Analysis of branch misses in Quicksort

Analysis of Branch Misses in Quicksort

Sebastian [email protected]

based on joint work with Conrado Martínez and Markus E. Nebel

04 January 2015

Meeting on Analytic Algorithmics and Combinatorics

Sebastian Wild Branch Misses in Quicksort 2015-01-04 1 / 15

Instruction Pipelines

Computers do not executeinstructions fully sequentially

Instead they use an “assembly line”

Example:

424344454647

41

48

...i := i + 1a := A[i]IF a < p GOTO 42j := j - 1a := A[j]IF a > p GOTO 45

...

each instruction broken in 4 stages

simpler steps shorter CPU cycles

one instruction per cycle finished . . .

. . . except for branches!

1 undo wrong instructions2 fill pipeline anew

Pipeline stalls are costly . . . can we avoid (some of) them?

Sebastian Wild Branch Misses in Quicksort 2015-01-04 2 / 15

Instruction Pipelines

Computers do not executeinstructions fully sequentially

Instead they use an “assembly line”

Example:

424344454647

41

48

...i := i + 1a := A[i]IF a < p GOTO 42j := j - 1a := A[j]IF a > p GOTO 45

...

each instruction broken in 4 stages

simpler steps shorter CPU cycles

one instruction per cycle finished . . .

. . . except for branches!

1 undo wrong instructions2 fill pipeline anew

Pipeline stalls are costly . . . can we avoid (some of) them?

Sebastian Wild Branch Misses in Quicksort 2015-01-04 2 / 15

Instruction Pipelines

Computers do not executeinstructions fully sequentially

Instead they use an “assembly line”

Example:

424344454647

41

48

...i := i + 1a := A[i]IF a < p GOTO 42j := j - 1a := A[j]IF a > p GOTO 45

...

each instruction broken in 4 stages

simpler steps shorter CPU cycles

one instruction per cycle finished . . .

. . . except for branches!

1 undo wrong instructions2 fill pipeline anew

Pipeline stalls are costly . . . can we avoid (some of) them?

Sebastian Wild Branch Misses in Quicksort 2015-01-04 2 / 15

Instruction Pipelines

Computers do not executeinstructions fully sequentially

Instead they use an “assembly line”

Example:

424344454647

41

48

...i := i + 1a := A[i]IF a < p GOTO 42j := j - 1a := A[j]IF a > p GOTO 45

...

each instruction broken in 4 stages

simpler steps shorter CPU cycles

one instruction per cycle finished . . .

. . . except for branches!

1 undo wrong instructions2 fill pipeline anew

Pipeline stalls are costly . . . can we avoid (some of) them?

Sebastian Wild Branch Misses in Quicksort 2015-01-04 2 / 15

Instruction Pipelines

Computers do not executeinstructions fully sequentially

Instead they use an “assembly line”

Example:

424344454647

41

48

...i := i + 1a := A[i]IF a < p GOTO 42j := j - 1a := A[j]IF a > p GOTO 45

...

each instruction broken in 4 stages

simpler steps shorter CPU cycles

one instruction per cycle finished . . .

. . . except for branches!

1 undo wrong instructions2 fill pipeline anew

Pipeline stalls are costly . . . can we avoid (some of) them?

Sebastian Wild Branch Misses in Quicksort 2015-01-04 2 / 15

Instruction Pipelines

Computers do not executeinstructions fully sequentially

Instead they use an “assembly line”

Example:

424344454647

41

48

...i := i + 1a := A[i]IF a < p GOTO 42j := j - 1a := A[j]IF a > p GOTO 45

...

each instruction broken in 4 stages

simpler steps shorter CPU cycles

one instruction per cycle finished . . .

. . . except for branches!

1 undo wrong instructions2 fill pipeline anew

Pipeline stalls are costly . . . can we avoid (some of) them?

Sebastian Wild Branch Misses in Quicksort 2015-01-04 2 / 15

Instruction Pipelines

Computers do not executeinstructions fully sequentially

Instead they use an “assembly line”

Example:

424344454647

41

48

...i := i + 1a := A[i]IF a < p GOTO 42j := j - 1a := A[j]IF a > p GOTO 45

...

each instruction broken in 4 stages

simpler steps shorter CPU cycles

one instruction per cycle finished . . .

. . . except for branches!

1 undo wrong instructions2 fill pipeline anew

Pipeline stalls are costly . . . can we avoid (some of) them?

Sebastian Wild Branch Misses in Quicksort 2015-01-04 2 / 15

Instruction Pipelines

Computers do not executeinstructions fully sequentially

Instead they use an “assembly line”

Example:

424344454647

41

48

...i := i + 1a := A[i]IF a < p GOTO 42j := j - 1a := A[j]IF a > p GOTO 45

...

each instruction broken in 4 stages

simpler steps shorter CPU cycles

one instruction per cycle finished . . .

. . . except for branches!

1 undo wrong instructions2 fill pipeline anew

Pipeline stalls are costly . . . can we avoid (some of) them?

Sebastian Wild Branch Misses in Quicksort 2015-01-04 2 / 15

Instruction Pipelines

Computers do not executeinstructions fully sequentially

Instead they use an “assembly line”

Example:

424344454647

41

48

...i := i + 1a := A[i]IF a < p GOTO 42j := j - 1a := A[j]IF a > p GOTO 45

...

each instruction broken in 4 stages

simpler steps shorter CPU cycles

one instruction per cycle finished . . .

. . . except for branches!

1 undo wrong instructions2 fill pipeline anew

Pipeline stalls are costly . . . can we avoid (some of) them?

Sebastian Wild Branch Misses in Quicksort 2015-01-04 2 / 15

Instruction Pipelines

Computers do not executeinstructions fully sequentially

Instead they use an “assembly line”

Example:

424344454647

41

48

...i := i + 1a := A[i]IF a < p GOTO 42j := j - 1a := A[j]IF a > p GOTO 45

...

each instruction broken in 4 stages

simpler steps shorter CPU cycles

one instruction per cycle finished . . .

. . . except for branches!

1 undo wrong instructions2 fill pipeline anew

Pipeline stalls are costly . . . can we avoid (some of) them?

Sebastian Wild Branch Misses in Quicksort 2015-01-04 2 / 15

Instruction Pipelines

Computers do not executeinstructions fully sequentially

Instead they use an “assembly line”

Example:

424344454647

41

48

...i := i + 1a := A[i]IF a < p GOTO 42j := j - 1a := A[j]IF a > p GOTO 45

...

each instruction broken in 4 stages

simpler steps shorter CPU cycles

one instruction per cycle finished . . .

. . . except for branches!

1 undo wrong instructions2 fill pipeline anew

Pipeline stalls are costly . . . can we avoid (some of) them?

Sebastian Wild Branch Misses in Quicksort 2015-01-04 2 / 15

Instruction Pipelines

Computers do not executeinstructions fully sequentially

Instead they use an “assembly line”

Example:

424344454647

41

48

...i := i + 1a := A[i]IF a < p GOTO 42j := j - 1a := A[j]IF a > p GOTO 45

...

each instruction broken in 4 stages

simpler steps shorter CPU cycles

one instruction per cycle finished . . .

. . . except for branches!

1 undo wrong instructions2 fill pipeline anew

Pipeline stalls are costly . . . can we avoid (some of) them?

Sebastian Wild Branch Misses in Quicksort 2015-01-04 2 / 15

Instruction Pipelines

Computers do not executeinstructions fully sequentially

Instead they use an “assembly line”

Example:

424344454647

41

48

...i := i + 1a := A[i]IF a < p GOTO 42j := j - 1a := A[j]IF a > p GOTO 45

...

each instruction broken in 4 stages

simpler steps shorter CPU cycles

one instruction per cycle finished . . .

. . . except for branches!

1 undo wrong instructions2 fill pipeline anew

Pipeline stalls are costly . . . can we avoid (some of) them?

Sebastian Wild Branch Misses in Quicksort 2015-01-04 2 / 15

Instruction Pipelines

Computers do not executeinstructions fully sequentially

Instead they use an “assembly line”

Example:

424344454647

41

48

...i := i + 1a := A[i]IF a < p GOTO 42j := j - 1a := A[j]IF a > p GOTO 45

...

each instruction broken in 4 stages

simpler steps shorter CPU cycles

one instruction per cycle finished . . .

. . . except for branches!

1 undo wrong instructions2 fill pipeline anew

Pipeline stalls are costly . . . can we avoid (some of) them?

Sebastian Wild Branch Misses in Quicksort 2015-01-04 2 / 15

Instruction Pipelines

Computers do not executeinstructions fully sequentially

Instead they use an “assembly line”

Example:

424344454647

41

48

...i := i + 1a := A[i]IF a < p GOTO 42j := j - 1a := A[j]IF a > p GOTO 45

...

each instruction broken in 4 stages

simpler steps shorter CPU cycles

one instruction per cycle finished . . .

. . . except for branches!

1 undo wrong instructions2 fill pipeline anew

Pipeline stalls are costly . . . can we avoid (some of) them?

Sebastian Wild Branch Misses in Quicksort 2015-01-04 2 / 15

Instruction Pipelines

Computers do not executeinstructions fully sequentially

Instead they use an “assembly line”

Example:

424344454647

41

48

...i := i + 1a := A[i]IF a < p GOTO 42j := j - 1a := A[j]IF a > p GOTO 45

...

each instruction broken in 4 stages

simpler steps shorter CPU cycles

one instruction per cycle finished . . .

. . . except for branches!

1 undo wrong instructions2 fill pipeline anew

Pipeline stalls are costly . . . can we avoid (some of) them?

Sebastian Wild Branch Misses in Quicksort 2015-01-04 2 / 15

Instruction Pipelines

Computers do not executeinstructions fully sequentially

Instead they use an “assembly line”

Example:

424344454647

41

48

...i := i + 1a := A[i]IF a < p GOTO 42j := j - 1a := A[j]IF a > p GOTO 45

...

each instruction broken in 4 stages

simpler steps shorter CPU cycles

one instruction per cycle finished . . .

. . . except for branches!

1 undo wrong instructions2 fill pipeline anew

Pipeline stalls are costly . . . can we avoid (some of) them?

