Funnelselect: Cache-Oblivious Multiple Selection Sebastian Wild joint work with Gerth Stølting Brodal European Symposium on Algorithms 2023 CWI Amsterdam Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 0 / 12

Outline 1 Multiple Selection 1 Multiple Selection 2 Cache Oblivious Algorithms 2 Cache Oblivious Algorithms 3 Funnelselect 3 Funnelselect 4 Conclusion 4 Conclusion Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 0 / 12

1 Multiple Selection 1 Multiple Selection Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 0 / 12

“What’s this about” in 2min Two ancient comparison-based problems on unordered list of N elements Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 1 / 12

“What’s this about” in 2min Two ancient comparison-based problems on unordered list of N elements Sorting Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 1 / 12

“What’s this about” in 2min Two ancient comparison-based problems on unordered list of N elements Sorting (Single) Selection e.g. find median by Rank Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 1 / 12

“What’s this about” in 2min Two ancient comparison-based problems on unordered list of N elements Sorting (Single) Selection e.g. find median by Rank O(N log2 N) comparisons, time O(N B logM/B N B ) I/Os internal: Mergesort IO: Multiway Mergesort CO: Funnelsort O(N) comparisons, time O(N/B) I/Os internal: Median-of-medians algorithm IO: Median-of-medians algorithm CO: Median-of-medians algorithm Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 1 / 12

“What’s this about” in 2min Two ancient comparison-based problems on unordered list of N elements Sorting (Single) Selection e.g. find median by Rank O(N log2 N) comparisons, time O(N B logM/B N B ) I/Os internal: Mergesort IO: Multiway Mergesort CO: Funnelsort O(N) comparisons, time O(N/B) I/Os internal: Median-of-medians algorithm IO: Median-of-medians algorithm CO: Median-of-medians algorithm more sorted more expensive Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 1 / 12

“What’s this about” in 2min Two ancient comparison-based problems on unordered list of N elements Sorting (Single) Selection e.g. find median by Rank O(N log2 N) comparisons, time O(N B logM/B N B ) I/Os internal: Mergesort IO: Multiway Mergesort CO: Funnelsort O(N) comparisons, time O(N/B) I/Os internal: Median-of-medians algorithm IO: Median-of-medians algorithm CO: Median-of-medians algorithm more sorted more expensive M u l t i p l e S e l e c t i o n Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 1 / 12

“What’s this about” in 2min Two ancient comparison-based problems on unordered list of N elements Sorting (Single) Selection e.g. find median by Rank O(N log2 N) comparisons, time O(N B logM/B N B ) I/Os internal: Mergesort IO: Multiway Mergesort CO: Funnelsort O(N) comparisons, time O(N/B) I/Os internal: Median-of-medians algorithm IO: Median-of-medians algorithm CO: Median-of-medians algorithm more sorted more expensive M u l t i p l e S e l e c t i o n This talk: What happens between selection and sorting in the cache-oblivious model? Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 1 / 12

The Multiple Selection Problem Goal: find q elements of ranks r1 < r2 < · · · < rq from unsorted elements here: all distinct x1 , . . . , xN ⇝ Report sought elements x(r1) , . . . , x(rq) in sorted order Example: 67 x1 30 x2 45 x3 33 x4 15 x5 99 x6 26 x7 90 x8 55 x9 9 x10 96 x11 45 x12 95 x13 31 x14 3 x15 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 r1 = 1 r2 = 2 r3 = 3 r4 = 8 Simple algorithms: 1 q calls to selection algorithm (quickselect, median of medians) ⇝ O(Nq) 2 Divide & conquer 1: (single) select r⌈q/2⌉ -th smallest and partition x1 , . . . , xN around it. Recursively select r1 , . . . , r⌈q/2⌉−1 and r⌈q/2⌉+1 , . . . , rq ⇝ O(N lg q) 3 Divide & conquer 2: Find median of x1 , . . . , xN and split query ranks. Recurse where subproblem contains query ranks. ⇝ O(N lg q) Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 2 / 12

