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Two Big Questions Derandomization Hierarchy Theorems
Two Big Questions:P vs. NP and P vs. BPP
Jeff Kinne
University of Wisconsin-Madison
Indiana State University, March 26, 2010
1 / 35
Two Big Questions Derandomization Hierarchy Theorems
Computational Complexity Theory
How much time, memory space, etc. are needed to solveproblems?
Is nondeterminism powerful? P vs. NP
Conjecture: P 6= NP
Techniques: hierarchy theorems, others
Is randomness powerful? P vs. BPP, L vs. BPL
Conjecture: P=BPP, L=BPL
My work: derandomization, hierarchy theorems
2 / 35
Two Big Questions Derandomization Hierarchy Theorems
Computational Complexity Theory
How much time, memory space, etc. are needed to solveproblems?
Is nondeterminism powerful?
P vs. NP
Conjecture: P 6= NP
Techniques: hierarchy theorems, others
Is randomness powerful? P vs. BPP, L vs. BPL
Conjecture: P=BPP, L=BPL
My work: derandomization, hierarchy theorems
2 / 35
Two Big Questions Derandomization Hierarchy Theorems
Computational Complexity Theory
How much time, memory space, etc. are needed to solveproblems?
Is nondeterminism powerful? P vs. NP
Conjecture: P 6= NP
Techniques: hierarchy theorems, others
Is randomness powerful? P vs. BPP, L vs. BPL
Conjecture: P=BPP, L=BPL
My work: derandomization, hierarchy theorems
2 / 35
Two Big Questions Derandomization Hierarchy Theorems
Computational Complexity Theory
How much time, memory space, etc. are needed to solveproblems?
Is nondeterminism powerful? P vs. NP
Conjecture: P 6= NP
Techniques: hierarchy theorems, others
Is randomness powerful? P vs. BPP, L vs. BPL
Conjecture: P=BPP, L=BPL
My work: derandomization, hierarchy theorems
2 / 35
Two Big Questions Derandomization Hierarchy Theorems
Computational Complexity Theory
How much time, memory space, etc. are needed to solveproblems?
Is nondeterminism powerful? P vs. NP
Conjecture: P 6= NP
Techniques: hierarchy theorems, others
Is randomness powerful? P vs. BPP, L vs. BPL
Conjecture: P=BPP, L=BPL
My work: derandomization, hierarchy theorems
2 / 35
Two Big Questions Derandomization Hierarchy Theorems
Computational Complexity Theory
How much time, memory space, etc. are needed to solveproblems?
Is nondeterminism powerful? P vs. NP
Conjecture: P 6= NP
Techniques: hierarchy theorems, others
Is randomness powerful?
P vs. BPP, L vs. BPL
Conjecture: P=BPP, L=BPL
My work: derandomization, hierarchy theorems
2 / 35
Two Big Questions Derandomization Hierarchy Theorems
Computational Complexity Theory
How much time, memory space, etc. are needed to solveproblems?
Is nondeterminism powerful? P vs. NP
Conjecture: P 6= NP
Techniques: hierarchy theorems, others
Is randomness powerful? P vs. BPP, L vs. BPL
Conjecture: P=BPP, L=BPL
My work: derandomization, hierarchy theorems
2 / 35
Two Big Questions Derandomization Hierarchy Theorems
Computational Complexity Theory
How much time, memory space, etc. are needed to solveproblems?
Is nondeterminism powerful? P vs. NP
Conjecture: P 6= NP
Techniques: hierarchy theorems, others
Is randomness powerful? P vs. BPP, L vs. BPL
Conjecture: P=BPP, L=BPL
My work: derandomization, hierarchy theorems
2 / 35
Two Big Questions Derandomization Hierarchy Theorems
Computational Complexity Theory
How much time, memory space, etc. are needed to solveproblems?