Sebastian Wild Branch Misses in Quicksort 2015-01-04 2 / 15

Branch Prediction

We could avoid stalls if we knewwhether a branch will be taken or notin general not possible prediction with heuristics:

Predict same outcome as last time.(1-bit predictor) 1 2

predict taken predict not taken

taken

not t. not t.

taken

Predict most frequent outcome withfinite memory (2-bit saturating counter) 1 2 3 4

predict taken predict not taken

taken

not t. not t. not t. not t.

takentakentaken

Flip prediction only after twoconsecutive errors (2-bit flip-consecutive)

pred

ictt

aken

predictnottaken

1

2

3

4

taken

not t.taken

not t.not t.

takennot t.

taken

wilder heuristics exist out there . . .not considered here

prediction can be wrong branch miss (BM)

Sebastian Wild Branch Misses in Quicksort 2015-01-04 3 / 15

Branch Prediction

We could avoid stalls if we knewwhether a branch will be taken or notin general not possible prediction with heuristics:

Predict same outcome as last time.(1-bit predictor) 1 2

predict taken predict not taken

taken

not t. not t.

taken

Predict most frequent outcome withfinite memory (2-bit saturating counter) 1 2 3 4

predict taken predict not taken

taken

not t. not t. not t. not t.

takentakentaken

Flip prediction only after twoconsecutive errors (2-bit flip-consecutive)

pred

ictt

aken

predictnottaken

1

2

3

4

taken

not t.taken

not t.not t.

takennot t.

taken

wilder heuristics exist out there . . .not considered here

prediction can be wrong branch miss (BM)

Sebastian Wild Branch Misses in Quicksort 2015-01-04 3 / 15

Branch Prediction

We could avoid stalls if we knewwhether a branch will be taken or notin general not possible prediction with heuristics:

Predict same outcome as last time.(1-bit predictor) 1 2

predict taken predict not taken

taken

not t. not t.

taken

Predict most frequent outcome withfinite memory (2-bit saturating counter) 1 2 3 4

predict taken predict not taken

taken

not t. not t. not t. not t.

takentakentaken

Flip prediction only after twoconsecutive errors (2-bit flip-consecutive)

pred

ictt

aken

predictnottaken

1

2

3

4

taken

not t.taken

not t.not t.

takennot t.

taken

wilder heuristics exist out there . . .not considered here

prediction can be wrong branch miss (BM)

Sebastian Wild Branch Misses in Quicksort 2015-01-04 3 / 15

Branch Prediction

We could avoid stalls if we knewwhether a branch will be taken or notin general not possible prediction with heuristics:

Predict same outcome as last time.(1-bit predictor) 1 2

predict taken predict not taken

taken

not t. not t.

taken

Predict most frequent outcome withfinite memory (2-bit saturating counter) 1 2 3 4

predict taken predict not taken

taken

not t. not t. not t. not t.

takentakentaken

Flip prediction only after twoconsecutive errors (2-bit flip-consecutive)

pred

ictt

aken

predictnottaken

1

2

3

4

taken

not t.taken

not t.not t.

takennot t.

taken

wilder heuristics exist out there . . .not considered here

prediction can be wrong branch miss (BM)

Sebastian Wild Branch Misses in Quicksort 2015-01-04 3 / 15

Branch Prediction

We could avoid stalls if we knewwhether a branch will be taken or notin general not possible prediction with heuristics:

Predict same outcome as last time.(1-bit predictor) 1 2

predict taken predict not taken

taken

not t. not t.

taken

Predict most frequent outcome withfinite memory (2-bit saturating counter) 1 2 3 4

predict taken predict not taken

taken

not t. not t. not t. not t.

takentakentaken

Flip prediction only after twoconsecutive errors (2-bit flip-consecutive)

pred

ictt

aken

predictnottaken

1

2

3

4

taken

not t.taken

not t.not t.

takennot t.

taken

wilder heuristics exist out there . . .not considered here

prediction can be wrong branch miss (BM)

Sebastian Wild Branch Misses in Quicksort 2015-01-04 3 / 15

Branch Prediction

We could avoid stalls if we knewwhether a branch will be taken or notin general not possible prediction with heuristics:

Predict same outcome as last time.(1-bit predictor) 1 2

predict taken predict not taken

taken

not t. not t.

taken

Predict most frequent outcome withfinite memory (2-bit saturating counter) 1 2 3 4

predict taken predict not taken

taken

not t. not t. not t. not t.

takentakentaken

Flip prediction only after twoconsecutive errors (2-bit flip-consecutive)

pred

ictt

aken

predictnottaken

1

2

3

4

taken

not t.taken

not t.not t.

takennot t.

taken

wilder heuristics exist out there . . .not considered here

prediction can be wrong branch miss (BM)

Sebastian Wild Branch Misses in Quicksort 2015-01-04 3 / 15

Why Should We Care?

misprediction rates of “typical” programs < 10%

(Comparison-based) sorting is different!Branch based on comparison resultComparisons reduce entropy (uncertainty about input)

The less comparisons we use, the less predictable they becomefor classic Quicksort: misprediction rate > 25 %with median-of-3: > 31.25 %

Practical Importance (KALIGOSI & SANDERS, ESA 2006):

on Pentium 4 Prescott: very skewed pivot faster than median branch misses dominated running time

Sebastian Wild Branch Misses in Quicksort 2015-01-04 4 / 15

Why Should We Care?

misprediction rates of “typical” programs < 10%

(Comparison-based) sorting is different!Branch based on comparison resultComparisons reduce entropy (uncertainty about input)

The less comparisons we use, the less predictable they becomefor classic Quicksort: misprediction rate > 25 %with median-of-3: > 31.25 %

Practical Importance (KALIGOSI & SANDERS, ESA 2006):

on Pentium 4 Prescott: very skewed pivot faster than median branch misses dominated running time

Sebastian Wild Branch Misses in Quicksort 2015-01-04 4 / 15

Why Should We Care?

misprediction rates of “typical” programs < 10%

(Comparison-based) sorting is different!Branch based on comparison resultComparisons reduce entropy (uncertainty about input)

The less comparisons we use, the less predictable they becomefor classic Quicksort: misprediction rate > 25 %with median-of-3: > 31.25 %

Practical Importance (KALIGOSI & SANDERS, ESA 2006):

on Pentium 4 Prescott: very skewed pivot faster than median branch misses dominated running time

Sebastian Wild Branch Misses in Quicksort 2015-01-04 4 / 15

Why Should We Care?

misprediction rates of “typical” programs < 10%

(Comparison-based) sorting is different!Branch based on comparison resultComparisons reduce entropy (uncertainty about input)

The less comparisons we use, the less predictable they becomefor classic Quicksort: misprediction rate > 25 %with median-of-3: > 31.25 %

Practical Importance (KALIGOSI & SANDERS, ESA 2006):

on Pentium 4 Prescott: very skewed pivot faster than median branch misses dominated running time

Sebastian Wild Branch Misses in Quicksort 2015-01-04 4 / 15

Track Record of Dual-Pivot Quicksort

Since 2009, Java uses YAROSLAVSKIY’s dual-pivot Quicksort (YQS)faster than previously used classic Quicksort (CQS) in practicetraditional cost measures do not explain this!

CQS YQS Relative

Running Time (from various experiments) −10±2%

Comparisons 2 1.9 −5%Swaps 0.3 0.6 +80%

Bytecode Instructions 18 21.7 +20.6%MMIX oops υ 11 13.1 +19.1%

MMIX mems µ 2.6 2.8 +5%

scanned elements1

(≈ cache misses)2 1.6 −20%

·n lnn+O(n) , average case results

What about branch misses? Can they explain YQS’s success? . . . stay tuned.

1KUSHAGRA, LÓPEZ-ORTIZ, MUNRO, QIAO; ALENEX 2014Sebastian Wild Branch Misses in Quicksort 2015-01-04 5 / 15

Track Record of Dual-Pivot Quicksort

Since 2009, Java uses YAROSLAVSKIY’s dual-pivot Quicksort (YQS)faster than previously used classic Quicksort (CQS) in practicetraditional cost measures do not explain this!

CQS YQS Relative

Running Time (from various experiments) −10±2%

Comparisons 2 1.9 −5%Swaps 0.3 0.6 +80%

Bytecode Instructions 18 21.7 +20.6%MMIX oops υ 11 13.1 +19.1%

MMIX mems µ 2.6 2.8 +5%

scanned elements1

(≈ cache misses)2 1.6 −20%

·n lnn+O(n) , average case results

What about branch misses? Can they explain YQS’s success? . . . stay tuned.

1KUSHAGRA, LÓPEZ-ORTIZ, MUNRO, QIAO; ALENEX 2014Sebastian Wild Branch Misses in Quicksort 2015-01-04 5 / 15

Track Record of Dual-Pivot Quicksort

Since 2009, Java uses YAROSLAVSKIY’s dual-pivot Quicksort (YQS)faster than previously used classic Quicksort (CQS) in practicetraditional cost measures do not explain this!

CQS YQS Relative

Running Time (from various experiments) −10±2%

Comparisons 2 1.9 −5%Swaps 0.3 0.6 +80%

Bytecode Instructions 18 21.7 +20.6%MMIX oops υ 11 13.1 +19.1%

MMIX mems µ 2.6 2.8 +5%

scanned elements1

(≈ cache misses)2 1.6 −20%

·n lnn+O(n) , average case results

What about branch misses? Can they explain YQS’s success? . . . stay tuned.

1KUSHAGRA, LÓPEZ-ORTIZ, MUNRO, QIAO; ALENEX 2014Sebastian Wild Branch Misses in Quicksort 2015-01-04 5 / 15

Track Record of Dual-Pivot Quicksort

Since 2009, Java uses YAROSLAVSKIY’s dual-pivot Quicksort (YQS)faster than previously used classic Quicksort (CQS) in practicetraditional cost measures do not explain this!

CQS YQS Relative

Running Time (from various experiments) −10±2%

Comparisons 2 1.9 −5%Swaps 0.3 0.6 +80%

Bytecode Instructions 18 21.7 +20.6%MMIX oops υ 11 13.1 +19.1%

MMIX mems µ 2.6 2.8 +5%

scanned elements1

(≈ cache misses)2 1.6 −20%

·n lnn+O(n) , average case results

What about branch misses? Can they explain YQS’s success? . . . stay tuned.

1KUSHAGRA, LÓPEZ-ORTIZ, MUNRO, QIAO; ALENEX 2014Sebastian Wild Branch Misses in Quicksort 2015-01-04 5 / 15

Track Record of Dual-Pivot Quicksort

Since 2009, Java uses YAROSLAVSKIY’s dual-pivot Quicksort (YQS)faster than previously used classic Quicksort (CQS) in practicetraditional cost measures do not explain this!