The Multiple Selection Problem Goal: find q elements of ranks r1 < r2 < · · · < rq from unsorted elements here: all distinct x1 , . . . , xN ⇝ Report sought elements x(r1) , . . . , x(rq) in sorted order Example: 67 x1 30 x2 45 x3 33 x4 15 x5 99 x6 26 x7 90 x8 55 x9 9 x10 96 x11 45 x12 95 x13 31 x14 3 x15 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 r1 = 1 r2 = 2 r3 = 3 r4 = 8 Simple algorithms: 1 q calls to selection algorithm (quickselect, median of medians) ⇝ O(Nq) 2 Divide & conquer 1: (single) select r⌈q/2⌉ -th smallest and partition x1 , . . . , xN around it. Recursively select r1 , . . . , r⌈q/2⌉−1 and r⌈q/2⌉+1 , . . . , rq ⇝ O(N lg q) 3 Divide & conquer 2: Find median of x1 , . . . , xN and split query ranks. Recurse where subproblem contains query ranks. ⇝ O(N lg q) Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 2 / 12

The Multiple Selection Problem Goal: find q elements of ranks r1 < r2 < · · · < rq from unsorted elements here: all distinct x1 , . . . , xN ⇝ Report sought elements x(r1) , . . . , x(rq) in sorted order Example: 67 x1 30 x2 45 x3 33 x4 15 x5 99 x6 26 x7 90 x8 55 x9 9 x10 96 x11 45 x12 95 x13 31 x14 3 x15 3 1 9 2 15 3 26 4 30 5 31 6 33 7 45 8 45 9 55 10 67 11 90 12 95 13 96 14 99 15 r1 = 1 r2 = 2 r3 = 3 r4 = 8 Simple algorithms: 1 q calls to selection algorithm (quickselect, median of medians) ⇝ O(Nq) 2 Divide & conquer 1: (single) select r⌈q/2⌉ -th smallest and partition x1 , . . . , xN around it. Recursively select r1 , . . . , r⌈q/2⌉−1 and r⌈q/2⌉+1 , . . . , rq ⇝ O(N lg q) 3 Divide & conquer 2: Find median of x1 , . . . , xN and split query ranks. Recurse where subproblem contains query ranks. ⇝ O(N lg q) Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 2 / 12

The Multiple Selection Problem Goal: find q elements of ranks r1 < r2 < · · · < rq from unsorted elements here: all distinct x1 , . . . , xN ⇝ Report sought elements x(r1) , . . . , x(rq) in sorted order Example: 67 x1 30 x2 45 x3 33 x4 15 x5 99 x6 26 x7 90 x8 55 x9 9 x10 96 x11 45 x12 95 x13 31 x14 3 x15 3 1 9 2 15 3 26 4 30 5 31 6 33 7 45 8 45 9 55 10 67 11 90 12 95 13 96 14 99 15 r1 = 1 r2 = 2 r3 = 3 r4 = 8 Answer: 3, 9, 15, 45 Simple algorithms: 1 q calls to selection algorithm (quickselect, median of medians) ⇝ O(Nq) 2 Divide & conquer 1: (single) select r⌈q/2⌉ -th smallest and partition x1 , . . . , xN around it. Recursively select r1 , . . . , r⌈q/2⌉−1 and r⌈q/2⌉+1 , . . . , rq ⇝ O(N lg q) 3 Divide & conquer 2: Find median of x1 , . . . , xN and split query ranks. Recurse where subproblem contains query ranks. ⇝ O(N lg q) Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 2 / 12

The Multiple Selection Problem Goal: find q elements of ranks r1 < r2 < · · · < rq from unsorted elements here: all distinct x1 , . . . , xN ⇝ Report sought elements x(r1) , . . . , x(rq) in sorted order Example: 67 x1 30 x2 45 x3 33 x4 15 x5 99 x6 26 x7 90 x8 55 x9 9 x10 96 x11 45 x12 95 x13 31 x14 3 x15 3 1 9 2 15 3 26 4 30 5 31 6 33 7 45 8 45 9 55 10 67 11 90 12 95 13 96 14 99 15 r1 = 1 r2 = 2 r3 = 3 r4 = 8 Answer: 3, 9, 15, 45 Simple algorithms: 1 q calls to selection algorithm (quickselect, median of medians) ⇝ O(Nq) 2 Divide & conquer 1: (single) select r⌈q/2⌉ -th smallest and partition x1 , . . . , xN around it. Recursively select r1 , . . . , r⌈q/2⌉−1 and r⌈q/2⌉+1 , . . . , rq ⇝ O(N lg q) 3 Divide & conquer 2: Find median of x1 , . . . , xN and split query ranks. Recurse where subproblem contains query ranks. ⇝ O(N lg q) Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 2 / 12