Is nondeterminism powerful? P vs. NP
Conjecture: P 6= NP
Techniques: hierarchy theorems, others
Is randomness powerful? P vs. BPP, L vs. BPL
Conjecture: P=BPP, L=BPL
My work: derandomization, hierarchy theorems
2 / 35
Two Big Questions Derandomization Hierarchy Theorems
Introducing two Big Questions
3 / 35
Two Big Questions Derandomization Hierarchy Theorems
Time Complexity
Decision Problem: yes/no questions
4 / 35
Two Big Questions Derandomization Hierarchy Theorems
Time Complexity
Poly Time (P)
Exp Time (EXP)
Decision Problem: yes/no questions
4 / 35
Two Big Questions Derandomization Hierarchy Theorems
Time Complexity
Poly Time (P)
Exp Time (EXP)
factoring,NP-complete sorting
shortest path
Decision Problem: yes/no questions
4 / 35
Two Big Questions Derandomization Hierarchy Theorems
Time Complexity
Poly Time (P)
Exp Time (EXP)
sorting?
factoring,NP-complete
shortest path
Decision Problem: yes/no questions
4 / 35
Two Big Questions Derandomization Hierarchy Theorems
Time Complexity
Poly Time (P)
Exp Time (EXP)
sorting?
factoring,NP-complete
shortest path
Decision Problem: yes/no questions
4 / 35
Two Big Questions Derandomization Hierarchy Theorems
Memory Space Complexity
Amount of working space memory needed
5 / 35
Two Big Questions Derandomization Hierarchy Theorems
Memory Space Complexity
Log Space (L)
Poly Space (PSPACE)
Amount of working space memory needed
5 / 35
Two Big Questions Derandomization Hierarchy Theorems
Memory Space Complexity
Log Space (L)
Poly Space (PSPACE)
Amount of working space memory needed
5 / 35
Two Big Questions Derandomization Hierarchy Theorems
Memory Space Complexity
Log Space (L)
Poly Space (PSPACE)
factoring,NP-complete arithmetic
Amount of working space memory needed
5 / 35
Two Big Questions Derandomization Hierarchy Theorems
Memory Space Complexity
Log Space (L)
Poly Space (PSPACE)
arithmetic?factoring,
NP-complete
Amount of working space memory needed
5 / 35
Two Big Questions Derandomization Hierarchy Theorems
Randomized Algorithm
Bounded error: Correct with probability > 99%
6 / 35
Two Big Questions Derandomization Hierarchy Theorems
Randomized Algorithm
01 10 10 1011 01 10
yes/no
Bounded error: Correct with probability > 99%
6 / 35
Two Big Questions Derandomization Hierarchy Theorems
Randomized Algorithm
Random Strings
good
bad
01 10 10 1011 01 10
yes/no
Bounded error: Correct with probability > 99%
6 / 35
Two Big Questions Derandomization Hierarchy Theorems
Randomized Algorithm
Random Strings
good
bad
01 10 10 1011 01 10
yes/no
Bounded error: Correct with probability > 99%
6 / 35
Two Big Questions Derandomization Hierarchy Theorems
Polynomial Identity Testing
p(x) = x3 · (3x − x2)2 − x4 · (2x3 + 5x) + x · (4x2 − x)3
Do all terms cancel?
7 / 35
Two Big Questions Derandomization Hierarchy Theorems
Polynomial Identity Testing
p(x) = x3 · (3x − x2)2 − x4 · (2x3 + 5x) + x · (4x2 − x)3
Do all terms cancel?
7 / 35
Two Big Questions Derandomization Hierarchy Theorems
Polynomial Identity Testing
p(x) = x3 · (3x − x2)2 − x4 · (2x3 + 5x) + x · (4x2 − x)3
Do all terms cancel?
7 / 35
Two Big Questions Derandomization Hierarchy Theorems
Polynomial Identity Testing
p(x) = x3 · (3x − x2)2 − x4 · (2x3 + 5x) + x · (4x2 − x)3
Do all terms cancel?
7 / 35
Two Big Questions Derandomization Hierarchy Theorems
Polynomial Identity Testing
p(x) = x3 · (3x − x2)2 − x4 · (2x3 + 5x) + x · (4x2 − x)3
Do all terms cancel?
7 / 35
Two Big Questions Derandomization Hierarchy Theorems
Multi-variate Polynomial Identity Testing
p(x1, x2, x3, x4) = x54 · (x1 − x2 − x3)
30 + (x4 + x3 − x1)15 ·
(x3 − x2 + x1)20 − (x2 − x3 + x4)
20 · (x2 + x1)15
Do all terms cancel?
Randomized Algorithm
Pick point (x1, ..., x4), each xi ∈R S
p non-zero, degree d ⇒ Pr[p(x1, ..., x4) = 0] ≤ d|S |
8 / 35
Two Big Questions Derandomization Hierarchy Theorems
Multi-variate Polynomial Identity Testing
p(x1, x2, x3, x4) = x54 · (x1 − x2 − x3)
30 + (x4 + x3 − x1)15 ·
(x3 − x2 + x1)20 − (x2 − x3 + x4)
20 · (x2 + x1)15
Do all terms cancel?