CQS YQS Relative

Running Time (from various experiments) −10±2%

Comparisons 2 1.9 −5%Swaps 0.3 0.6 +80%

Bytecode Instructions 18 21.7 +20.6%MMIX oops υ 11 13.1 +19.1%

MMIX mems µ 2.6 2.8 +5%

scanned elements1

(≈ cache misses)2 1.6 −20%

·n lnn+O(n) , average case results

What about branch misses? Can they explain YQS’s success? . . . stay tuned.

1KUSHAGRA, LÓPEZ-ORTIZ, MUNRO, QIAO; ALENEX 2014Sebastian Wild Branch Misses in Quicksort 2015-01-04 5 / 15

Track Record of Dual-Pivot Quicksort

Since 2009, Java uses YAROSLAVSKIY’s dual-pivot Quicksort (YQS)faster than previously used classic Quicksort (CQS) in practicetraditional cost measures do not explain this!

CQS YQS Relative

Running Time (from various experiments) −10±2%

Comparisons 2 1.9 −5%Swaps 0.3 0.6 +80%

Bytecode Instructions 18 21.7 +20.6%MMIX oops υ 11 13.1 +19.1%

MMIX mems µ 2.6 2.8 +5%

scanned elements1

(≈ cache misses)2 1.6 −20%

·n lnn+O(n) , average case results

What about branch misses? Can they explain YQS’s success? . . . stay tuned.

1KUSHAGRA, LÓPEZ-ORTIZ, MUNRO, QIAO; ALENEX 2014Sebastian Wild Branch Misses in Quicksort 2015-01-04 5 / 15

Track Record of Dual-Pivot Quicksort

Since 2009, Java uses YAROSLAVSKIY’s dual-pivot Quicksort (YQS)faster than previously used classic Quicksort (CQS) in practicetraditional cost measures do not explain this!

CQS YQS Relative

Running Time (from various experiments) −10±2%

Comparisons 2 1.9 −5%Swaps 0.3 0.6 +80%

Bytecode Instructions 18 21.7 +20.6%MMIX oops υ 11 13.1 +19.1%

MMIX mems µ 2.6 2.8 +5%

scanned elements1

(≈ cache misses)2 1.6 −20%

·n lnn+O(n) , average case results

What about branch misses? Can they explain YQS’s success? . . . stay tuned.

1KUSHAGRA, LÓPEZ-ORTIZ, MUNRO, QIAO; ALENEX 2014Sebastian Wild Branch Misses in Quicksort 2015-01-04 5 / 15

Random Model

n i. i. d. elements chosen uniformly in [0, 1]

0 1

U1 U2U3 U4U5U6 U7U8

pairwise distinct almost surely

relative ranking is a random permutation

equivalent to classic model

Consider pivot value P fixed:

Pr[U < P

]= P

= D1

Pr[U > P

]= 1− P

= D2

0 1P

Similarly for dual-pivot Quicksort with pivots P 6 QPr[

U < P]= D1

Pr[P < U < Q

]= D2

Pr[

U > Q]= D3

0 1P Q

These probabilities hold for all elements U,independent of all other elements!

Sebastian Wild Branch Misses in Quicksort 2015-01-04 6 / 15

Random Model

n i. i. d. elements chosen uniformly in [0, 1]

0 1

U1 U2U3 U4U5U6 U7U8

pairwise distinct almost surely

relative ranking is a random permutation

equivalent to classic model

Consider pivot value P fixed:

Pr[U < P

]= P

= D1

Pr[U > P

]= 1− P

= D2

0 1P

Similarly for dual-pivot Quicksort with pivots P 6 QPr[

U < P]= D1

Pr[P < U < Q

]= D2

Pr[

U > Q]= D3

0 1P Q

These probabilities hold for all elements U,independent of all other elements!

Sebastian Wild Branch Misses in Quicksort 2015-01-04 6 / 15

Random Model

n i. i. d. elements chosen uniformly in [0, 1]

0 1

U1 U2U3 U4U5U6 U7U8

pairwise distinct almost surely

relative ranking is a random permutation

equivalent to classic model

Consider pivot value P fixed:

Pr[U < P

]= P

= D1

Pr[U > P

]= 1− P

= D2

0 1P

Similarly for dual-pivot Quicksort with pivots P 6 QPr[

U < P]= D1

Pr[P < U < Q

]= D2

Pr[

U > Q]= D3

0 1P Q

These probabilities hold for all elements U,independent of all other elements!

Sebastian Wild Branch Misses in Quicksort 2015-01-04 6 / 15

Random Model

n i. i. d. elements chosen uniformly in [0, 1]

0 1

U1 U2U3 U4U5U6 U7U8

pairwise distinct almost surely

relative ranking is a random permutation

equivalent to classic model

Consider pivot value P fixed:

Pr[U < P

]= P

= D1

Pr[U > P

]= 1− P

= D2

0 1P

Similarly for dual-pivot Quicksort with pivots P 6 QPr[

U < P]= D1

Pr[P < U < Q

]= D2

Pr[

U > Q]= D3

0 1P Q

These probabilities hold for all elements U,independent of all other elements!

Sebastian Wild Branch Misses in Quicksort 2015-01-04 6 / 15

Random Model

n i. i. d. elements chosen uniformly in [0, 1]

0 1

U1 U2U3 U4U5U6 U7U8

pairwise distinct almost surely

relative ranking is a random permutation

equivalent to classic model

Consider pivot value P fixed:

Pr[U < P

]= P

= D1

Pr[U > P

]= 1− P

= D2

0 1P

Similarly for dual-pivot Quicksort with pivots P 6 QPr[

U < P]= D1

Pr[P < U < Q

]= D2

Pr[

U > Q]= D3

0 1P Q

These probabilities hold for all elements U,independent of all other elements!

Sebastian Wild Branch Misses in Quicksort 2015-01-04 6 / 15

Random Model

n i. i. d. elements chosen uniformly in [0, 1]

0 1

U1 U2U3 U4U5U6 U7U8

pairwise distinct almost surely

relative ranking is a random permutation

equivalent to classic model

Consider pivot value P fixed:

Pr[U < P

]= P

= D1

Pr[U > P

]= 1− P

= D2

0 1P

Similarly for dual-pivot Quicksort with pivots P 6 QPr[

U < P]= D1

Pr[P < U < Q

]= D2

Pr[

U > Q]= D3

0 1P Q

These probabilities hold for all elements U,independent of all other elements!

Sebastian Wild Branch Misses in Quicksort 2015-01-04 6 / 15

Random Model

n i. i. d. elements chosen uniformly in [0, 1]

0 1

U1 U2U3 U4U5U6 U7U8

pairwise distinct almost surely

relative ranking is a random permutation

equivalent to classic model

Consider pivot value P fixed:

Pr[U < P

]= P = D1

Pr[U > P

]= 1− P = D2

0 1P

D1 D2

Similarly for dual-pivot Quicksort with pivots P 6 QPr[

U < P]= D1

Pr[P < U < Q

]= D2

Pr[

U > Q]= D3

0 1P Q

D1 D2 D3

These probabilities hold for all elements U,independent of all other elements!

Sebastian Wild Branch Misses in Quicksort 2015-01-04 6 / 15

Random Model

n i. i. d. elements chosen uniformly in [0, 1]

0 1

U1 U2U3 U4U5U6 U7U8

pairwise distinct almost surely

relative ranking is a random permutation

equivalent to classic model

Consider pivot value P fixed:

Pr[U < P

]= P = D1

Pr[U > P

]= 1− P = D2

0 1P

D1 D2

Similarly for dual-pivot Quicksort with pivots P 6 QPr[

U < P]= D1

Pr[P < U < Q

]= D2

Pr[

U > Q]= D3

0 1P Q

D1 D2 D3

These probabilities hold for all elements U,independent of all other elements!

Sebastian Wild Branch Misses in Quicksort 2015-01-04 6 / 15

Random Model

n i. i. d. elements chosen uniformly in [0, 1]

0 1

U1 U2U3 U4U5U6 U7U8

pairwise distinct almost surely

relative ranking is a random permutation

equivalent to classic model

Consider pivot value P fixed:

Pr[U < P

]= P = D1

Pr[U > P

]= 1− P = D2

0 1P

D1 D2

Similarly for dual-pivot Quicksort with pivots P 6 QPr[

U < P]= D1

Pr[P < U < Q

]= D2

Pr[

U > Q]= D3

0 1P Q

D1 D2 D3

These probabilities hold for all elements U,independent of all other elements!

Sebastian Wild Branch Misses in Quicksort 2015-01-04 6 / 15

Branches in CQS

How many branches in first partitioning step of CQS?

Consider pivot value P fixed. D = (D1, D2) = (P, 1− P) fixed.

one comparison branch per element U:

U < P left partition

U > P right partition

}

branch taken with prob. Pi. i. d. for all elements U! memoryless source

other branches (loop logic etc.)easy to predictonly constant number of mispredictions

can be ignored (for leading term asymptotics)

Sebastian Wild Branch Misses in Quicksort 2015-01-04 7 / 15

Branches in CQS

How many branches in first partitioning step of CQS?

Consider pivot value P fixed. D = (D1, D2) = (P, 1− P) fixed.

one comparison branch per element U:

U < P left partition

U > P right partition

}

branch taken with prob. Pi. i. d. for all elements U! memoryless source

other branches (loop logic etc.)easy to predictonly constant number of mispredictions

can be ignored (for leading term asymptotics)

Sebastian Wild Branch Misses in Quicksort 2015-01-04 7 / 15

Branches in CQS

How many branches in first partitioning step of CQS?

Consider pivot value P fixed. D = (D1, D2) = (P, 1− P) fixed.

one comparison branch per element U:

U < P left partition

U > P right partition

}

branch taken with prob. Pi. i. d. for all elements U! memoryless source

other branches (loop logic etc.)easy to predictonly constant number of mispredictions

can be ignored (for leading term asymptotics)

Sebastian Wild Branch Misses in Quicksort 2015-01-04 7 / 15

Branches in CQS

How many branches in first partitioning step of CQS?

Consider pivot value P fixed. D = (D1, D2) = (P, 1− P) fixed.

one comparison branch per element U:

U < P left partition

U > P right partition

}

branch taken with prob. Pi. i. d. for all elements U! memoryless source

other branches (loop logic etc.)

easy to predictonly constant number of mispredictions

can be ignored (for leading term asymptotics)

Sebastian Wild Branch Misses in Quicksort 2015-01-04 7 / 15

Branches in CQS

How many branches in first partitioning step of CQS?