Fine-grained bounds “Gap Entropy”: B = q+1 i=1 ∆i log2 N ∆i with ∆i = ri − ri−1 (1 ⩽ i ⩽ q + 1, r0 = 0 and rq+1 = N + 1) r1 = 1 r2 = 2 r3 = 3 r4 = 8 ∆ 0 = 1 ∆ 1 = 1 ∆ 2 = 1 ∆0 = 5 ∆1 = 8 ⇝ B ≈ 26.9 lower bound of B − O(N) comparisons via “finish sorting” argument upper bound of B + o(B) + O(N) comparisons Kaligosi, Mehlhorn, Munro & Sanders: Towards Optimal Multiple Selection, ICALP 2005 follows from nontrivial analysis; much easier: repeated median selection & recursion gets O(B + N) (→ paper) ⇝ In internal memory, multiple-selection problem solved✓ Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 3 / 12

Fine-grained bounds “Gap Entropy”: B = q+1 i=1 ∆i log2 N ∆i with ∆i = ri − ri−1 (1 ⩽ i ⩽ q + 1, r0 = 0 and rq+1 = N + 1) r1 = 1 r2 = 2 r3 = 3 r4 = 8 ∆ 0 = 1 ∆ 1 = 1 ∆ 2 = 1 ∆0 = 5 ∆1 = 8 ⇝ B ≈ 26.9 lower bound of B − O(N) comparisons via “finish sorting” argument upper bound of B + o(B) + O(N) comparisons Kaligosi, Mehlhorn, Munro & Sanders: Towards Optimal Multiple Selection, ICALP 2005 follows from nontrivial analysis; much easier: repeated median selection & recursion gets O(B + N) (→ paper) ⇝ In internal memory, multiple-selection problem solved✓ Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 3 / 12

Fine-grained bounds “Gap Entropy”: B = q+1 i=1 ∆i log2 N ∆i with ∆i = ri − ri−1 (1 ⩽ i ⩽ q + 1, r0 = 0 and rq+1 = N + 1) r1 = 1 r2 = 2 r3 = 3 r4 = 8 ∆ 0 = 1 ∆ 1 = 1 ∆ 2 = 1 ∆0 = 5 ∆1 = 8 ⇝ B ≈ 26.9 lower bound of B − O(N) comparisons via “finish sorting” argument upper bound of B + o(B) + O(N) comparisons Kaligosi, Mehlhorn, Munro & Sanders: Towards Optimal Multiple Selection, ICALP 2005 follows from nontrivial analysis; much easier: repeated median selection & recursion gets O(B + N) (→ paper) ⇝ In internal memory, multiple-selection problem solved✓ Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 3 / 12

Outline 1 Multiple Selection 1 Multiple Selection 2 Cache Oblivious Algorithms 2 Cache Oblivious Algorithms 3 Funnelselect 3 Funnelselect 4 Conclusion 4 Conclusion Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 3 / 12

2 Cache Oblivious Algorithms 2 Cache Oblivious Algorithms Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 3 / 12

IO Model & CO Model CPU fast random access slow access only in blocks of B cells IO Model (External Memory Model): Cost of computation = #“I/Os” = #blocks transfered between cache & external memory Cache size M ... external memory – size unbounded Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 4 / 12

IO Model & CO Model CPU fast random access slow access only in blocks of B cells IO Model (External Memory Model): Cost of computation = #“I/Os” = #blocks transfered between cache & external memory Cache size M ... external memory – size unbounded Cache Oblivious (CO) Model algorithm can’t use M and B (I/Os automatic, OPT paging policy) ⇝ CO algorithm works for all even hierarchies! M and B Frigo, Leiserson, Prokop, Ramachandran: Cache-oblivious algorithms, TALG 2012 Tall Cache Assumption: M ⩾ B1+ε think: ε = 1 ⇝ cache fits many cache lines necessary for existence of I/O-optimal (comparison-based) cache-oblivious sorting algorithms Brodal, Fagerberg: On the limits of cache-obliviousness, STOC 2003 Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 4 / 12