Randomized Algorithm
Pick point (x1, ..., x4), each xi ∈R S
p non-zero, degree d ⇒ Pr[p(x1, ..., x4) = 0] ≤ d|S |
8 / 35
Two Big Questions Derandomization Hierarchy Theorems
Multi-variate Polynomial Identity Testing
p(x1, x2, x3, x4) = x54 · (x1 − x2 − x3)
30 + (x4 + x3 − x1)15 ·
(x3 − x2 + x1)20 − (x2 − x3 + x4)
20 · (x2 + x1)15
Do all terms cancel?
Randomized Algorithm
Pick point (x1, ..., x4), each xi ∈R S
p non-zero, degree d ⇒ Pr[p(x1, ..., x4) = 0] ≤ d|S |
8 / 35
Two Big Questions Derandomization Hierarchy Theorems
Multi-variate Polynomial Identity Testing
p(x1, x2, x3, x4) = x54 · (x1 − x2 − x3)
30 + (x4 + x3 − x1)15 ·
(x3 − x2 + x1)20 − (x2 − x3 + x4)
20 · (x2 + x1)15
Do all terms cancel?
Randomized Algorithm
Pick point (x1, ..., x4), each xi ∈R S
p non-zero, degree d ⇒ Pr[p(x1, ..., x4) = 0] ≤ d|S |
8 / 35
Two Big Questions Derandomization Hierarchy Theorems
Multi-variate Polynomial Identity Testing
p(x1, x2, x3, x4) = x54 · (x1 − x2 − x3)
30 + (x4 + x3 − x1)15 ·
(x3 − x2 + x1)20 − (x2 − x3 + x4)
20 · (x2 + x1)15
Do all terms cancel?
Randomized Algorithm
Pick point (x1, ..., x4), each xi ∈R S
p non-zero, degree d ⇒ Pr[p(x1, ..., x4) = 0] ≤ d|S |
8 / 35
Two Big Questions Derandomization Hierarchy Theorems
Multi-variate Polynomial Identity Testing
p(x1, x2, x3, x4) = x54 · (x1 − x2 − x3)
30 + (x4 + x3 − x1)15 ·
(x3 − x2 + x1)20 − (x2 − x3 + x4)
20 · (x2 + x1)15
Do all terms cancel?
Randomized Algorithm
Pick point (x1, ..., x4), each xi ∈R S
p non-zero, degree d
⇒ Pr[p(x1, ..., x4) = 0] ≤ d|S |
8 / 35
Two Big Questions Derandomization Hierarchy Theorems
Multi-variate Polynomial Identity Testing
p(x1, x2, x3, x4) = x54 · (x1 − x2 − x3)
30 + (x4 + x3 − x1)15 ·
(x3 − x2 + x1)20 − (x2 − x3 + x4)
20 · (x2 + x1)15
Do all terms cancel?
Randomized Algorithm
Pick point (x1, ..., x4), each xi ∈R S
p non-zero, degree d ⇒ Pr[p(x1, ..., x4) = 0] ≤ d|S |
8 / 35
Two Big Questions Derandomization Hierarchy Theorems
Randomized Algorithm
Random Strings
good
bad
01 10 10 1011 01 10
yes/no
p(x1, x2, x3, x4)
x1, ..., x4
BPP: Bounded-error Probabilistic Poly time , BPL: log space
P?= BPP
9 / 35
Two Big Questions Derandomization Hierarchy Theorems
Randomized Algorithm
Random Strings
good
bad
01 10 10 1011 01 10
yes/no
p(x1, x2, x3, x4)
x1, ..., x4
BPP: Bounded-error Probabilistic Poly time
, BPL: log space
P?= BPP
9 / 35
Two Big Questions Derandomization Hierarchy Theorems
Randomized Algorithm
Random Strings
good
bad
01 10 10 1011 01 10
yes/no
p(x1, x2, x3, x4)
x1, ..., x4
BPP: Bounded-error Probabilistic Poly time , BPL: log space
P?= BPP
9 / 35
Two Big Questions Derandomization Hierarchy Theorems
Randomized Algorithm
Random Strings
good
bad
01 10 10 1011 01 10
yes/no
p(x1, x2, x3, x4)
x1, ..., x4
BPP: Bounded-error Probabilistic Poly time , BPL: log space
P?= BPP
9 / 35
Two Big Questions Derandomization Hierarchy Theorems
Nondeterministic Algorithm
10 / 35
Two Big Questions Derandomization Hierarchy Theorems
Graph 3-Coloring
11 / 35
Two Big Questions Derandomization Hierarchy Theorems
Graph 3-Coloring
11 / 35
Two Big Questions Derandomization Hierarchy Theorems
Nondeterministic Algorithm
NP: Nondeterministic Polynomial time
“NP-Complete” problems: 3-Coloring, TSP, Knapsack, ...
