Equivalence Between Priority Queues and Sorting in External Memory Zhewei Wei Renmin University of China MADALGO, Aarhus University Ke Yi The Hong Kong University of Science and Technology h

Priority Queue Maintain a set of keys Support insertions, deletions and findmin (deletemin) Fundamental data structure Used as subroutines in greedy algorithms Dijkstra’s single source shortest path algorithm Prim’s minimum spanning tree algorithm

Sorting to Priority Queue Priority queue can do sorting Given N unsorted keys Insert the keys to the priority queue Perform N deletemin operations (find minimum and delete it) If a priority queue can support insertion, deletion, findmin in S(N) time, then the sorting algorithm runs in O(NS(N)) time.

Priority Queue to Sorting Thorup [2007]: sorting can do priority queue! A sorting algorithm sorts N keys in N*S(N) time in RAM model A priority queue support all operations in O(S(N)) time Use sorting algorithm as a black box O(Nloglog N) sorting -> O(loglog N) priority queue O(N loglog N ) sorting -> O( loglog N ) priority queue

The I/O Model [Aggarwal and Vitter 1988] Size: M Unlimited size Size: B Disk Memory CPU Block Complexity: # of block transfers (I/Os) CPU computations and memory accesses are free

Cache-Oblivious Model Size: ? Unlimited size Size: ? Memory Block Disk CPU Optimal without knowledge of M and B Optimal for all M and B

Sorting in the I/O Model Sorting bound: Upper bound: external merge sort Lower bound: holds for comparison model or indivisibility assumption Conjecture: lower bound holds for B not too small, even without indivisibility assumption Sort(N)= Θ(N/B * logM/BN ) I/Os Treat keys as atoms N/B is the # of I/Os to make one pass of scanning, # of passes needed

Priority Queue in External Memory I/O model Buffer tree [Arge 1995] M/B-ary heaps [Fadel et. al. 1999] Array heaps[Brodal and Katajainen 1998] O(1/B*logM/BN ) amortized cost Tree-based: do not give any priority queue-to-sorting reduction

Priority Queue in External Memory Cache-oblivious priority queue [Arge et.al. 2002] Keys are moving around in loglog N levels M>B2 O(1/B*logM/BN) with tall cache assumption Reduction: Given an external sorting algorithm that sorts N keys in NS(N)/B I/Os, there is an external priority queue that support all operations in O(S(N)loglog N/B) amortized I/Os

Our Results S(N)/B for S(N) = Ω(2log*N), or M = Ω(B*log(c)N) A sorting algorithm sorts N keys in N*S(N)/B time in the I/O model Use sorting algorithm as a black box S(N) + S(B*log N) + S(B*loglog N)) + … A priority queue support all operations in 1/B*Σi≥0S(Blog(i)(N/B)) amortized I/Os S(N)/B for S(N) = Ω(2log*N), or M = Ω(B*log(c)N) Other wise O((S(N) log*N) /B) No new bounds for external priority queue External priority queue lower bound -> external sorting lower bound

Outline How Thorup did it (on a high level) How we extend it in external memory (on a high level) Open problems

Thorup’s Reduction Word RAM model: each word consists of w ≥ log N bits constant number of registers, each with capacity for one word Atomic heap [Han 2004]: support insertions, deletions, and predecessor queries in set of O(log2 N) size in constant time So it serves as a priority queue that supports all operations in constant time

Thorup’s Reduction – O(S(N)*log N) O(log N) levels N keys N/2 keys c keys 2c keys N/4 keys Invariant: Keys in higher level are larger than keys in Lower level … First I will present a S(N)*log N reduction. Result-wise it does make any sense to consider such a reduction, since even if we can sort in linear time, this reduction only gives us a priority queue with cost log N, no better than binary tree Keys in higher levels are larger than keys in lower levels When a level gets unbalanced, that is, it expand or shrink by a constant factor Rebalance it by taking its neighbouring levels, sort and merge them, and redistribute keys Keep min in the head

Thorup’s Reduction – O(S(N)*log N) N keys N/2 keys c keys 2c keys N/4 keys O(log N) levels Rebalance cost for level 2j: 2j*S(N) # of sorts in N updates: N/2j Amortized cost in level 2j: S(N) log N levels … Cost: O(S(N)*logN)

O(S(N)) Amortized cost Thorup’s Reduction N/log N base sets N/2log N 1 base sets 2 base sets N/4log N Base sets log N O(log N) levels Split/merge base sets: S(N) amortized Rebalancing level 2j: 2jS(N)/log N # of rebalance in N updates: N/2j Amortized cost for level 2j: S(N)/log N … The base sets are sorted relative to each other, but we don’t sort keys inside a base set. We use the maximum key in a base set to represent it. O(S(N)) Amortized cost

