The Weak-Heap Family of Priority Queues in Theory ... - The CPH STL

Our research programProvide the best bounds on the comparison complexity of priorityqueueoperations (n the current size of the data structure):find-min: no element comparisonsdelete,delete-min: ∼β lg n element comparisonsOther operations: ∼κ element comparisonswhere β and κ are constants.worst case per operationFull operation repertoire: insert, borrow, decrease, meldc○ Performance Engineering Laboratory 18th CATS, Melbourne, 2 Feb. 2012 (2)

New gameApplication = Request sequence(Think graph applications)What is the best bound when handling arequest sequence consisting of n insert, ndelete-min, and m decrease operations?worst case per sequencec○ Performance Engineering Laboratory 18th CATS, Melbourne, 2 Feb. 2012 (3)

Weak heap• A binary tree• The root only has a right child if any• Elements obey the weak-heap ordering:for element x, every element in the rightsubtree is ≥ x• With the exception of the root, the nodesthat have at most one child are at the lasttwo levels onlyminimumleaf registryOur representation pointer-based!c○ Performance Engineering Laboratory 18th CATS, Melbourne, 2 Feb. 2012 (5)

Normal referee comments• Element comparisons are not relevant in practice!• Pairing heaps are much faster in practice!• No one would ever implement Fibonacci heaps since they are socomplicated.This is you guys! Thank you! You pushed us to do some practicalexperiments!c○ Performance Engineering Laboratory 18th CATS, Melbourne, 2 Feb. 2012 (7)

Interactiongraph0101010100 11tentative distance010100 11 0100 11 01 priority queueminimumCombine the graph vertex and the priorityqueuenode [Knuth 1994]→ improves cache behavioura factor of two speed-upc○ Performance Engineering Laboratory 18th CATS, Melbourne, 2 Feb. 2012 (10)

TuningRunning time per n [µs]Element comparisons per nStructureCPH STLFibonacciheapLEDA 6.2FibonacciheapOperationinsertn: 10 000 0.10 0.18n: 100 000 0.09 0.15n: 1 000 000 0.09 0.15decreasen: 10 000 0.03 0.06n: 100 000 0.05 0.22n: 1 000 000 0.06 0.31delete-minn: 10 000 0.7 1.2n: 100 000 1.4 2.7n: 1 000 000 2.8 4.5On my computer (Ubuntu, gcc, with -O3)StructureCPH STLFibonacciheapLEDA 6.2FibonacciheapOperationinsertn: 10 000 0 1n: 100 000 0 1n: 1 000 000 0 1decreasen: 10 000 0 2n: 100 000 0 2n: 1 000 000 0 2delete-minn: 10 000 16.2 29.9n: 100 000 21.2 38.3n: 1 000 000 26.2 46.5a factor of two speed-upc○ Performance Engineering Laboratory 18th CATS, Melbourne, 2 Feb. 2012 (11)

Parameterized designweak heapelement typecomparator typeint, double, ...std::less, std::greater, ...node type weak−heap node, combined weak−heap node & graph node, ...modifier type relaxed heap modifier, ...level−registry typemark−registry typeleaf registry, ...naive mark registry, eager mark registry, lazy mark registry, ...• comparators shared• nodes shared• transformations shared• level registries shared• mark registries shareda factor of two less codec○ Performance Engineering Laboratory 18th CATS, Melbourne, 2 Feb. 2012 (12)

What is the best?Our reference sequenceTheory: rank-relaxed weak heapDijkstra—time: binary heap[Williams 1964]Dijkstra—comps: weak heap[Dutton 1993]Worst case per operationinsert—time: Fibonacci heap[Fredman & Tarjan 1987]insert—comps: Fibonacci heapdecrease—time: Fibonacci heapdecrease—comps: Fibonacci heapdelete-min—time: weak queue[Vuillemin 1978]delete-min—comps: weak heapc○ Performance Engineering Laboratory 18th CATS, Melbourne, 2 Feb. 2012 (13)

ConclusionsEarlier conjectures—see my SWAT-2010 slidesKatajainen’s third conjecture: Our reference sequence can be handledwith 2m + n lg n + o(n) element comparisons in O(m + n lg n)time.Cache effects: We have neglected the cache behaviour of priorityqueues for too long.Source code: Our programs are available via the CPH STL website.c○ Performance Engineering Laboratory 18th CATS, Melbourne, 2 Feb. 2012 (14)