IO Model & CO Model CPU fast random access slow access only in blocks of B cells IO Model (External Memory Model): Cost of computation = #“I/Os” = #blocks transfered between cache & external memory Cache size M ... external memory – size unbounded Cache Oblivious (CO) Model algorithm can’t use M and B (I/Os automatic, OPT paging policy) ⇝ CO algorithm works for all even hierarchies! M and B Frigo, Leiserson, Prokop, Ramachandran: Cache-oblivious algorithms, TALG 2012 Tall Cache Assumption: M ⩾ B1+ε think: ε = 1 ⇝ cache fits many cache lines necessary for existence of I/O-optimal (comparison-based) cache-oblivious sorting algorithms Brodal, Fagerberg: On the limits of cache-obliviousness, STOC 2003 Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 4 / 12

IO Model & CO Model CPU fast random access slow access only in blocks of B cells IO Model (External Memory Model): Cost of computation = #“I/Os” = #blocks transfered between cache & external memory Cache size M ... external memory – size unbounded Cache Oblivious (CO) Model algorithm can’t use M and B (I/Os automatic, OPT paging policy) ⇝ CO algorithm works for all even hierarchies! M and B Frigo, Leiserson, Prokop, Ramachandran: Cache-oblivious algorithms, TALG 2012 Tall Cache Assumption: M ⩾ B1+ε think: ε = 1 ⇝ cache fits many cache lines necessary for existence of I/O-optimal (comparison-based) cache-oblivious sorting algorithms Brodal, Fagerberg: On the limits of cache-obliviousness, STOC 2003 Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 4 / 12

Previous Work & Our Results Reference Comparisons I/Os Comments Single selection Blum et al. 1973 wc 5.4305N O(N/B) CO, deterministic Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 5 / 12

Previous Work & Our Results Reference Comparisons I/Os Comments Single selection Blum et al. 1973 wc 5.4305N O(N/B) CO, deterministic Multiple selection Dobkin & Munro 1981 wc 3B + O(N) O((B + N)/B) CO, deterministic Hu et al. 2014 wc O(B + N) O(BI/O + N/B) deterministic Barbay et al. 2016 wc O(B + N) O(BI/O + N/B) online, determ., M ⩾ B1+ε B = q+1 i=1 ∆i log2 N ∆i BI/O = q+1 i=1 ∆i B logM B N ∆i = B B log2 (M/B) ≪ (usually) B B Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 5 / 12

Previous Work & Our Results Reference Comparisons I/Os Comments Single selection Blum et al. 1973 wc 5.4305N O(N/B) CO, deterministic Multiple selection Dobkin & Munro 1981 wc 3B + O(N) O((B + N)/B) CO, deterministic Hu et al. 2014 wc O(B + N) O(BI/O + N/B) deterministic Barbay et al. 2016 wc O(B + N) O(BI/O + N/B) online, determ., M ⩾ B1+ε Funnelselect E O(B + N) O(BI/O + N/B) CO, randomized, M ⩾ B1+ε Lower bound B − O(N) Ω(BI/O ) − O N B M ⩾ B1+ε B = q+1 i=1 ∆i log2 N ∆i BI/O = q+1 i=1 ∆i B logM B N ∆i = B B log2 (M/B) ≪ (usually) B B Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 5 / 12

Previous Work & Our Results Reference Comparisons I/Os Comments Single selection Blum et al. 1973 wc 5.4305N O(N/B) CO, deterministic Multiple selection Dobkin & Munro 1981 wc 3B + O(N) O((B + N)/B) CO, deterministic Hu et al. 2014 wc O(B + N) O(BI/O + N/B) deterministic Barbay et al. 2016 wc O(B + N) O(BI/O + N/B) online, determ., M ⩾ B1+ε Funnelselect E O(B + N) O(BI/O + N/B) CO, randomized, M ⩾ B1+ε Lower bound B − O(N) Ω(BI/O ) − O N B M ⩾ B1+ε B = q+1 i=1 ∆i log2 N ∆i BI/O = q+1 i=1 ∆i B logM B N ∆i = B B log2 (M/B) ≪ (usually) B B Funnelselect is the first cache-oblivious I/O-optimal algorithm. Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 5 / 12