Consider pivot value P fixed. D = (D1, D2) = (P, 1− P) fixed.

one comparison branch per element U:

U < P left partition

U > P right partition

}

branch taken with prob. Pi. i. d. for all elements U! memoryless source

other branches (loop logic etc.)

easy to predictonly constant number of mispredictions

can be ignored (for leading term asymptotics)

Sebastian Wild Branch Misses in Quicksort 2015-01-04 7 / 15

Branches in CQS

How many branches in first partitioning step of CQS?

Consider pivot value P fixed. D = (D1, D2) = (P, 1− P) fixed.

one comparison branch per element U:

U < P left partition

U > P right partition

}

branch taken with prob. Pi. i. d. for all elements U! memoryless source

other branches (loop logic etc.)

easy to predictonly constant number of mispredictions

can be ignored (for leading term asymptotics)

Sebastian Wild Branch Misses in Quicksort 2015-01-04 7 / 15

Branches in CQS

How many branches in first partitioning step of CQS?

Consider pivot value P fixed. D = (D1, D2) = (P, 1− P) fixed.

one comparison branch per element U:

U < P left partition

U > P right partition

}

branch taken with prob. Pi. i. d. for all elements U! memoryless source

other branches (loop logic etc.)

easy to predictonly constant number of mispredictions

can be ignored (for leading term asymptotics)

Sebastian Wild Branch Misses in Quicksort 2015-01-04 7 / 15

Misprediction Rate for Memoryless Sources

Branches taken i. i. d. with probability p.

Information theoretic lower bound: Miss rate: fOPT(p) = min{p, 1− p}

Can approach lower bound by estimating p.

p̂ ≥ 12 taken p̂ < 1

2 not taken

But: Actual predictors have very little memory!

1-bit PredictorWrong prediction whenever value changes

Miss rate: f1bit(p) = 2p(1− p)

1 2

predict taken predict not taken

p

1−p 1−p

p

Sebastian Wild Branch Misses in Quicksort 2015-01-04 8 / 15

Misprediction Rate for Memoryless Sources

Branches taken i. i. d. with probability p.

Information theoretic lower bound: Miss rate: fOPT(p) = min{p, 1− p}

Can approach lower bound by estimating p.

p̂ ≥ 12 taken p̂ < 1

2 not taken

But: Actual predictors have very little memory!

1-bit PredictorWrong prediction whenever value changes

Miss rate: f1bit(p) = 2p(1− p)

1 2

predict taken predict not taken

p

1−p 1−p

p

Sebastian Wild Branch Misses in Quicksort 2015-01-04 8 / 15

Misprediction Rate for Memoryless Sources

Branches taken i. i. d. with probability p.

Information theoretic lower bound: Miss rate: fOPT(p) = min{p, 1− p}

Can approach lower bound by estimating p.

p̂ ≥ 12 taken p̂ < 1

2 not taken

But: Actual predictors have very little memory!

1-bit PredictorWrong prediction whenever value changes

Miss rate: f1bit(p) = 2p(1− p)

1 2

predict taken predict not taken

p

1−p 1−p

p

Sebastian Wild Branch Misses in Quicksort 2015-01-04 8 / 15

Misprediction Rate for Memoryless Sources

Branches taken i. i. d. with probability p.

Information theoretic lower bound: Miss rate: fOPT(p) = min{p, 1− p}

Can approach lower bound by estimating p.

p̂ ≥ 12 taken p̂ < 1

2 not taken

But: Actual predictors have very little memory!

1-bit PredictorWrong prediction whenever value changes

Miss rate: f1bit(p) = 2p(1− p)

1 2

predict taken predict not taken

p

1−p 1−p

p

Sebastian Wild Branch Misses in Quicksort 2015-01-04 8 / 15

Misprediction Rate for Memoryless Sources

Branches taken i. i. d. with probability p.

Information theoretic lower bound: Miss rate: fOPT(p) = min{p, 1− p}

Can approach lower bound by estimating p.

p̂ ≥ 12 taken p̂ < 1

2 not taken

But: Actual predictors have very little memory!

1-bit PredictorWrong prediction whenever value changes

Miss rate: f1bit(p) = 2p(1− p)

1 2

predict taken predict not taken

p

1−p 1−p

p

Sebastian Wild Branch Misses in Quicksort 2015-01-04 8 / 15

Misprediction Rate for Memoryless Sources

Branches taken i. i. d. with probability p.

Information theoretic lower bound: Miss rate: fOPT(p) = min{p, 1− p}

Can approach lower bound by estimating p.

p̂ ≥ 12 taken p̂ < 1

2 not taken

But: Actual predictors have very little memory!

1-bit PredictorWrong prediction whenever value changes

Miss rate: f1bit(p) = 2p(1− p)

1 2

predict taken predict not taken

p

1−p 1−p

p

Sebastian Wild Branch Misses in Quicksort 2015-01-04 8 / 15

Misprediction Rate for Memoryless Sources

Branches taken i. i. d. with probability p.

Information theoretic lower bound: Miss rate: fOPT(p) = min{p, 1− p}

Can approach lower bound by estimating p.

p̂ ≥ 12 taken p̂ < 1

2 not taken

But: Actual predictors have very little memory!

1-bit PredictorWrong prediction whenever value changes

Miss rate: f1bit(p) = 2p(1− p)

1 2

predict taken predict not taken

p

1−p 1−p

p

Sebastian Wild Branch Misses in Quicksort 2015-01-04 8 / 15

Misprediction Rate for Memoryless Sources

Branches taken i. i. d. with probability p.

Information theoretic lower bound: Miss rate: fOPT(p) = min{p, 1− p}

Can approach lower bound by estimating p.

p̂ ≥ 12 taken p̂ < 1

2 not taken

But: Actual predictors have very little memory!

1-bit PredictorWrong prediction whenever value changes

Miss rate: f1bit(p) = 2p(1− p)

1 2

predict taken predict not taken

p

1−p 1−p

p

Sebastian Wild Branch Misses in Quicksort 2015-01-04 8 / 15

Misprediction Rate for Memoryless Sources [2]

2-bit Saturating CounterMiss rate? . . . depends on state! 1 2 3 4

predict taken predict not taken

p

1−p 1−p 1−p 1−p

ppp

But: Very fast convergence to steady statedifferent initial state distributions20 iterations for p = 2

3

use steady-state miss-rate:expected miss rate over states in stationarydistributionhere: f2-bit-sc(p) =

q

1− 2qwith q = p(1− p).

similarly for 2-bit Flip-Consecutive

f2-bit-fc(p) =q(1+ 2q)

1− q.

Sebastian Wild Branch Misses in Quicksort 2015-01-04 9 / 15

Misprediction Rate for Memoryless Sources [2]

2-bit Saturating CounterMiss rate? . . . depends on state! 1 2 3 4

predict taken predict not taken

p

1−p 1−p 1−p 1−p

ppp

But: Very fast convergence to steady statedifferent initial state distributions20 iterations for p = 2

3

use steady-state miss-rate:expected miss rate over states in stationarydistributionhere: f2-bit-sc(p) =

q

1− 2qwith q = p(1− p).

similarly for 2-bit Flip-Consecutive

f2-bit-fc(p) =q(1+ 2q)

1− q.

Sebastian Wild Branch Misses in Quicksort 2015-01-04 9 / 15

Misprediction Rate for Memoryless Sources [2]

2-bit Saturating CounterMiss rate? . . . depends on state! 1 2 3 4

predict taken predict not taken

p

1−p 1−p 1−p 1−p

ppp

But: Very fast convergence to steady statedifferent initial state distributions20 iterations for p = 2

3

use steady-state miss-rate:expected miss rate over states in stationarydistributionhere: f2-bit-sc(p) =

q

1− 2qwith q = p(1− p).

similarly for 2-bit Flip-Consecutive

f2-bit-fc(p) =q(1+ 2q)

1− q.

Sebastian Wild Branch Misses in Quicksort 2015-01-04 9 / 15

Misprediction Rate for Memoryless Sources [2]

2-bit Saturating CounterMiss rate? . . . depends on state! 1 2 3 4

predict taken predict not taken

p

1−p 1−p 1−p 1−p

ppp

But: Very fast convergence to steady statedifferent initial state distributions20 iterations for p = 2

3

use steady-state miss-rate:expected miss rate over states in stationarydistributionhere: f2-bit-sc(p) =

q

1− 2qwith q = p(1− p).

similarly for 2-bit Flip-Consecutive

f2-bit-fc(p) =q(1+ 2q)

1− q.

Sebastian Wild Branch Misses in Quicksort 2015-01-04 9 / 15

Misprediction Rate for Memoryless Sources [2]

2-bit Saturating CounterMiss rate? . . . depends on state! 1 2 3 4

predict taken predict not taken

p

1−p 1−p 1−p 1−p

ppp

But: Very fast convergence to steady statedifferent initial state distributions20 iterations for p = 2

3

use steady-state miss-rate:expected miss rate over states in stationarydistributionhere: f2-bit-sc(p) =

q

1− 2qwith q = p(1− p).

similarly for 2-bit Flip-Consecutive

f2-bit-fc(p) =q(1+ 2q)

1− q.

Sebastian Wild Branch Misses in Quicksort 2015-01-04 9 / 15

Misprediction Rate for Memoryless Sources [2]

2-bit Saturating CounterMiss rate? . . . depends on state! 1 2 3 4

predict taken predict not taken

p

1−p 1−p 1−p 1−p

ppp

But: Very fast convergence to steady statedifferent initial state distributions20 iterations for p = 2

3

use steady-state miss-rate:expected miss rate over states in stationarydistributionhere: f2-bit-sc(p) =

q

1− 2qwith q = p(1− p).

similarly for 2-bit Flip-Consecutive

f2-bit-fc(p) =q(1+ 2q)

1− q.

Sebastian Wild Branch Misses in Quicksort 2015-01-04 9 / 15

Misprediction Rate for Memoryless Sources [2]

2-bit Saturating CounterMiss rate? . . . depends on state! 1 2 3 4

predict taken predict not taken

p

1−p 1−p 1−p 1−p

ppp

But: Very fast convergence to steady statedifferent initial state distributions20 iterations for p = 2

3

use steady-state miss-rate:expected miss rate over states in stationarydistributionhere: f2-bit-sc(p) =

q

1− 2qwith q = p(1− p).

similarly for 2-bit Flip-Consecutive

f2-bit-fc(p) =q(1+ 2q)

1− q.

