Optimal Partitioning for Dual Pivot Quicksort
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1 Optimal Partitioning for Dual Pivot Quicksort Martin Aumüller, Martin Dietzfelbinger Technische Universität Ilmenau, Germany ICALP 2013 Riga, July 12, 2013 M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 1/17
2 Dual Pivot Quicksort p M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 2/17
3 Dual Pivot Quicksort p Choose two pivots p, q with p < q. M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 2/17
4 Dual Pivot Quicksort p q Choose two pivots p, q with p < q. M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 2/17
5 Dual Pivot Quicksort p q Partition, i.e., re-arrange elements via swaps. Partition: p p p... q q q M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 2/17
6 Dual Pivot Quicksort p q Partition, i.e., re-arrange elements via swaps. Partition: p p p... q q q M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 2/17
7 Dual Pivot Quicksort p q Use recursion to sort these three subarrays. Partition: p p p... q q q M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 2/17
8 Dual Pivot Quicksort p q Use recursion to sort these three subarrays. M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 2/17
9 Dual Pivot Quicksort p Done. M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 2/17
10 Dual Pivot Quicksort: History Robert Sedgewick (1975): Analyzed a dual pivot algorithm that makes much more swaps than classical quicksort on average no further investigation. Pascal Hennequin (1991): Thorough analysis of quicksort with k 1 pivots. for k = 2, no improvements found. for k 3, slight improvements, partitioning too complicated. M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 3/17
11 Dual Pivot Quicksort: History Robert Sedgewick (1975): Analyzed a dual pivot algorithm that makes much more swaps than classical quicksort on average no further investigation. Pascal Hennequin (1991): Thorough analysis of quicksort with k 1 pivots. for k = 2, no improvements found. for k 3, slight improvements, partitioning too complicated. So, why bother? M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 3/17
12 Dual Pivot Quicksort: History Robert Sedgewick (1975): Analyzed a dual pivot algorithm that makes much more swaps than classical quicksort on average no further investigation. Pascal Hennequin (1991): Thorough analysis of quicksort with k 1 pivots. for k = 2, no improvements found. for k 3, slight improvements, partitioning too complicated. So, why bother? Java 7 (2009): Classical quicksort is replaced by a dual pivot quicksort variant due to Yaroslavskiy! Rigorous analysis of the core of this algorithm by Wild and Nebel (2012). M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 3/17
13 Classical QS vs. Yaroslavskiy s Dual Pivot QS Classical QS Yaroslavskiy Average Comparison Count 2n ln n (= 1.38n log n) 1.9n ln n Experiments: Yaroslavskiy s algorithm around 10% faster! M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 4/17
14 Classical QS vs. Yaroslavskiy s Dual Pivot QS Classical QS Yaroslavskiy Optimal DP Average Comparison Count 2n ln n (= 1.38n log n) 1.9n ln n?? Experiments: Yaroslavskiy s algorithm around 10% faster! Question: What is possible? M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 4/17
15 Focus of This Talk introduce a model that captures all dual pivot algorithms, give a unified analysis of the average comparison count, identify optimal algorithms. For analysis: input is random permutation of {1,..., n} left-most and right-most elements are the two pivots analyze average sorting cost M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 5/17
16 Reduce Sorting to Partitioning Fact (Hennequin, 1991) For dual pivot quicksort, average partitioning cost of a n + O(1) leads to average sorting cost 6 5a n ln n + O(n) a n + O(1) 6 5 a n ln n + O(n) p p p... q q q M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 6/17
17 Reduce Partitioning to Classification simplification: focus on classifying elements as small, medium, or large independent of this: clever swap strategy to obtain a partition p p p... q q q M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 7/17