Our technical challenge 1 Key algorithmic change from internal ⇝ external-memory: (for sorting and multiple selection) binary partitioning ⇝ ≈ M B -way partitioning can’t do that cache obliviously 2 cache-oblivious sorting uses Nδ-way partitioning by “funnels” ⇝ first partitioning round already has sorting complexity ⇝ ω(BI/O ) I/Os can’t use that for multiple selection Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 6 / 12

Our technical challenge 1 Key algorithmic change from internal ⇝ external-memory: (for sorting and multiple selection) binary partitioning ⇝ ≈ M B -way partitioning can’t do that cache obliviously 2 cache-oblivious sorting uses Nδ-way partitioning by “funnels” ⇝ first partitioning round already has sorting complexity ⇝ ω(BI/O ) I/Os can’t use that for multiple selection Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 6 / 12

Our technical challenge 1 Key algorithmic change from internal ⇝ external-memory: (for sorting and multiple selection) binary partitioning ⇝ ≈ M B -way partitioning can’t do that cache obliviously 2 cache-oblivious sorting uses Nδ-way partitioning by “funnels” ⇝ first partitioning round already has sorting complexity ⇝ ω(BI/O ) I/Os can’t use that for multiple selection Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 6 / 12

Our technical challenge 1 Key algorithmic change from internal ⇝ external-memory: (for sorting and multiple selection) binary partitioning ⇝ ≈ M B -way partitioning can’t do that cache obliviously 2 cache-oblivious sorting uses Nδ-way partitioning by “funnels” ⇝ first partitioning round already has sorting complexity ⇝ ω(BI/O ) I/Os can’t use that for multiple selection Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 6 / 12

Outline 1 Multiple Selection 1 Multiple Selection 2 Cache Oblivious Algorithms 2 Cache Oblivious Algorithms 3 Funnelselect 3 Funnelselect 4 Conclusion 4 Conclusion Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 6 / 12

3 Funnelselect 3 Funnelselect Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 6 / 12

Funnels for Partitioning Can use funnels from Funnelsort in reverse! Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 7 / 12

Funnels for Partitioning Can use funnels from Funnelsort in reverse! Funnel: k = 4 √ N-way merger (recursive binary merges) judiciously sized buffers for intermediate results Simplifying assumptions: N = 22i and d = 4 output array input arrays Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 7 / 12

Funnels for Partitioning Can use funnels from Funnelsort in reverse! Funnel: k = 4 √ N-way merger (recursive binary merges) judiciously sized buffers for intermediate results Simplifying assumptions: N = 22i and d = 4 output array input arrays P1 P3 P5 P7 P9 P11 P13 P15 P2 P6 P10 P14 P4 P12 P8 input array output arrays Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 7 / 12

Funnels for Partitioning Can use funnels from Funnelsort in reverse! Funnel: k = 4 √ N-way merger (recursive binary merges) judiciously sized buffers for intermediate results Simplifying assumptions: N = 22i and d = 4 P1 P3 P5 P7 P9 P11 P13 P15 P2 P6 P10 P14 P4 P12 P8 input array output arrays √ k-partitioner √ k-partitioners k-partitioner Funnel recursion (van Emde Boas) Funnel buffer sizes (largest in middle layer) ⇝ overall space O(k5/2) = O(N5/8) Nodes partition around pivot value P Partition = push input down when output buffer full, recursively push need final flush of all buffers ≈ I/O-optimal cache-oblivious gadget for ˆ k-way partitioning with ˆ k ≈ M0.3 ⇝ Can be used for expected I/O-optimal CO quicksort (but better options available) Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 7 / 12