Sebastian Wild Branch Misses in Quicksort 2015-01-04 9 / 15

Misprediction Rate for Memoryless Sources [2]

2-bit Saturating CounterMiss rate? . . . depends on state! 1 2 3 4

predict taken predict not taken

p

1−p 1−p 1−p 1−p

ppp

But: Very fast convergence to steady statedifferent initial state distributions20 iterations for p = 2

3

use steady-state miss-rate:expected miss rate over states in stationarydistributionhere: f2-bit-sc(p) =

q

1− 2qwith q = p(1− p).

similarly for 2-bit Flip-Consecutive

f2-bit-fc(p) =q(1+ 2q)

1− q.

Sebastian Wild Branch Misses in Quicksort 2015-01-04 9 / 15

Misprediction Rate for Memoryless Sources [2]

2-bit Saturating CounterMiss rate? . . . depends on state! 1 2 3 4

predict taken predict not taken

p

1−p 1−p 1−p 1−p

ppp

But: Very fast convergence to steady statedifferent initial state distributions20 iterations for p = 2

3

use steady-state miss-rate:expected miss rate over states in stationarydistributionhere: f2-bit-sc(p) =

q

1− 2qwith q = p(1− p).

similarly for 2-bit Flip-Consecutive

f2-bit-fc(p) =q(1+ 2q)

1− q.

Sebastian Wild Branch Misses in Quicksort 2015-01-04 9 / 15

Misprediction Rate for Memoryless Sources [2]

2-bit Saturating CounterMiss rate? . . . depends on state! 1 2 3 4

predict taken predict not taken

p

1−p 1−p 1−p 1−p

ppp

But: Very fast convergence to steady statedifferent initial state distributions20 iterations for p = 2

3

use steady-state miss-rate:expected miss rate over states in stationarydistributionhere: f2-bit-sc(p) =

q

1− 2qwith q = p(1− p).

similarly for 2-bit Flip-Consecutive

f2-bit-fc(p) =q(1+ 2q)

1− q.

Sebastian Wild Branch Misses in Quicksort 2015-01-04 9 / 15

Misprediction Rate for Memoryless Sources [2]

2-bit Saturating CounterMiss rate? . . . depends on state! 1 2 3 4

predict taken predict not taken

p

1−p 1−p 1−p 1−p

ppp

But: Very fast convergence to steady statedifferent initial state distributions20 iterations for p = 2

3

use steady-state miss-rate:expected miss rate over states in stationarydistributionhere: f2-bit-sc(p) =

q

1− 2qwith q = p(1− p).

similarly for 2-bit Flip-Consecutive

f2-bit-fc(p) =q(1+ 2q)

1− q.

Sebastian Wild Branch Misses in Quicksort 2015-01-04 9 / 15

Misprediction Rate for Memoryless Sources [2]

2-bit Saturating CounterMiss rate? . . . depends on state! 1 2 3 4

predict taken predict not taken

p

1−p 1−p 1−p 1−p

ppp

But: Very fast convergence to steady statedifferent initial state distributions20 iterations for p = 2

3

use steady-state miss-rate:expected miss rate over states in stationarydistributionhere: f2-bit-sc(p) =

q

1− 2qwith q = p(1− p).

similarly for 2-bit Flip-Consecutive

f2-bit-fc(p) =q(1+ 2q)

1− q.

Sebastian Wild Branch Misses in Quicksort 2015-01-04 9 / 15

Misprediction Rate for Memoryless Sources [2]

2-bit Saturating CounterMiss rate? . . . depends on state! 1 2 3 4

predict taken predict not taken

p

1−p 1−p 1−p 1−p

ppp

But: Very fast convergence to steady statedifferent initial state distributions20 iterations for p = 2

3

use steady-state miss-rate:expected miss rate over states in stationarydistributionhere: f2-bit-sc(p) =

q

1− 2qwith q = p(1− p).

similarly for 2-bit Flip-Consecutive

f2-bit-fc(p) =q(1+ 2q)

1− q.

Sebastian Wild Branch Misses in Quicksort 2015-01-04 9 / 15

Misprediction Rate for Memoryless Sources [2]

2-bit Saturating CounterMiss rate? . . . depends on state! 1 2 3 4

predict taken predict not taken

p

1−p 1−p 1−p 1−p

ppp

But: Very fast convergence to steady statedifferent initial state distributions20 iterations for p = 2

3

use steady-state miss-rate:expected miss rate over states in stationarydistributionhere: f2-bit-sc(p) =

q

1− 2qwith q = p(1− p).

similarly for 2-bit Flip-Consecutive

f2-bit-fc(p) =q(1+ 2q)

1− q.

Sebastian Wild Branch Misses in Quicksort 2015-01-04 9 / 15

Misprediction Rate for Memoryless Sources [2]

2-bit Saturating CounterMiss rate? . . . depends on state! 1 2 3 4

predict taken predict not taken

p

1−p 1−p 1−p 1−p

ppp

But: Very fast convergence to steady statedifferent initial state distributions20 iterations for p = 2

3

use steady-state miss-rate:expected miss rate over states in stationarydistributionhere: f2-bit-sc(p) =

q

1− 2qwith q = p(1− p).

similarly for 2-bit Flip-Consecutive

f2-bit-fc(p) =q(1+ 2q)

1− q.

Sebastian Wild Branch Misses in Quicksort 2015-01-04 9 / 15

Misprediction Rate for Memoryless Sources [2]

2-bit Saturating CounterMiss rate? . . . depends on state! 1 2 3 4

predict taken predict not taken

p

1−p 1−p 1−p 1−p

ppp

But: Very fast convergence to steady statedifferent initial state distributions20 iterations for p = 2

3

use steady-state miss-rate:expected miss rate over states in stationarydistributionhere: f2-bit-sc(p) =

q

1− 2qwith q = p(1− p).

similarly for 2-bit Flip-Consecutive

f2-bit-fc(p) =q(1+ 2q)

1− q.

Sebastian Wild Branch Misses in Quicksort 2015-01-04 9 / 15

Misprediction Rate for Memoryless Sources [2]

2-bit Saturating CounterMiss rate? . . . depends on state! 1 2 3 4

predict taken predict not taken

p

1−p 1−p 1−p 1−p

ppp

But: Very fast convergence to steady statedifferent initial state distributions20 iterations for p = 2

3

use steady-state miss-rate:expected miss rate over states in stationarydistributionhere: f2-bit-sc(p) =

q

1− 2qwith q = p(1− p).

similarly for 2-bit Flip-Consecutive

f2-bit-fc(p) =q(1+ 2q)

1− q.

Sebastian Wild Branch Misses in Quicksort 2015-01-04 9 / 15

Misprediction Rate for Memoryless Sources [2]

2-bit Saturating CounterMiss rate? . . . depends on state! 1 2 3 4

predict taken predict not taken

p

1−p 1−p 1−p 1−p

ppp

But: Very fast convergence to steady statedifferent initial state distributions20 iterations for p = 2

3

use steady-state miss-rate:expected miss rate over states in stationarydistributionhere: f2-bit-sc(p) =

q

1− 2qwith q = p(1− p).

similarly for 2-bit Flip-Consecutive

f2-bit-fc(p) =q(1+ 2q)

1− q.

Sebastian Wild Branch Misses in Quicksort 2015-01-04 9 / 15

Misprediction Rate for Memoryless Sources [2]

2-bit Saturating CounterMiss rate? . . . depends on state! 1 2 3 4

predict taken predict not taken

p

1−p 1−p 1−p 1−p

ppp

But: Very fast convergence to steady statedifferent initial state distributions20 iterations for p = 2

3

use steady-state miss-rate:expected miss rate over states in stationarydistributionhere: f2-bit-sc(p) =

q

1− 2qwith q = p(1− p).

similarly for 2-bit Flip-Consecutive

f2-bit-fc(p) =q(1+ 2q)

1− q.

Sebastian Wild Branch Misses in Quicksort 2015-01-04 9 / 15

Misprediction Rate for Memoryless Sources [2]

2-bit Saturating CounterMiss rate? . . . depends on state! 1 2 3 4

predict taken predict not taken

p

1−p 1−p 1−p 1−p

ppp

But: Very fast convergence to steady statedifferent initial state distributions20 iterations for p = 2

3

use steady-state miss-rate:expected miss rate over states in stationarydistributionhere: f2-bit-sc(p) =

q

1− 2qwith q = p(1− p).

similarly for 2-bit Flip-Consecutive

f2-bit-fc(p) =q(1+ 2q)

1− q.

Sebastian Wild Branch Misses in Quicksort 2015-01-04 9 / 15

Misprediction Rate for Memoryless Sources [2]

2-bit Saturating CounterMiss rate? . . . depends on state! 1 2 3 4

predict taken predict not taken

p

1−p 1−p 1−p 1−p

ppp

But: Very fast convergence to steady statedifferent initial state distributions20 iterations for p = 2

3

use steady-state miss-rate:expected miss rate over states in stationarydistributionhere: f2-bit-sc(p) =

q

1− 2qwith q = p(1− p).

similarly for 2-bit Flip-Consecutive

f2-bit-fc(p) =q(1+ 2q)

1− q.

Sebastian Wild Branch Misses in Quicksort 2015-01-04 9 / 15

Misprediction Rate for Memoryless Sources [2]

2-bit Saturating CounterMiss rate? . . . depends on state! 1 2 3 4

predict taken predict not taken

p

1−p 1−p 1−p 1−p

ppp

But: Very fast convergence to steady statedifferent initial state distributions20 iterations for p = 2

3

use steady-state miss-rate:expected miss rate over states in stationarydistributionhere: f2-bit-sc(p) =

q

1− 2qwith q = p(1− p).

similarly for 2-bit Flip-Consecutive

f2-bit-fc(p) =q(1+ 2q)

1− q.

Sebastian Wild Branch Misses in Quicksort 2015-01-04 9 / 15

Misprediction Rate for Memoryless Sources [2]

2-bit Saturating CounterMiss rate? . . . depends on state! 1 2 3 4

predict taken predict not taken

p

1−p 1−p 1−p 1−p

ppp

But: Very fast convergence to steady statedifferent initial state distributions20 iterations for p = 2

3

use steady-state miss-rate:expected miss rate over states in stationarydistributionhere: f2-bit-sc(p) =

q

1− 2qwith q = p(1− p).

similarly for 2-bit Flip-Consecutive

f2-bit-fc(p) =q(1+ 2q)

1− q.

