2(2.23x slower) Fibonacci 225.9(3.31x slower) rb_heap 215.2(3.48x slower) ruby 3.3.0 (2023-12-25 revision 5124f9ac75) +YJIT [x86_64-darwin23] Library Iterations per second pairing_heap 1276 pairing_heap(SimplePairingHeap) 625.6(2.04x slower) lazy_priority_queue 533.36(2.39x slower) Fibonacci 495.519(2.57x slower) rb_heap 453.323(2.81x slower) jruby 9.4.5.0 (3.1.4) 2023-11-02 1abae2700f OpenJDK 64-Bit Server VM 21+35-2513 on 21+35-2513 +indy +jit [x86_64-darwin] Library Iterations per second pairing_heap(PairingHeap) 1377 Fibonacci 1088(1.27x slower) lazy_priority_queue 953.935(1.44x slower) pairing_heap(SimplePairingHeap) 621.35(2.22x slower) rb_heap 595.11(2.31x slower) truffleruby 23.1.2, like ruby 3.2.2, Oracle GraalVM JVM [x86_64-darwin] Library Iterations per second pairing_heap(PairingHeap) 12712 pairing_heap(SimplePairingHeap) 9447(1.35x slower) rb_heap 4009(3.17x slower) Fibonacci 2793(4.55x slower) lazy_priority_queue 1188(10.7x slower)
Simple Dijkstra's algorithm implementation source code
Heaps that support change_priority operation use it. Heaps that do not support it use dijkstra implementation that do not rely on change_priority instead and do additional pops and pushes instead(see Dijkstra-NoDec from Priority Queues and Dijkstra’s Algorithm).
| ruby 3.3.0 (2023-12-25 revision 5124f9ac75) [x86_64-darwin23] | |||
|---|---|---|---|
| Library | Iterations | Seconds | Iterations per second |
| pairing_heap (SimplePairingHeap) | 41 | 60.100316 | 0.682 |
| pairing_heap (PairingHeap) | 38 | 61.003607 | 0.623(1.09x slower) |
| rb_heap | 30 | 61.486216 | 0.488(1.40x slower) |
| lazy_priority_queue | 23 | 60.251764 | 0.382(1.79x slower) |
| Fibonacci | 13 | 61.158622 | 0.213(3.21x slower) |
| ruby 3.3.0 (2023-12-25 revision 5124f9ac75) +YJIT [x86_64-darwin23] | |||
| Library | Iterations | Seconds | Iterations per second |
| pairing_heap (SimplePairingHeap) | 65 | 60.805882 | 1.070 |
| pairing_heap (PairingHeap) | 60 | 60.790842 | 0.987(1.08x slower) |
| rb_heap | 54 | 60.917679 | 0.887(1.21x slower) |
| lazy_priority_queue | 30 | 60.712786 | 0.495(2.16x slower) |
| Fibonacci | 24 | 61.900715 | 0.388(2.76x slower) |
| jruby 9.4.5.0 (3.1.4) 2023-11-02 1abae2700f OpenJDK 64-Bit Server VM 21+35-2513 on 21+35-2513 +indy +jit [x86_64-darwin] | |||
| Library | Iterations | Seconds | Iterations per second |
| rb_heap | 70 | 60.077928 | 1.168 |
| pairing_heap (SimplePairingHeap) | 66 | 60.420935 | 1.094(1.07x slower) |
| pairing_heap (PairingHeap) | 64 | 60.114467 | 1.066(1.1x slower) |
| Fibonacci | 54 | 60.426971 | 0.898(1.30x slower) |
| lazy_priority_queue | 49 | 60.636963 | 0.809(1.44x slower) |
| truffleruby 23.1.2, like ruby 3.2.2, Oracle GraalVM JVM [x86_64-darwin] | |||
| Library | Iterations | Seconds | Iterations per second |
| pairing_heap (SimplePairingHeap) | 288 | 60.102278 | 4.936 |
| pairing_heap (PairingHeap) | 232 | 60.159057 | 3.936(1.25x slower) |
| rb_heap | 227 | 60.082482 | 3.881(1.27x slower) |
| Fibonacci | 101 | 60.076691 | 1.721(2.87x slower) |