18 Model: Decision Tree 3 : p σ µ λ 2 : q 4 : p 2 : p σ µ λ σ µ λ σ µ λ 4 : p 4 : q 4 : q 2 : q 2 : p 2 : p 4 : p 4 : q 4 : p σ σ µ λ λ a2 : σ a3 : σ a4 : σ a2 : σ a2 : µ a2 : λ... a3 : µ a3 : µ a3 : µ... a4 : µ a4 : µ a4 : µ a2 : λ a3 : λ a4 : λ Example: M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 8/17
19 Model: Decision Tree 3 : p σ µ λ 2 : q 4 : p 2 : p σ µ λ σ µ λ σ µ λ 4 : p 4 : q 4 : q 2 : q 2 : p 2 : p 4 : p 4 : q 4 : p σ σ µ λ λ a2 : σ a3 : σ a4 : σ a2 : σ a2 : µ a2 : λ... a3 : µ a3 : µ a3 : µ... a4 : µ a4 : µ a4 : µ a2 : λ a3 : λ a4 : λ Example: p q M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 8/17
20 Model: Decision Tree 3 : p σ µ λ 2 : q 4 : p 2 : p σ µ λ σ µ λ σ µ λ 4 : p 4 : q 4 : q 2 : q 2 : p 2 : p 4 : p 4 : q 4 : p σ σ µ λ λ a2 : σ a3 : σ a4 : σ a2 : σ a2 : µ a2 : λ... a3 : µ a3 : µ a3 : µ... a4 : µ a4 : µ a4 : µ a2 : λ a3 : λ a4 : λ Example: p q M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 8/17
21 Model: Decision Tree +1 3 : p σ µ λ 2 : q 4 : p 2 : p σ µ λ σ µ λ σ µ λ 4 : p 4 : q 4 : q 2 : q 2 : p 2 : p 4 : p 4 : q 4 : p σ σ µ λ λ a2 : σ a3 : σ a4 : σ a2 : σ a2 : µ a2 : λ... a3 : µ a3 : µ a3 : µ... a4 : µ a4 : µ a4 : µ a2 : λ a3 : λ a4 : λ Example: p σ q M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 8/17
22 Model: Decision Tree +1 3 : p σ µ λ 2 : q 4 : p 2 : p σ µ λ σ µ λ σ µ λ 4 : p 4 : q 4 : q 2 : q 2 : p 2 : p 4 : p 4 : q 4 : p σ σ µ λ λ a2 : σ a3 : σ a4 : σ a2 : σ a2 : µ a2 : λ... a3 : µ a3 : µ a3 : µ... a4 : µ a4 : µ a4 : µ a2 : λ a3 : λ a4 : λ Example: p σ q M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 8/17
23 Model: Decision Tree +1 3 : p +2 σ µ λ 2 : q 4 : p 2 : p σ µ λ σ µ λ σ µ λ 4 : p 4 : q 4 : q 2 : q 2 : p 2 : p 4 : p 4 : q 4 : p σ σ µ λ λ a2 : σ a3 : σ a4 : σ a2 : σ a2 : µ a2 : λ... a3 : µ a3 : µ a3 : µ... a4 : µ a4 : µ a4 : µ a2 : λ a3 : λ a4 : λ Example: p µ σ q M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 8/17
24 Model: Decision Tree +1 3 : p +2 σ µ λ 2 : q 4 : p 2 : p σ µ λ σ µ λ σ µ λ 4 : p 4 : q 4 : q 2 : q 2 : p 2 : p 4 : p 4 : q 4 : p σ σ µ λ λ a2 : σ a3 : σ a4 : σ a2 : σ a2 : µ a2 : λ... a3 : µ a3 : µ a3 : µ... a4 : µ a4 : µ a4 : µ a2 : λ a3 : λ a4 : λ Example: p µ σ q M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 8/17
25 Model: Decision Tree +1 3 : p +2 σ µ λ 2 : q 4 : p 2 : p σ µ λ σ µ λ σ µ λ +2 4 : p 4 : q 4 : q 2 : q 2 : p 2 : p 4 : p 4 : q 4 : p σ σ µ λ λ a2 : σ a3 : σ a4 : σ a2 : σ a2 : µ a2 : λ... a3 : µ a3 : µ a3 : µ... a4 : µ a4 : µ a4 : µ a2 : λ a3 : λ a4 : λ Example: p µ σ σ q M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 8/17
26 Model: Decision Tree +1 3 : p +2 σ µ λ 2 : q 4 : p 2 : p σ µ λ σ µ λ σ µ λ +2 4 : p 4 : q 4 : q 2 : q 2 : p 2 : p 4 : p 4 : q 4 : p σ σ µ λ λ a2 : σ a3 : σ a4 : σ a2 : σ a2 : µ a2 : λ... a3 : µ a3 : µ a3 : µ... a4 : µ a4 : µ a4 : µ a2 : λ a3 : λ a4 : λ Example: p µ σ σ q 5 comparisons. M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 8/17
27 Average Cost For the average partitioning/classification cost p n we get: p n 4/3n + average number of additional comparisons For all classification algorithms: each element has to be compared at least once against a pivot each medium element has to be compared to both ( n/3 on avg.) Additional comparisons: a small element is compared with the larger pivot first a large element is compared with the smaller pivot first M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 9/17
28 Strategy 1 (unrealistic) Optimal Strategies σ λ σ µ???? Next element Let s, l denote the number of small/large elements in area (resp.). M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 10/17
29 Strategy 1 (unrealistic) Optimal Strategies σ λ σ µ???? Next element Let s, l denote the number of small/large elements in Classification l > s: compare with larger pivot first. l s: compare with smaller pivot first. area (resp.). M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 10/17