Funnels for Partitioning Can use funnels from Funnelsort in reverse! Funnel: k = 4 √ N-way merger (recursive binary merges) judiciously sized buffers for intermediate results Simplifying assumptions: N = 22i and d = 4 P1 P3 P5 P7 P9 P11 P13 P15 P2 P6 P10 P14 P4 P12 P8 input array output arrays √ k-partitioner √ k-partitioners k2 per buffer k-partitioner Funnel recursion (van Emde Boas) Funnel buffer sizes (largest in middle layer) ⇝ overall space O(k5/2) = O(N5/8) Nodes partition around pivot value P Partition = push input down when output buffer full, recursively push need final flush of all buffers ≈ I/O-optimal cache-oblivious gadget for ˆ k-way partitioning with ˆ k ≈ M0.3 ⇝ Can be used for expected I/O-optimal CO quicksort (but better options available) Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 7 / 12

Funnels for Partitioning Can use funnels from Funnelsort in reverse! Funnel: k = 4 √ N-way merger (recursive binary merges) judiciously sized buffers for intermediate results Simplifying assumptions: N = 22i and d = 4 P1 P3 P5 P7 P9 P11 P13 P15 P2 P6 P10 P14 P4 P12 P8 input array output arrays √ k-partitioner √ k-partitioners k2 per buffer k-partitioner Funnel recursion (van Emde Boas) Funnel buffer sizes (largest in middle layer) ⇝ overall space O(k5/2) = O(N5/8) Nodes partition around pivot value P Partition = push input down when output buffer full, recursively push need final flush of all buffers ≈ I/O-optimal cache-oblivious gadget for ˆ k-way partitioning with ˆ k ≈ M0.3 ⇝ Can be used for expected I/O-optimal CO quicksort (but better options available) Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 7 / 12

Funnels for Partitioning Can use funnels from Funnelsort in reverse! Funnel: k = 4 √ N-way merger (recursive binary merges) judiciously sized buffers for intermediate results Simplifying assumptions: N = 22i and d = 4 P1 P3 P5 P7 P9 P11 P13 P15 P2 P6 P10 P14 P4 P12 P8 input array output arrays √ k-partitioner √ k-partitioners k2 per buffer k-partitioner Funnel recursion (van Emde Boas) Funnel buffer sizes (largest in middle layer) ⇝ overall space O(k5/2) = O(N5/8) Nodes partition around pivot value P Partition = push input down when output buffer full, recursively push need final flush of all buffers ≈ I/O-optimal cache-oblivious gadget for ˆ k-way partitioning with ˆ k ≈ M0.3 ⇝ Can be used for expected I/O-optimal CO quicksort (but better options available) Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 7 / 12

Funnels for Partitioning Can use funnels from Funnelsort in reverse! Funnel: k = 4 √ N-way merger (recursive binary merges) judiciously sized buffers for intermediate results Simplifying assumptions: N = 22i and d = 4 P1 P3 P5 P7 P9 P11 P13 P15 P2 P6 P10 P14 P4 P12 P8 input array output arrays √ k-partitioner √ k-partitioners k2 per buffer k-partitioner Funnel recursion (van Emde Boas) Funnel buffer sizes (largest in middle layer) ⇝ overall space O(k5/2) = O(N5/8) Nodes partition around pivot value P Partition = push input down when output buffer full, recursively push need final flush of all buffers ≈ I/O-optimal cache-oblivious gadget for ˆ k-way partitioning with ˆ k ≈ M0.3 ⇝ Can be used for expected I/O-optimal CO quicksort (but better options available) Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 7 / 12

Early Truncation Recall: for multiple selection (in general), can’t use full k-partitioner (already sorting complexity) Early truncation: Don’t split buckets that don’t contain any query ranks! Buckets depend on (random) pivots ⇝ Only know a query’s bucket after partitioning ... Assume bucket boundaries close within safety margin ±ξ = N1/2+δ to expected location. ⇝ If bucket is expected query free, don’t split further. “Expected query free” known up front (Depends only on N & query ranks) ⇝ Can remove unwanted partitioning nodes from funnel in preprocessing. Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 8 / 12

Early Truncation Recall: for multiple selection (in general), can’t use full k-partitioner (already sorting complexity) Early truncation: Don’t split buckets that don’t contain any query ranks! P1 P3 P5 P7 P9 P11 P13 P15 P2 P6 P10 P14 P4 P12 P8 Buckets depend on (random) pivots ⇝ Only know a query’s bucket after partitioning ... Assume bucket boundaries close within safety margin ±ξ = N1/2+δ to expected location. ⇝ If bucket is expected query free, don’t split further. “Expected query free” known up front (Depends only on N & query ranks) ⇝ Can remove unwanted partitioning nodes from funnel in preprocessing. Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 8 / 12