O(S(N)) Amortized cost Thorup’s Reduction O(1) cost N/log N base sets N/2log N 1 base sets 2 base sets N/4log N Base sets log N Split/merge base sets: S(N) amortized Rebalancing level 2j: 2jS(N)/log N # of rebalance in N updates: N/2j Amortized cost for level 2j: S(N)/log N … The base sets are sorted relative to each other, but we don’t sort keys inside a base set. We use the maximum key in a base set to represent it. Atomic heap of size log N O(S(N)) Amortized cost

Thorup’s Reduction O(S(N)) Amortized cost … Atomic Buffer size: N/log N Buffer size: N/2log N Buffer size: N/4log N N/log N base sets N/2log N base sets O(S(N)) Amortized cost N/4log N Base sets … Atomic heap of size log N Amortized Cost: O(S(N)) O(1) cost 2 base sets Atomic heap of size log N 1 base sets

Externalize Thorup’s Reduction Where does B come in? How to replace atomic heap? How to handle deletions in external memory?

Where does B come in? B*log N … Buffer of size B*log N Buffer size: N/log N Buffer size: N/2log N Buffer size: N/4log N N/Blog N base sets N/2Blog N 1 base sets 2 base sets N/4Blog N Base sets B*log N … Buffer of size B*log N

I/O-efficient Flush Operation Buffer size |R| k substructures Sort keys in buffer: O(R*S(R)/B) Distribute keys to k substructures: O(R/B+k) Total I/O cost: O(RS(N)/B + k) If k =O(R/B), total flush cost is O(RS(N)/B), amortized cost is O(S(N)/B)

Amortized I/O cost for flushing level buffers: O(S(N)/B) Where does B come in? Base sets: 2j/(Blog N) Buffer size: 2j/log N B*log N … Amortized I/O cost for flushing level buffers: O(S(N)/B) If a level holds 2j keys Largest buffer size: 2j/log N Largest # of base sets: 2j/Blog N Smallest base set (head) size: B*log N

Replacing Atomic Heap R = B*log N k = log N … Buffer of size B*log N

Replacing Atomic Heap Amortized I/O cost: O(S(N)/B) … Recursively build the structure in the head … Buffer of size B*log N Head of size O(Blog N)

Recursively Build Layers O(log* N) Layers N keys B*log (N/B) keys cB keys 2^c*B keys B*loglog(N/B) keys Levels rebalancing - Move base sets around - Redistribute buffer - S(N)/(Blog N) for one level - S(N)/B for one layer - S(N)log* N/B amortized I/O cost … # of logs to take on N/B before it gets to constant,

Recursively Build Layers O(log* N) Layers N keys B*log (N/B) keys cB keys 2^c*B keys B*loglog(N/B) keys … Layers Rebalancing - Rebuild the first (last) level - S(N)/B for one layer - S (N)log* N/B amortized I/O cost

Recursively Build Layers N keys B*log (N/B) keys cB keys 2^c*B keys B*loglog(N/B) keys O(log* N) Layers …

Recursively Build Layers R = B k = log* N N keys B*log (N/B) keys B*loglog(N/B) keys … Memory buffer of size O(B) 2^c*B keys cB keys

Recursively Build Layers Amortized cost: log* N/B N keys B*log (N/B) keys B*loglog(N/B) keys … I/O cost per update: O(S(N)log* N/B) Memory buffer of size O(B) 2^c*B keys cB keys

Delete x -> Insert (-, x) Handle Deletions Follow a pointer to perform deletion takes 1 I/O per deletion Deleting signals: Delete x -> Insert (-, x) Perform actual deletion afterwards Unlike buffer tree, we don’t have access to the “leaves”(base sets) Invariant: Only process deleting signals in the head

Schedule Avoid repeated sorting If head or memory buffer unbalanced: Flush stage: flush all overflowed buffers and rebalance all unbalanced base sets Push stage: rebalance all overflowed layers and levels (expand) Pull stage: deal with delete signals and rebalance all underflowed layers and levels (shrink)

Open problems Optimal reduction? Priority queue that support insertions/deletions in O(1/B) I/O cost for set of size O(B*log(c) N) New reduction framework Better (than loglog N) reduction in Cache-oblivious model? Hard to do I/O-efficient flushing and rebalancing without knowing B

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