Sebastian Wild Branch Misses in Quicksort 2015-01-04 9 / 15

Misprediction Rate for Memoryless Sources [2]

2-bit Saturating CounterMiss rate? . . . depends on state! 1 2 3 4

predict taken predict not taken

p

1−p 1−p 1−p 1−p

ppp

But: Very fast convergence to steady statedifferent initial state distributions20 iterations for p = 2

3

use steady-state miss-rate:expected miss rate over states in stationarydistributionhere: f2-bit-sc(p) =

q

1− 2qwith q = p(1− p).

similarly for 2-bit Flip-Consecutive

f2-bit-fc(p) =q(1+ 2q)

1− q.

Sebastian Wild Branch Misses in Quicksort 2015-01-04 9 / 15

Misprediction Rate for Memoryless Sources [2]

2-bit Saturating CounterMiss rate? . . . depends on state! 1 2 3 4

predict taken predict not taken

p

1−p 1−p 1−p 1−p

ppp

But: Very fast convergence to steady statedifferent initial state distributions20 iterations for p = 2

3

use steady-state miss-rate:expected miss rate over states in stationarydistributionhere: f2-bit-sc(p) =

q

1− 2qwith q = p(1− p).

similarly for 2-bit Flip-Consecutive

f2-bit-fc(p) =q(1+ 2q)

1− q.

Sebastian Wild Branch Misses in Quicksort 2015-01-04 9 / 15

Misprediction Rate for Memoryless Sources [2]

2-bit Saturating CounterMiss rate? . . . depends on state! 1 2 3 4

predict taken predict not taken

p

1−p 1−p 1−p 1−p

ppp

But: Very fast convergence to steady statedifferent initial state distributions20 iterations for p = 2

3

use steady-state miss-rate:expected miss rate over states in stationarydistributionhere: f2-bit-sc(p) =

q

1− 2qwith q = p(1− p).

similarly for 2-bit Flip-Consecutive

f2-bit-fc(p) =q(1+ 2q)

1− q.

Sebastian Wild Branch Misses in Quicksort 2015-01-04 9 / 15

Misprediction Rate for Memoryless Sources [2]

2-bit Saturating CounterMiss rate? . . . depends on state! 1 2 3 4

predict taken predict not taken

p

1−p 1−p 1−p 1−p

ppp

But: Very fast convergence to steady statedifferent initial state distributions20 iterations for p = 2

3

use steady-state miss-rate:expected miss rate over states in stationarydistributionhere: f2-bit-sc(p) =

q

1− 2qwith q = p(1− p).

similarly for 2-bit Flip-Consecutive

f2-bit-fc(p) =q(1+ 2q)

1− q.

Sebastian Wild Branch Misses in Quicksort 2015-01-04 9 / 15

Distribution of Pivot Values

In (classic) Quicksort branch probability is random expected miss rate: E[f(P)]. (expectation over pivot values P)

What is the distribution of P?without sampling: P D

= Uniform(0, 1)

Typical pivot choice: median of k (in practice: k = 3)or pseudomedian of 9 (“ninther”)

Here: more general scheme with parameter t = (t1, t2)

Example: k = 6 and t = (3, 2):

P

t1 t2

t = (0, 0) no samplingt = (t, t) gives median-of-(2t+ 1)can also sample skewed pivots

Distribution of pivot value: PD= Beta(t1 + 1, t2 + 1)

Sebastian Wild Branch Misses in Quicksort 2015-01-04 10 / 15

Distribution of Pivot Values

In (classic) Quicksort branch probability is random expected miss rate: E[f(P)]. (expectation over pivot values P)

What is the distribution of P?without sampling: P D

= Uniform(0, 1)

Typical pivot choice: median of k (in practice: k = 3)or pseudomedian of 9 (“ninther”)

Here: more general scheme with parameter t = (t1, t2)

Example: k = 6 and t = (3, 2):

P

t1 t2

t = (0, 0) no samplingt = (t, t) gives median-of-(2t+ 1)can also sample skewed pivots

Distribution of pivot value: PD= Beta(t1 + 1, t2 + 1)

Sebastian Wild Branch Misses in Quicksort 2015-01-04 10 / 15

Distribution of Pivot Values

In (classic) Quicksort branch probability is random expected miss rate: E[f(P)]. (expectation over pivot values P)

What is the distribution of P?without sampling: P D

= Uniform(0, 1)

Typical pivot choice: median of k (in practice: k = 3)or pseudomedian of 9 (“ninther”)

Here: more general scheme with parameter t = (t1, t2)

Example: k = 6 and t = (3, 2):

P

t1 t2

t = (0, 0) no samplingt = (t, t) gives median-of-(2t+ 1)can also sample skewed pivots

Distribution of pivot value: PD= Beta(t1 + 1, t2 + 1)

Sebastian Wild Branch Misses in Quicksort 2015-01-04 10 / 15

Distribution of Pivot Values

In (classic) Quicksort branch probability is random expected miss rate: E[f(P)]. (expectation over pivot values P)

What is the distribution of P?without sampling: P D

= Uniform(0, 1)

Typical pivot choice: median of k (in practice: k = 3)or pseudomedian of 9 (“ninther”)

Here: more general scheme with parameter t = (t1, t2)

Example: k = 6 and t = (3, 2):

P

t1 t2

t = (0, 0) no samplingt = (t, t) gives median-of-(2t+ 1)can also sample skewed pivots

Distribution of pivot value: PD= Beta(t1 + 1, t2 + 1)

Sebastian Wild Branch Misses in Quicksort 2015-01-04 10 / 15

Distribution of Pivot Values

In (classic) Quicksort branch probability is random expected miss rate: E[f(P)]. (expectation over pivot values P)

What is the distribution of P?without sampling: P D

= Uniform(0, 1)

Typical pivot choice: median of k (in practice: k = 3)or pseudomedian of 9 (“ninther”)

Here: more general scheme with parameter t = (t1, t2)

Example: k = 6 and t = (3, 2):

P

t1 t2

t = (0, 0) no samplingt = (t, t) gives median-of-(2t+ 1)can also sample skewed pivots

Distribution of pivot value: PD= Beta(t1 + 1, t2 + 1)

Sebastian Wild Branch Misses in Quicksort 2015-01-04 10 / 15

Distribution of Pivot Values

In (classic) Quicksort branch probability is random expected miss rate: E[f(P)]. (expectation over pivot values P)

What is the distribution of P?without sampling: P D

= Uniform(0, 1)

Typical pivot choice: median of k (in practice: k = 3)or pseudomedian of 9 (“ninther”)

Here: more general scheme with parameter t = (t1, t2)

Example: k = 6 and t = (3, 2):

P

t1 t2

t = (0, 0) no samplingt = (t, t) gives median-of-(2t+ 1)can also sample skewed pivots

Distribution of pivot value: PD= Beta(t1 + 1, t2 + 1)

Sebastian Wild Branch Misses in Quicksort 2015-01-04 10 / 15

Distribution of Pivot Values

In (classic) Quicksort branch probability is random expected miss rate: E[f(P)]. (expectation over pivot values P)

What is the distribution of P?without sampling: P D

= Uniform(0, 1)

Typical pivot choice: median of k (in practice: k = 3)or pseudomedian of 9 (“ninther”)

Here: more general scheme with parameter t = (t1, t2)

Example: k = 6 and t = (3, 2):

P

t1 t2

t = (0, 0) no samplingt = (t, t) gives median-of-(2t+ 1)can also sample skewed pivots

Distribution of pivot value: PD= Beta(t1 + 1, t2 + 1)

Sebastian Wild Branch Misses in Quicksort 2015-01-04 10 / 15

Distribution of Pivot Values

In (classic) Quicksort branch probability is random expected miss rate: E[f(P)]. (expectation over pivot values P)

What is the distribution of P?without sampling: P D

= Uniform(0, 1)

Typical pivot choice: median of k (in practice: k = 3)or pseudomedian of 9 (“ninther”)

Here: more general scheme with parameter t = (t1, t2)

Example: k = 6 and t = (3, 2):

P

t1 t2

t = (0, 0) no samplingt = (t, t) gives median-of-(2t+ 1)can also sample skewed pivots

Distribution of pivot value: PD= Beta(t1 + 1, t2 + 1)

Sebastian Wild Branch Misses in Quicksort 2015-01-04 10 / 15

Distribution of Pivot Values

In (classic) Quicksort branch probability is random expected miss rate: E[f(P)]. (expectation over pivot values P)

What is the distribution of P?without sampling: P D

= Uniform(0, 1)

Typical pivot choice: median of k (in practice: k = 3)or pseudomedian of 9 (“ninther”)

Here: more general scheme with parameter t = (t1, t2)

Example: k = 6 and t = (3, 2):

P

t1 t2

t = (0, 0) no samplingt = (t, t) gives median-of-(2t+ 1)can also sample skewed pivots

Distribution of pivot value: PD= Beta(t1 + 1, t2 + 1)

Sebastian Wild Branch Misses in Quicksort 2015-01-04 10 / 15

Distribution of Pivot Values

In (classic) Quicksort branch probability is random expected miss rate: E[f(P)]. (expectation over pivot values P)

What is the distribution of P?without sampling: P D

= Uniform(0, 1)

Typical pivot choice: median of k (in practice: k = 3)or pseudomedian of 9 (“ninther”)

Here: more general scheme with parameter t = (t1, t2)

Example: k = 6 and t = (3, 2):

P

t1 t2

t = (0, 0) no samplingt = (t, t) gives median-of-(2t+ 1)can also sample skewed pivots

Distribution of pivot value: PD= Beta(t1 + 1, t2 + 1)

Sebastian Wild Branch Misses in Quicksort 2015-01-04 10 / 15

Miss Rates for Quicksort Branch

expected miss rate given by integral

E[f(P)] =

ˆ 10

f(p) · pt1(1− p)t2

B(t+ 1)dp

e. g. for 1-bit predictor

E[f1-bit(P)] =

ˆ 10

2p(1− p) · pt1(1− p)t2

B(t+ 1)dp

= 2(t1 + 1)(t2 + 1)

(k+ 2)(k+ 1)

no concise representation for other integrals . . . (see paper)

but: exact values for fixed t

Sebastian Wild Branch Misses in Quicksort 2015-01-04 11 / 15

Miss Rates for Quicksort Branch

expected miss rate given by integral

E[f(P)] =

ˆ 10

f(p) · pt1(1− p)t2

B(t+ 1)dp

e. g. for 1-bit predictor

E[f1-bit(P)] =

ˆ 10

2p(1− p) · pt1(1− p)t2

B(t+ 1)dp

= 2(t1 + 1)(t2 + 1)