Early Truncation Recall: for multiple selection (in general), can’t use full k-partitioner (already sorting complexity) Early truncation: Don’t split buckets that don’t contain any query ranks! P1 P3 P7 P9 P2 P6 P10 P4 P12 P8 Buckets depend on (random) pivots ⇝ Only know a query’s bucket after partitioning ... Assume bucket boundaries close within safety margin ±ξ = N1/2+δ to expected location. ⇝ If bucket is expected query free, don’t split further. “Expected query free” known up front (Depends only on N & query ranks) ⇝ Can remove unwanted partitioning nodes from funnel in preprocessing. Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 8 / 12

Early Truncation Recall: for multiple selection (in general), can’t use full k-partitioner (already sorting complexity) Early truncation: Don’t split buckets that don’t contain any query ranks! P1 P3 P7 P9 P2 P6 P10 P4 P12 P8 Buckets depend on (random) pivots ⇝ Only know a query’s bucket after partitioning ... Assume bucket boundaries close within safety margin ±ξ = N1/2+δ to expected location. ⇝ If bucket is expected query free, don’t split further. “Expected query free” known up front (Depends only on N & query ranks) ⇝ Can remove unwanted partitioning nodes from funnel in preprocessing. Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 8 / 12

Early Truncation Recall: for multiple selection (in general), can’t use full k-partitioner (already sorting complexity) Early truncation: Don’t split buckets that don’t contain any query ranks! P1 P3 P7 P9 P2 P6 P10 P4 P12 P8 Buckets depend on (random) pivots ⇝ Only know a query’s bucket after partitioning ... Assume bucket boundaries close within safety margin ±ξ = N1/2+δ to expected location. ⇝ If bucket is expected query free, don’t split further. “Expected query free” known up front (Depends only on N & query ranks) ⇝ Can remove unwanted partitioning nodes from funnel in preprocessing. Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 8 / 12

Pivot Sampling Recall this? Assume bucket boundaries close within safety margin ±ξ = N1/2+δ to expected location. We have to make that true by choosing pivots well The following randomized choice works with high probability: by standard Chernoff bound arguments 1 Include each element in sample ¯ S with prob. p = 1/ log2 (N). 2 Sort the sample ¯ S. 3 Pick pivot Pi as ≈ ipN/kth smallest in ¯ S (i = 1, . . . , k − 1) Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 9 / 12

Pivot Sampling Recall this? Assume bucket boundaries close within safety margin ±ξ = N1/2+δ to expected location. We have to make that true by choosing pivots well The following randomized choice works with high probability: by standard Chernoff bound arguments 1 Include each element in sample ¯ S with prob. p = 1/ log2 (N). 2 Sort the sample ¯ S. 3 Pick pivot Pi as ≈ ipN/kth smallest in ¯ S (i = 1, . . . , k − 1) Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 9 / 12

Overall Algorithm Funnelselect: 1 Sample pivots P1 , . . . , Pk−1 2 Build k-partitioner using P1 , . . . , Pk−1 3 Mark expected query free buckets & rewire their parent’s buffer to output 4 For each bucket with queries: i Sort the bucket (Funnelsort) ii Report sought elements Observations: No (top-level) recursion needed Algorithm can fail at several places sample too small, sample too large, pivots too skewed, query in expected query free bucket ⇝ restart but: with high probability, no fails ⇝ no effect on expected running time Can be augmented to produce input partitioned around sought elements in contiguous external memory (default: only return sought elements in order) Can be made to handle equal elements Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 10 / 12

Overall Algorithm Funnelselect: 1 Sample pivots P1 , . . . , Pk−1 2 Build k-partitioner using P1 , . . . , Pk−1 3 Mark expected query free buckets & rewire their parent’s buffer to output 4 For each bucket with queries: i Sort the bucket (Funnelsort) ii Report sought elements Observations: No (top-level) recursion needed Algorithm can fail at several places sample too small, sample too large, pivots too skewed, query in expected query free bucket ⇝ restart but: with high probability, no fails ⇝ no effect on expected running time Can be augmented to produce input partitioned around sought elements in contiguous external memory (default: only return sought elements in order) Can be made to handle equal elements Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 10 / 12