| lazy_priority_queue | 66 | 60.771569 | 1.1(4.49x slower) |
Summary
Change priority required
| ruby 3.3.0 (2023-12-25 revision 5124f9ac75) [x86_64-darwin23] | |||
|---|---|---|---|
| Library | Slower geometric mean | ||
| pairing_heap | 1 | ||
| lazy_priority_queue | 2.1x slower | ||
| Fibonacci | 3.38x slower | ||
| ruby 3.3.0 (2023-12-25 revision 5124f9ac75) +YJIT [x86_64-darwin23] | |||
| Library | Slower geometric mean | ||
| pairing_heap | 1 | ||
| lazy_priority_queue | 2.55x slower | ||
| Fibonacci | 2.74x slower | ||
| jruby 9.4.5.0 (3.1.4) 2023-11-02 1abae2700f OpenJDK 64-Bit Server VM 21+35-2513 on 21+35-2513 +indy +jit [x86_64-darwin] | |||
| Library | Slower geometric mean | ||
| pairing_heap | 1 | ||
| Fibonacci | 1.267x slower | ||
| lazy_priority_queue | 1.396x slower | ||
| truffleruby 23.1.2, like ruby 3.2.2, Oracle GraalVM JVM [x86_64-darwin] | |||
| Library | Slower geometric mean | ||
| pairing_heap | 1 | ||
| lazy_priority_queue | 3.54x slower | ||
| Fibonacci | 5.86x slower | ||
Change priority not required
| ruby 3.3.0 (2023-12-25 revision 5124f9ac75) [x86_64-darwin23] | |||
|---|---|---|---|
| Library | Slower geometric mean | ||
| pairing_heap (SimplePairingHeap) | 1 | ||
| pairing_heap (PairingHeap) | 1.29x slower | ||
| rb_heap | 1.53x slower | ||
| lazy_priority_queue | 2.6x slower | ||
| Fibonacci | 4.29x slower | ||
| ruby 3.3.0 (2023-12-25 revision 5124f9ac75) +YJIT [x86_64-darwin23] | |||
| Library | Slower geometric mean | ||
| pairing_heap (SimplePairingHeap) | 1 | ||
| rb_heap | 1.227x slower | ||
| pairing_heap (PairingHeap) | 1.316x slower | ||
| lazy_priority_queue | 3.094x slower | ||
| Fibonacci | 3.79x slower | ||
| jruby 9.4.5.0 (3.1.4) 2023-11-02 1abae2700f OpenJDK 64-Bit Server VM 21+35-2513 on 21+35-2513 +indy +jit [x86_64-darwin] | |||
| Library | Slower geometric mean | ||
| pairing_heap (SimplePairingHeap) | 1.033x slower | ||
| rb_heap | 1.133x slower | ||
| pairing_heap (PairingHeap) | 1.302x slower | ||
| Fibonacci | 1.602x slower | ||
| lazy_priority_queue | 1.777x slower | ||
| truffleruby 23.1.2, like ruby 3.2.2, Oracle GraalVM JVM [x86_64-darwin] | |||
| Library | Slower geometric mean | ||
| pairing_heap (SimplePairingHeap) | 1 | ||
| rb_heap | 1.35x slower | ||
| pairing_heap (PairingHeap) | 1.58x slower | ||
| lazy_priority_queue | 5.46x slower | ||
| Fibonacci | 7.54x slower | ||
Development
After checking out the repo, run bin/setup to install dependencies. Then, run rake test to run the tests. You can also run bin/console for an interactive prompt that will allow you to experiment.
To install this gem onto your local machine, run bundle exec rake install. To release a new version, update the version number in version.rb, and then run bundle exec rake release, which will create a git tag for the version, push git commits and the created tag, and push the .gem file to rubygems.org.
Contributing
Bug reports and pull requests are welcome on GitHub at https://github.com/mhib/pairing_heap.
License
The gem is available as open source under the terms of the MIT License.