P?= NP
NP?= coNP
12 / 35
Two Big Questions Derandomization Hierarchy Theorems
Nondeterministic Algorithm
01 10 10 1011 01 10
yes/no
Proof/Certificate
graph
coloring
NP: Nondeterministic Polynomial time
“NP-Complete” problems: 3-Coloring, TSP, Knapsack, ...
P?= NP
NP?= coNP
12 / 35
Two Big Questions Derandomization Hierarchy Theorems
Nondeterministic Algorithm
Potential Proofs
good
bad
01 10 10 1011 01 10
yes/no
Proof/Certificate
graph
coloring
NP: Nondeterministic Polynomial time
“NP-Complete” problems: 3-Coloring, TSP, Knapsack, ...
P?= NP
NP?= coNP
12 / 35
Two Big Questions Derandomization Hierarchy Theorems
Nondeterministic Algorithm
Potential Proofs
good
bad
01 10 10 1011 01 10
yes/no
Proof/Certificate
graph
coloring
NP: Nondeterministic Polynomial time
“NP-Complete” problems: 3-Coloring, TSP, Knapsack, ...
P?= NP
NP?= coNP
12 / 35
Two Big Questions Derandomization Hierarchy Theorems
Nondeterministic Algorithm
Potential Proofs
good
bad
01 10 10 1011 01 10
yes/no
Proof/Certificate
graph
coloring
NP: Nondeterministic Polynomial time
“NP-Complete” problems: 3-Coloring, TSP, Knapsack, ...
P?= NP
NP?= coNP
12 / 35
Two Big Questions Derandomization Hierarchy Theorems
Nondeterministic Algorithm
Potential Proofs
good
bad
01 10 10 1011 01 10
yes/no
Proof/Certificate
graph
coloring
NP: Nondeterministic Polynomial time
“NP-Complete” problems: 3-Coloring, TSP, Knapsack, ...
P?= NP
NP?= coNP
12 / 35
Two Big Questions Derandomization Hierarchy Theorems
Nondeterministic Algorithm
Potential Proofs
good
bad
01 10 10 1011 01 10
yes/no
Proof/Certificate
graph
coloring
NP: Nondeterministic Polynomial time
“NP-Complete” problems: 3-Coloring, TSP, Knapsack, ...
P?= NP NP
?= coNP
12 / 35
Two Big Questions Derandomization Hierarchy Theorems
Two Big Questions
P?= NP
Is finding proofs as easy as verifying them?
Is 3-coloring in Polynomial Time?
P?= BPP
Does randomness truly add power?
13 / 35
Two Big Questions Derandomization Hierarchy Theorems
Two Big Questions
P?= NP
Is finding proofs as easy as verifying them?
Is 3-coloring in Polynomial Time?
P?= BPP
Does randomness truly add power?
13 / 35
Two Big Questions Derandomization Hierarchy Theorems
Computational Complexity Theory
How much time, memory space, etc. are needed to solveproblems?