(k+ 2)(k+ 1)

no concise representation for other integrals . . . (see paper)

but: exact values for fixed t

Sebastian Wild Branch Misses in Quicksort 2015-01-04 11 / 15

Miss Rates for Quicksort Branch

expected miss rate given by integral

E[f(P)] =

ˆ 10

f(p) · pt1(1− p)t2

B(t+ 1)dp

e. g. for 1-bit predictor

E[f1-bit(P)] =

ˆ 10

2p(1− p) · pt1(1− p)t2

B(t+ 1)dp = 2

(t1 + 1)(t2 + 1)

(k+ 2)(k+ 1)

no concise representation for other integrals . . . (see paper)

but: exact values for fixed t

Sebastian Wild Branch Misses in Quicksort 2015-01-04 11 / 15

Miss Rates for Quicksort Branch

expected miss rate given by integral

E[f(P)] =

ˆ 10

f(p) · pt1(1− p)t2

B(t+ 1)dp

e. g. for 1-bit predictor

E[f1-bit(P)] =

ˆ 10

2p(1− p) · pt1(1− p)t2

B(t+ 1)dp = 2

(t1 + 1)(t2 + 1)

(k+ 2)(k+ 1)

no concise representation for other integrals . . . (see paper)

but: exact values for fixed t

Sebastian Wild Branch Misses in Quicksort 2015-01-04 11 / 15

Miss Rate and Branch Misses

Miss Rate for CQS with median of 2t+1:

0 2 4 6 8

0.3

0.4

0.50.5

t

miss rate

OPT 1-bit

2-bit sc 2-bit fc

miss rates quickly get bad(close to guessing!)but: less comparisons in total!

0 2 4 6 8

1.4

1.6

1.8

2

1/ ln2

·n lnn+O(n)

t

#cmps

Consider number of branch misses:

#BM = #comparisons · miss rate

Overall BM still grows with t.

0 2 4 6 8

0.5

0.6

0.7

0.5/ ln2

·n lnn+O(n)

t

#BM

Sebastian Wild Branch Misses in Quicksort 2015-01-04 12 / 15

Miss Rate and Branch Misses

Miss Rate for CQS with median of 2t+1:

0 2 4 6 8

0.3

0.4

0.50.5

t

miss rate

OPT 1-bit

2-bit sc 2-bit fc

miss rates quickly get bad(close to guessing!)but: less comparisons in total!

0 2 4 6 8

1.4

1.6

1.8

2

1/ ln2

·n lnn+O(n)

t

#cmps

Consider number of branch misses:

#BM = #comparisons · miss rate

Overall BM still grows with t.

0 2 4 6 8

0.5

0.6

0.7

0.5/ ln2

·n lnn+O(n)

t

#BM

Sebastian Wild Branch Misses in Quicksort 2015-01-04 12 / 15

Miss Rate and Branch Misses

Miss Rate for CQS with median of 2t+1:

0 2 4 6 8

0.3

0.4

0.50.5

t

miss rate

OPT 1-bit

2-bit sc 2-bit fc

miss rates quickly get bad(close to guessing!)but: less comparisons in total!

0 2 4 6 8

1.4

1.6

1.8

2

1/ ln2

·n lnn+O(n)

t

#cmps

Consider number of branch misses:

#BM = #comparisons · miss rate

Overall BM still grows with t.

0 2 4 6 8

0.5

0.6

0.7

0.5/ ln2

·n lnn+O(n)

t

#BM

Sebastian Wild Branch Misses in Quicksort 2015-01-04 12 / 15

Miss Rate and Branch Misses

Miss Rate for CQS with median of 2t+1:

0 2 4 6 8

0.3

0.4

0.50.5

t

miss rate

OPT 1-bit

2-bit sc 2-bit fc

miss rates quickly get bad(close to guessing!)but: less comparisons in total!

0 2 4 6 8

1.4

1.6

1.8

2

1/ ln2

·n lnn+O(n)

t

#cmps

Consider number of branch misses:

#BM = #comparisons · miss rate

Overall BM still grows with t.

0 2 4 6 8

0.5

0.6

0.7

0.5/ ln2

·n lnn+O(n)

t

#BM

Sebastian Wild Branch Misses in Quicksort 2015-01-04 12 / 15

Miss Rate and Branch Misses

Miss Rate for CQS with median of 2t+1:

0 2 4 6 8

0.3

0.4

0.50.5

t

miss rate

OPT 1-bit

2-bit sc 2-bit fc

miss rates quickly get bad(close to guessing!)but: less comparisons in total!

0 2 4 6 8

1.4

1.6

1.8

2

1/ ln2

·n lnn+O(n)

t

#cmps

Consider number of branch misses:

#BM = #comparisons · miss rate

Overall BM still grows with t.

0 2 4 6 8

0.5

0.6

0.7

0.5/ ln2

·n lnn+O(n)

t

#BM

Sebastian Wild Branch Misses in Quicksort 2015-01-04 12 / 15

Miss Rate and Branch Misses

Miss Rate for CQS with median of 2t+1:

0 2 4 6 8

0.3

0.4

0.50.5

t

miss rate

OPT 1-bit

2-bit sc 2-bit fc

miss rates quickly get bad(close to guessing!)but: less comparisons in total!

0 2 4 6 8

1.4

1.6

1.8

2

1/ ln2

·n lnn+O(n)

t

#cmps

Consider number of branch misses:

#BM = #comparisons · miss rate

Overall BM still grows with t.

0 2 4 6 8

0.5

0.6

0.7

0.5/ ln2

·n lnn+O(n)

t

#BM

Sebastian Wild Branch Misses in Quicksort 2015-01-04 12 / 15

Branch Misses in YQS

Original question: Does YQS better than CQS w. r. t. branch misses?

Complication for analysis:4 branch locationshow often they areexecuted depends oninput

< P ?

swap ` < Q ?

skip swap g

3 7

3 7

>Q ?

< P ? skip

swap ` swap k

37

3 7

< P P ≤ ◦ ≤ Q ≥ QP QExample: C(y1)

executed ( D1 +D2 )n+O(1) times. (in expectation, conditional on D)

branch taken i. i. d. with prob D1 . (conditional on D)

expected #BM at C(y1) in first partitioning step:E[(D1 +D2) · f(D1)] · n+O(1)

Integrals even more “fun” . . . but doable

Sebastian Wild Branch Misses in Quicksort 2015-01-04 13 / 15

Branch Misses in YQS

Original question: Does YQS better than CQS w. r. t. branch misses?

Complication for analysis:4 branch locationshow often they areexecuted depends oninput

< P ?

swap ` < Q ?

skip swap g

3 7

3 7

>Q ?

< P ? skip

swap ` swap k

37

3 7

< P P ≤ ◦ ≤ Q ≥ QP QExample: C(y1)

executed ( D1 +D2 )n+O(1) times. (in expectation, conditional on D)

branch taken i. i. d. with prob D1 . (conditional on D)

expected #BM at C(y1) in first partitioning step:E[(D1 +D2) · f(D1)] · n+O(1)

Integrals even more “fun” . . . but doable

Sebastian Wild Branch Misses in Quicksort 2015-01-04 13 / 15

Branch Misses in YQS

Original question: Does YQS better than CQS w. r. t. branch misses?

Complication for analysis:4 branch locationshow often they areexecuted depends oninput

< P ?

swap ` < Q ?

skip swap g

3 7

3 7

>Q ?

< P ? skip

swap ` swap k

37

3 7

< P P ≤ ◦ ≤ Q ≥ QP QExample: C(y1)

executed ( D1 +D2 )n+O(1) times. (in expectation, conditional on D)

branch taken i. i. d. with prob D1 . (conditional on D)

expected #BM at C(y1) in first partitioning step:E[(D1 +D2) · f(D1)] · n+O(1)

Integrals even more “fun” . . . but doable

Sebastian Wild Branch Misses in Quicksort 2015-01-04 13 / 15

Branch Misses in YQS

Original question: Does YQS better than CQS w. r. t. branch misses?

Complication for analysis:4 branch locationshow often they areexecuted depends oninput

< P ?

swap ` < Q ?

skip swap g

3 7

3 7

>Q ?

< P ? skip

swap ` swap k

37

3 7

< P P ≤ ◦ ≤ Q ≥ QP QExample: C(y1)

executed ( D1 +D2 )n+O(1) times. (in expectation, conditional on D)

branch taken i. i. d. with prob D1 . (conditional on D)

expected #BM at C(y1) in first partitioning step:E[(D1 +D2) · f(D1)] · n+O(1)

Integrals even more “fun” . . . but doable

Sebastian Wild Branch Misses in Quicksort 2015-01-04 13 / 15

Branch Misses in YQS

Original question: Does YQS better than CQS w. r. t. branch misses?

Complication for analysis:4 branch locationshow often they areexecuted depends oninput

< P ?

swap ` < Q ?

skip swap g

3 7

3 7

>Q ?

< P ? skip

swap ` swap k

37

3 7

< P P ≤ ◦ ≤ Q ≥ QP QExample: C(y1)

executed ( D1 +D2 )n+O(1) times. (in expectation, conditional on D)

branch taken i. i. d. with prob D1 . (conditional on D)

expected #BM at C(y1) in first partitioning step:E[(D1 +D2) · f(D1)] · n+O(1)

Integrals even more “fun” . . . but doable

Sebastian Wild Branch Misses in Quicksort 2015-01-04 13 / 15

Branch Misses in YQS

Original question: Does YQS better than CQS w. r. t. branch misses?

Complication for analysis:4 branch locationshow often they areexecuted depends oninput

< P ?

swap ` < Q ?

skip swap g

3 7

3 7

>Q ?

< P ? skip

swap ` swap k

37

3 7

< P P ≤ ◦ ≤ Q ≥ QP QExample: C(y1)

executed ( D1 +D2 )n+O(1) times. (in expectation, conditional on D)

branch taken i. i. d. with prob D1 . (conditional on D)

expected #BM at C(y1) in first partitioning step:E[(D1 +D2) · f(D1)] · n+O(1)

Integrals even more “fun” . . . but doable

Sebastian Wild Branch Misses in Quicksort 2015-01-04 13 / 15

Branch Misses in YQS

Original question: Does YQS better than CQS w. r. t. branch misses?

Complication for analysis:4 branch locationshow often they areexecuted depends oninput

< P ?

swap ` < Q ?

skip swap g

3 7

3 7

>Q ?