30 Optimal Strategies Strategy 1 (unrealistic) σ λ σ µ???? Strategy 2 σ λ σ µ???? Next element Next element Let s, l denote the number of small/large elements in Classification l > s: compare with larger pivot first. l s: compare with smaller pivot first. area (resp.). M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 10/17
31 Optimal Strategies Strategy 1 (unrealistic) σ λ σ µ???? Next element Strategy 2 σ λ σ µ???? Next element Let s, l denote the number of small/large elements in area (resp.). Classification l > s: compare with larger pivot first. Theorem l s: compare with smaller pivot first. These strategies (turned into dp algorithms) have the minimum possible average sorting cost: 1.8n ln n + O(n). M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 10/17
32 Optimal Strategies Strategy 1 (unrealistic) σ λ σ µ σ λ σ σ Next element Strategy 2 σ λ σ µ σ λ σ σ Next element Let s, l denote the number of small/large elements in area (resp.). Classification l > s: compare with larger pivot first. Theorem l s: compare with smaller pivot first. These strategies (turned into dp algorithms) have the minimum possible average sorting cost: 1.8n ln n + O(n). M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 10/17
33 Optimal Strategies Strategy 1 (unrealistic) σ λ σ µ σ λ σ σ Next element Strategy 2 σ λ σ µ σ λ σ σ Next element Let s, l denote the number of small/large elements in area (resp.). Classification l > s: compare with larger pivot first. Theorem l s: compare with smaller pivot first. These strategies (turned into dp algorithms) have the minimum possible average sorting cost: 1.8n ln n + O(n). M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 10/17
34 Cost of an Arbitrary Decision Tree p n 4/3n + average number of additional comparisons M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 11/17
35 Cost of an Arbitrary Decision Tree p n 4/3n + average number of additional comparisons Central: f q s,l average number of comparisons to the larger pivot first. Lemma For the average partition cost p n of a decision tree T we have: p n = 4 3 n + 1 ( ) ( n ) f q s,l s + (n 2 f q s,l ) l + o(n). 2 (n 2) s+l n 2 dependence introduces an error term of o(n). M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 11/17
36 Cost of an Arbitrary Decision Tree p n 4/3n + average number of additional comparisons Central: f q s,l average number of comparisons to the larger pivot first. Lemma For the average partition cost p n of a decision tree T we have: p n = 4 3 n + 1 ( ) ( n ) f q s,l s + (n 2 f q s,l ) l + o(n). 2 (n 2) s+l n 2 dependence introduces an error term of o(n). M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 11/17
37 Cost of an Arbitrary Decision Tree p n 4/3n + average number of additional comparisons Central: f q s,l average number of comparisons to the larger pivot first. Lemma For the average partition cost p n of a decision tree T we have: p n = 4 3 n + 1 ( ) ( n ) f q s,l s + (n 2 f q s,l ) l + o(n). 2 (n 2) s+l n 2 dependence introduces an error term of o(n). M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 11/17
38 Cost of an Arbitrary Decision Tree p n 4/3n + average number of additional comparisons Central: f q s,l average number of comparisons to the larger pivot first. Lemma For the average partition cost p n of a decision tree T we have: p n = 4 3 n + 1 ( ) ( n ) f q s,l s + (n 2 f q s,l ) l + o(n). 2 (n 2) s+l n 2 dependence introduces an error term of o(n). M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 11/17
39 Cost of an Arbitrary Decision Tree p n 4/3n + average number of additional comparisons Central: f q s,l average number of comparisons to the larger pivot first. Lemma For the average partition cost p n of a decision tree T we have: p n = 4 3 n + 1 ( ) ( n ) f q s,l s + (n 2 f q s,l ) l + o(n). 2 (n 2) s+l n 2 dependence introduces an error term of o(n). Message: f q s,l fully describes the average cost (up to lower order terms.) M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 11/17