Outline 1 Multiple Selection 1 Multiple Selection 2 Cache Oblivious Algorithms 2 Cache Oblivious Algorithms 3 Funnelselect 3 Funnelselect 4 Conclusion 4 Conclusion Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 10 / 12

4 Conclusion 4 Conclusion Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 10 / 12

Conclusion We presented Funnelselect the first I/O-optimal, cache-oblivious multiple-selection algorithm The presented algorithm is inherently randomized, but we meanwhile found a deterministic method! (but too late for proceedings ... but stay tuned) Open Problems: 1 Funnelselect assumes a tall cache. Cannot be avoided since I/O-opt. CO sorting (⊂ multiple selection) requires it. But single selection doesn’t! ⇝ What happens in between? 2 Can online multiple selection be solved I/O-optimal cache obliviously? Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 11 / 12

Conclusion We presented Funnelselect the first I/O-optimal, cache-oblivious multiple-selection algorithm The presented algorithm is inherently randomized, but we meanwhile found a deterministic method! (but too late for proceedings ... but stay tuned) Open Problems: 1 Funnelselect assumes a tall cache. Cannot be avoided since I/O-opt. CO sorting (⊂ multiple selection) requires it. But single selection doesn’t! ⇝ What happens in between? 2 Can online multiple selection be solved I/O-optimal cache obliviously? Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 11 / 12

Conclusion We presented Funnelselect the first I/O-optimal, cache-oblivious multiple-selection algorithm The presented algorithm is inherently randomized, but we meanwhile found a deterministic method! (but too late for proceedings ... but stay tuned) Open Problems: 1 Funnelselect assumes a tall cache. Cannot be avoided since I/O-opt. CO sorting (⊂ multiple selection) requires it. But single selection doesn’t! ⇝ What happens in between? 2 Can online multiple selection be solved I/O-optimal cache obliviously? Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 11 / 12

We’re hiring! for Computing over compressed graph-structured data 3 year postdoc PhD student Liverpool sounds cool? → Talk to me! Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 12 / 12

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I/O Lower Bound Recall: B = q+1 i=1 ∆i log2 N ∆i with ∆i = ri − ri−1 (1 ⩽ i ⩽ q + 1, r0 = 0, rq+1 = N + 1) BI/O = q+1 i=1 ∆i B logM B N ∆i = B B log2 (M/B) ≪ (usually) B B Theorem (Lower bound) External-memory multiple selection in expectation requires Ω(BI/O ) − O N B logM/B B I/Os. Follows from general reduction (cmps bound ⇝ I/Os bound) Arge, Knudsen, Larsen: A general lower bound on the I/O-complexity of comparison-based algorithms, WADS 1993 “finish-sorting” argument no longer rigorous Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 14 / 12

Recap: (Lazy) Funnelsort Funnelsort is a k = 4 √ N-way Mergesort (outer recursion) each realized by recursive binary merging (inner recursion) with judiciously sized buffers for intermediate results (funnel) Simplifying assumptions: N = 22i and d = 4 (i.e., ε ⩾ 2 3 in tall cache assumption) d > 2 controls fanout (≈ N1/d-way merging) output array k2 per buffer input arrays √ k-merger √ k-mergers k-merger Recursive structure (cf. van Emde Boas trees) largest buffers in middle layer ⇝ overall space O(k5/2) = O(N5/8) Merge = fill output buffer when input buffer empty, recursively fill it ≈ I/O-optimal cache-oblivious gadget for ˆ k-way merging with ˆ k ≈ M0.3 Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 15 / 12

Tall Caches CPU Cache size M T a l l ... external memory – size unbounded Tall Cache Assumption: M ⩾ B1+ε think: ε = 1 ⇝ M/B ⩾ Bε ⇝ cache fits many cache lines necessary for existence of I/O-optimal (comparison-based) cache-oblivious sorting algorithms Brodal, Fagerberg: On the limits of cache-obliviousness, STOC 2003 Sebastian Wild Funnelselect: Cache-Oblivious Multiple Selection 2023-09-05 16 / 12