< P ? skip

swap ` swap k

37

3 7

< P P ≤ ◦ ≤ Q ≥ QP QExample: C(y1)

executed ( D1 +D2 )n+O(1) times. (in expectation, conditional on D)

branch taken i. i. d. with prob D1 . (conditional on D)

expected #BM at C(y1) in first partitioning step:E[(D1 +D2) · f(D1)] · n+O(1)

Integrals even more “fun” . . . but doable

Sebastian Wild Branch Misses in Quicksort 2015-01-04 13 / 15

Results CQS vs. YQS

Original question: Does YQS better than CQS w. r. t. branch misses?

Expected number of branch misses

without pivot sampling

CQS YQS Relative

OPT 0.5 0.513 +2.6%

1-bit 0.6 0.673 +1.0%

2-bit sc 0.571 0.585 +2.5%

2-bit fc 0.589 0.602 +2.2%

·n lnn+O(n)

CQS median-of-3 vs. YQS tertiles-of-5

CQS YQS Relative

OPT 0.536 0.538 +0.4%

1-bit 0.686 0.687 +0.1%

2-bit sc 0.611 0.613 +0.3%

2-bit fc 0.627 0.629 +0.3%

·n lnn+O(n)

essentially same number of BM. Branch misses not a plausible explanation for YQS’s success.

Sebastian Wild Branch Misses in Quicksort 2015-01-04 14 / 15

Results CQS vs. YQS

Original question: Does YQS better than CQS w. r. t. branch misses?

Expected number of branch misses

without pivot sampling

CQS YQS Relative

OPT 0.5 0.513 +2.6%

1-bit 0.6 0.673 +1.0%

2-bit sc 0.571 0.585 +2.5%

2-bit fc 0.589 0.602 +2.2%

·n lnn+O(n)

CQS median-of-3 vs. YQS tertiles-of-5

CQS YQS Relative

OPT 0.536 0.538 +0.4%

1-bit 0.686 0.687 +0.1%

2-bit sc 0.611 0.613 +0.3%

2-bit fc 0.627 0.629 +0.3%

·n lnn+O(n)

essentially same number of BM. Branch misses not a plausible explanation for YQS’s success.

Sebastian Wild Branch Misses in Quicksort 2015-01-04 14 / 15

Results CQS vs. YQS

Original question: Does YQS better than CQS w. r. t. branch misses?

Expected number of branch misses

without pivot sampling

CQS YQS Relative

OPT 0.5 0.513 +2.6%

1-bit 0.6 0.673 +1.0%

2-bit sc 0.571 0.585 +2.5%

2-bit fc 0.589 0.602 +2.2%

·n lnn+O(n)

CQS median-of-3 vs. YQS tertiles-of-5

CQS YQS Relative

OPT 0.536 0.538 +0.4%

1-bit 0.686 0.687 +0.1%

2-bit sc 0.611 0.613 +0.3%

2-bit fc 0.627 0.629 +0.3%

·n lnn+O(n)

essentially same number of BM. Branch misses not a plausible explanation for YQS’s success.

Sebastian Wild Branch Misses in Quicksort 2015-01-04 14 / 15

Results CQS vs. YQS

Original question: Does YQS better than CQS w. r. t. branch misses?

Expected number of branch misses

without pivot sampling

CQS YQS Relative

OPT 0.5 0.513 +2.6%

1-bit 0.6 0.673 +1.0%

2-bit sc 0.571 0.585 +2.5%

2-bit fc 0.589 0.602 +2.2%

·n lnn+O(n)

CQS median-of-3 vs. YQS tertiles-of-5

CQS YQS Relative

OPT 0.536 0.538 +0.4%

1-bit 0.686 0.687 +0.1%

2-bit sc 0.611 0.613 +0.3%

2-bit fc 0.627 0.629 +0.3%

·n lnn+O(n)

essentially same number of BM. Branch misses not a plausible explanation for YQS’s success.

Sebastian Wild Branch Misses in Quicksort 2015-01-04 14 / 15

Results CQS vs. YQS

Original question: Does YQS better than CQS w. r. t. branch misses?

Expected number of branch misses

without pivot sampling

CQS YQS Relative

OPT 0.5 0.513 +2.6%

1-bit 0.6 0.673 +1.0%

2-bit sc 0.571 0.585 +2.5%

2-bit fc 0.589 0.602 +2.2%

·n lnn+O(n)

CQS median-of-3 vs. YQS tertiles-of-5

CQS YQS Relative

OPT 0.536 0.538 +0.4%

1-bit 0.686 0.687 +0.1%

2-bit sc 0.611 0.613 +0.3%

2-bit fc 0.627 0.629 +0.3%

·n lnn+O(n)

essentially same number of BM. Branch misses not a plausible explanation for YQS’s success.

Sebastian Wild Branch Misses in Quicksort 2015-01-04 14 / 15

Conclusion

Precise analysis of branch misses in Quicksort (CQS and YQS)including pivot samplinglower bounds on branch miss rates

CQS and YQS cause very similar number of BM Strengthened evidence for the hypothesis that

YQS is faster because of better usage of memory hierarchy.

Sebastian Wild Branch Misses in Quicksort 2015-01-04 15 / 15

Conclusion

Precise analysis of branch misses in Quicksort (CQS and YQS)including pivot samplinglower bounds on branch miss rates

CQS and YQS cause very similar number of BM Strengthened evidence for the hypothesis that

YQS is faster because of better usage of memory hierarchy.

Sebastian Wild Branch Misses in Quicksort 2015-01-04 15 / 15

Conclusion

Precise analysis of branch misses in Quicksort (CQS and YQS)including pivot samplinglower bounds on branch miss rates

CQS and YQS cause very similar number of BM Strengthened evidence for the hypothesis that

YQS is faster because of better usage of memory hierarchy.

Sebastian Wild Branch Misses in Quicksort 2015-01-04 15 / 15

Miss Rate for Branches in Quicksort

without sampling: P D= Uniform(0, 1)

E[fOPT(P)] =

ˆ 10

min{p, 1− p}dp

= 0.25

E[f1-bit(P)] =

ˆ 10

2p(1− p)dp

= 0.3

E[f2-bit-sc(P)] =

ˆ 10

p(1− p)

1− 2p(1− p)dp =

π

4−1

2≈ 0.285

E[f2-bit-fc(P)] =

ˆ 10

2p2(1− p)2 + p(1− p)

1− 2p(1− p)dp =

2π√3−10

3≈ 0.294

Sebastian Wild Branch Misses in Quicksort 2015-01-04 16 / 15

Miss Rate for Branches in Quicksort

without sampling: P D= Uniform(0, 1)

E[fOPT(P)] =

ˆ 10

min{p, 1− p}dp

= 0.25

E[f1-bit(P)] =

ˆ 10

2p(1− p)dp

= 0.3

E[f2-bit-sc(P)] =

ˆ 10

p(1− p)

1− 2p(1− p)dp =

π

4−1

2≈ 0.285

E[f2-bit-fc(P)] =

ˆ 10

2p2(1− p)2 + p(1− p)

1− 2p(1− p)dp =

2π√3−10

3≈ 0.294

Sebastian Wild Branch Misses in Quicksort 2015-01-04 16 / 15

Miss Rate for Branches in Quicksort

without sampling: P D= Uniform(0, 1)

E[fOPT(P)] =

ˆ 10

min{p, 1− p}dp = 0.25

E[f1-bit(P)] =

ˆ 10

2p(1− p)dp

= 0.3

E[f2-bit-sc(P)] =

ˆ 10

p(1− p)

1− 2p(1− p)dp =

π

4−1

2≈ 0.285

E[f2-bit-fc(P)] =

ˆ 10

2p2(1− p)2 + p(1− p)

1− 2p(1− p)dp =

2π√3−10

3≈ 0.294

Sebastian Wild Branch Misses in Quicksort 2015-01-04 16 / 15

Miss Rate for Branches in Quicksort

without sampling: P D= Uniform(0, 1)

E[fOPT(P)] =

ˆ 10

min{p, 1− p}dp = 0.25

E[f1-bit(P)] =

ˆ 10

2p(1− p)dp

= 0.3

E[f2-bit-sc(P)] =

ˆ 10

p(1− p)

1− 2p(1− p)dp =

π

4−1

2≈ 0.285

E[f2-bit-fc(P)] =

ˆ 10

2p2(1− p)2 + p(1− p)

1− 2p(1− p)dp =

2π√3−10

3≈ 0.294

Sebastian Wild Branch Misses in Quicksort 2015-01-04 16 / 15

Miss Rate for Branches in Quicksort

without sampling: P D= Uniform(0, 1)

E[fOPT(P)] =

ˆ 10

min{p, 1− p}dp = 0.25

E[f1-bit(P)] =

ˆ 10

2p(1− p)dp = 0.3

E[f2-bit-sc(P)] =

ˆ 10

p(1− p)

1− 2p(1− p)dp =

π

4−1

2≈ 0.285

E[f2-bit-fc(P)] =

ˆ 10

2p2(1− p)2 + p(1− p)

1− 2p(1− p)dp =

2π√3−10

3≈ 0.294

Sebastian Wild Branch Misses in Quicksort 2015-01-04 16 / 15

Miss Rate for Branches in Quicksort

without sampling: P D= Uniform(0, 1)

E[fOPT(P)] =

ˆ 10

min{p, 1− p}dp = 0.25

E[f1-bit(P)] =

ˆ 10

2p(1− p)dp = 0.3

E[f2-bit-sc(P)] =

ˆ 10

p(1− p)

1− 2p(1− p)dp =

π

4−1

2≈ 0.285

E[f2-bit-fc(P)] =

ˆ 10

2p2(1− p)2 + p(1− p)

1− 2p(1− p)dp =

2π√3−10

3≈ 0.294

Sebastian Wild Branch Misses in Quicksort 2015-01-04 16 / 15

Miss Rate for Branches in Quicksort

without sampling: P D= Uniform(0, 1)

E[fOPT(P)] =

ˆ 10

min{p, 1− p}dp = 0.25

E[f1-bit(P)] =

ˆ 10

2p(1− p)dp = 0.3

E[f2-bit-sc(P)] =

ˆ 10

p(1− p)

1− 2p(1− p)dp =

π

4−1

2≈ 0.285

E[f2-bit-fc(P)] =

ˆ 10

2p2(1− p)2 + p(1− p)

1− 2p(1− p)dp =

2π√3−10

3≈ 0.294

Sebastian Wild Branch Misses in Quicksort 2015-01-04 16 / 15