40 Cost of an Arbitrary Decision Tree p n 4/3n + average number of additional comparisons Central: f q s,l average number of comparisons to the larger pivot first. Lemma For the average partition cost p n of a decision tree T we have: p n = 4 3 n + 1 ( ) ( n ) f q s,l s + (n 2 f q s,l ) l + o(n). 2 (n 2) s+l n 2 dependence introduces an error term of o(n). Message: f q s,l fully describes the average cost (up to lower order terms.) Example: Yaroslavskiy s algorithm: f q s,l = l. (Leads to: 1.9n ln n.) M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 11/17
41 An Optimal Strategy Using An Oracle Assume: An oracle tells us whether or not l > s for an input: Classification l > s: Compare all elements to the larger pivot first l s: Compare all elements to the smaller pivot first M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 12/17
42 An Optimal Strategy Using An Oracle Assume: An oracle tells us whether or not l > s for an input: Classification l > s: Compare all elements to the larger pivot first l s: Compare all elements to the smaller pivot first Why is it optimal? It minimizes p n = 4 3 n + 1 ( n ) 2 (n 2) s+l n 2 ( f q s,l s + (n 2 f q s,l ) l ) + o(n). M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 12/17
43 An Optimal Strategy Using An Oracle Assume: An oracle tells us whether or not l > s for an input: Classification l > s: Compare all elements to the larger pivot first (f q s,l = n 2). l s: Compare all elements to the smaller pivot first (f q s,l = 0). Why is it optimal? It minimizes p n = 4 3 n + 1 ( n ) 2 (n 2) s+l n 2 ( f q s,l s + (n 2 f q s,l ) l ) + o(n). M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 12/17
44 An Optimal Strategy Using An Oracle Assume: An oracle tells us whether or not l > s for an input: Classification l > s: Compare all elements to the larger pivot first (f q s,l = n 2). l s: Compare all elements to the smaller pivot first (f q s,l = 0). Why is it optimal? It minimizes p n = 4 3 n + 1 ( n ) 2 (n 2) s+l n 2 ( f q s,l s + (n 2 f q s,l ) l ) + o(n). Average sorting cost: 1.8n ln n + o(n ln n) M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 12/17
45 Random Sampling Just look at, e.g., n 2/3 elements. Classification Seen more large elements than small elements? Compare all elements with the larger pivot first, otherwise with the smaller pivot first. proof that this works: standard machinery (tail estimates). difference to oracle estimation is o(n). Average sorting cost: 1.8n ln n + o(n ln n). M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 13/17
46 Experiments Our Method (N) Yaroslavskiy Clever Quicksort time [ms] n/10^6 Sorting integers: About 3% slower than Yaroslavskiy. M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 14/17
47 Experiments (2) Our Method (N) Yaroslavskiy Clever Quicksort 8000 time [ms] n/10^6 Sorting strings: About 2% faster than Yaroslavskiy. M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 15/17
48 Conclusion and Open Questions So far: gave a unified model for dual pivot quicksort algorithms, showed how to calculate the average cost for general algorithms, identified three (but only two algorithmic ) optimal dual pivot quicksort algorithms. Optimum: 1.8n ln n. (QS: 2n ln n, Yaroslavskiy: 1.9n ln n.) Open questions: Cost measure that describes running time behavior accurately? Optimal algorithms for multi-pivot quicksort with k 3 pivots? (Best (known) 3-pivot algorithm: 1.846n ln n comparisons on average) M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 16/17
49 Thanks! Any questions? M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 17/17
50 Theoretical Result Classical QS Yaroslavskiy Our algorithm Average Comparison Count 2n ln n 1.9n ln n 1.8n ln n Average Swap Count n ln n 0.6n ln n n ln n LB: 0.3n ln n M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 16/17
51 p p p... q q?? q current element M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 17/17
52 p p p... q q?? q small element M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 17/17
53 p p p... q q?? q exchange M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 17/17
54 p p p... q?? q q exchange M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 17/17
55 p p p... q?? q q inspect next element M. Aumüller Optimal Partitioning for Dual Pivot Quicksort 17/17
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