Is nondeterminism powerful? P vs. NP
Conjecture: P 6= NP
Techniques: hierarchy theorems, others
Is randomness powerful? P vs. BPP, L vs. BPL
Conjecture: P=BPP, L=BPL
My work: derandomization, hierarchy theorems
14 / 35
Two Big Questions Derandomization Hierarchy Theorems
Derandomization
15 / 35
Two Big Questions Derandomization Hierarchy Theorems
Naive Derandomization
Random Strings
good
bad
01 10 10 1011 01 10
yes/no
p(x1, x2, x3, x4)
x1, ..., x4
Try all possible random bit strings – exponentially many
16 / 35
Two Big Questions Derandomization Hierarchy Theorems
Naive Derandomization
Random Strings
good
bad
01 10 10 1011 01 10
yes/no
p(x1, x2, x3, x4)
x1, ..., x4
Try all possible random bit strings – exponentially many
16 / 35
Two Big Questions Derandomization Hierarchy Theorems
Naive Derandomization
Random Strings
good
bad
01 10 10 1011 01 10
yes/no
p(x1, x2, x3, x4)
x1, ..., x4
Try all possible random bit strings
– exponentially many
16 / 35
Two Big Questions Derandomization Hierarchy Theorems
Naive Derandomization
Random Strings
good
bad
01 10 10 1011 01 10
yes/no
p(x1, x2, x3, x4)
x1, ..., x4
Try all possible random bit strings – exponentially many
16 / 35
Two Big Questions Derandomization Hierarchy Theorems
Derandomization – the Standard PRG Approach
Poly many strings to try
⇒ O(log n) seed, exp stretch
17 / 35
Two Big Questions Derandomization Hierarchy Theorems
Derandomization – the Standard PRG Approach
Random Strings
good
bad
01 10 10 1011 01 10
yes/no
p(x1, x2, x3, x4)
x1, ..., x4
Poly many strings to try
⇒ O(log n) seed, exp stretch
17 / 35
Two Big Questions Derandomization Hierarchy Theorems
Derandomization – the Standard PRG Approach
Random Strings
good
bad
pseudo–random
01 10 10 1011 01 10
yes/no
p(x1, x2, x3, x4)
x1, ..., x4
Poly many strings to try
⇒ O(log n) seed, exp stretch
17 / 35
Two Big Questions Derandomization Hierarchy Theorems
Derandomization – the Standard PRG Approach
Random Strings
good
bad
pseudo–random
01 10 10 1011 01 10
yes/no
PRG
p(x1, x2, x3, x4)
x1, ..., x4
Poly many strings to try
⇒ O(log n) seed, exp stretch
17 / 35
Two Big Questions Derandomization Hierarchy Theorems
Derandomization – the Standard PRG Approach
Random Strings
good
bad
pseudo–random
01 10 10 1011 01 10
yes/no
PRG
p(x1, x2, x3, x4)
x1, ..., x4
Poly many strings to try
⇒ O(log n) seed, exp stretch
17 / 35
Two Big Questions Derandomization Hierarchy Theorems
Derandomization – the Standard PRG Approach
Random Strings
good
bad
pseudo–random
01 10 10 1011 01 10
yes/no
PRG
p(x1, x2, x3, x4)
x1, ..., x4
Poly many strings to try ⇒ O(log n) seed, exp stretch
17 / 35
Two Big Questions Derandomization Hierarchy Theorems
Derandomization – the Standard PRG Approach
Random Strings
good
bad
pseudo–random
01 10 10 1011 01 10
yes/no
PRGhard
functionh
p(x1, x2, x3, x4)
x1, ..., x4
Poly many strings to try ⇒ O(log n) seed, exp stretch
17 / 35
Two Big Questions Derandomization Hierarchy Theorems
A New Approach – Typically-Correct Derandomization
Seed length n, poly stretch
18 / 35
Two Big Questions Derandomization Hierarchy Theorems
A New Approach – Typically-Correct Derandomization
Random Strings
good
bad
pseudo–random
01 10 10 1011 01 10
yes/no
PRG
p(x1, x2, x3, x4)
x1, ..., x4
Seed length n, poly stretch
18 / 35
Two Big Questions Derandomization Hierarchy Theorems
A New Approach – Typically-Correct Derandomization
Random Strings
good
bad
pseudo–random
01 10 10 1011 01 10
yes/no
PRG
11 01 10
p(x1, x2, x3, x4)
x1, ..., x4
Seed length n, poly stretch
18 / 35
Two Big Questions Derandomization Hierarchy Theorems
A New Approach – Typically-Correct Derandomization
Random Strings
good
bad
pseudo–random
01 10 10 1011 01 10
yes/no
PRG
11 01 10
p(x1, x2, x3, x4)
x1, ..., x4
Seed length n, poly stretch
18 / 35
Two Big Questions Derandomization Hierarchy Theorems
Standard Use of PRG’s vs. Typ-Correct
Standard Derandomization Typ-Correct Derandomization[Nisan & Wigderson, ...] [Kinne, Van Melkebeek, Shaltiel]
• Always correct • Small # mistakes• Run PRG many times • Run PRG only once• Need exponential stretch • Need only poly stretch• Conditional results • Unconditional results:
fast parallel time, streaming,communication protocols
19 / 35
Two Big Questions Derandomization Hierarchy Theorems
Standard Use of PRG’s vs. Typ-Correct
Standard Derandomization Typ-Correct Derandomization[Nisan & Wigderson, ...] [Kinne, Van Melkebeek, Shaltiel]
• Always correct • Small # mistakes• Run PRG many times • Run PRG only once• Need exponential stretch • Need only poly stretch• Conditional results • Unconditional results:
fast parallel time, streaming,communication protocols
19 / 35
Two Big Questions Derandomization Hierarchy Theorems
Standard Use of PRG’s vs. Typ-Correct
Standard Derandomization Typ-Correct Derandomization[Nisan & Wigderson, ...] [Kinne, Van Melkebeek, Shaltiel]
• Always correct • Small # mistakes
• Run PRG many times • Run PRG only once• Need exponential stretch • Need only poly stretch• Conditional results • Unconditional results:
fast parallel time, streaming,communication protocols
19 / 35
Two Big Questions Derandomization Hierarchy Theorems
Standard Use of PRG’s vs. Typ-Correct
Standard Derandomization Typ-Correct Derandomization[Nisan & Wigderson, ...] [Kinne, Van Melkebeek, Shaltiel]
• Always correct • Small # mistakes• Run PRG many times • Run PRG only once
• Need exponential stretch • Need only poly stretch• Conditional results • Unconditional results:
fast parallel time, streaming,communication protocols
19 / 35
Two Big Questions Derandomization Hierarchy Theorems
Standard Use of PRG’s vs. Typ-Correct
Standard Derandomization Typ-Correct Derandomization[Nisan & Wigderson, ...] [Kinne, Van Melkebeek, Shaltiel]
• Always correct • Small # mistakes• Run PRG many times • Run PRG only once• Need exponential stretch • Need only poly stretch
• Conditional results • Unconditional results:fast parallel time, streaming,communication protocols
19 / 35
Two Big Questions Derandomization Hierarchy Theorems
Standard Use of PRG’s vs. Typ-Correct
Standard Derandomization Typ-Correct Derandomization[Nisan & Wigderson, ...] [Kinne, Van Melkebeek, Shaltiel]
• Always correct • Small # mistakes• Run PRG many times • Run PRG only once• Need exponential stretch • Need only poly stretch• Conditional results • Unconditional results:
fast parallel time, streaming,communication protocols
19 / 35
Two Big Questions Derandomization Hierarchy Theorems
Computational Complexity Theory
How much time, memory space, etc. are needed to solveproblems?
Is nondeterminism powerful? P vs. NP
Conjecture: P 6= NP
Techniques: hierarchy theorems, others
Is randomness powerful? P vs. BPP, L vs. BPL
Conjecture: P=BPP, L=BPL
My work: derandomization, hierarchy theorems
20 / 35
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems
21 / 35
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems
Fix a model of computing(deterministic, randomized, nondeterministic)
Can we achieve more given more resources?
My work: hierarchy theorems for randomized algorithms
22 / 35
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems
Fix a model of computing
(deterministic, randomized, nondeterministic)
Can we achieve more given more resources?
My work: hierarchy theorems for randomized algorithms
22 / 35
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems
Fix a model of computing(deterministic, randomized, nondeterministic)
Can we achieve more given more resources?
My work: hierarchy theorems for randomized algorithms
22 / 35
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems
Fix a model of computing(deterministic, randomized, nondeterministic)
Can we achieve more given more resources?
My work: hierarchy theorems for randomized algorithms
22 / 35
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems
Fix a model of computing(deterministic, randomized, nondeterministic)
Can we achieve more given more resources?
n time
My work: hierarchy theorems for randomized algorithms
22 / 35
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems
Fix a model of computing(deterministic, randomized, nondeterministic)
Can we achieve more given more resources?
n time
n2 time
My work: hierarchy theorems for randomized algorithms
22 / 35
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems
Fix a model of computing(deterministic, randomized, nondeterministic)
Can we achieve more given more resources?
n time
n2 timen3 time
My work: hierarchy theorems for randomized algorithms
22 / 35
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems
Fix a model of computing(deterministic, randomized, nondeterministic)
Can we achieve more given more resources?
n time
n2 time
Poly Timen3 time
My work: hierarchy theorems for randomized algorithms
22 / 35
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems
Fix a model of computing(deterministic, randomized, nondeterministic)
Can we achieve more given more resources?
n timePoly time =
My work: hierarchy theorems for randomized algorithms
22 / 35
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems
Fix a model of computing(deterministic, randomized, nondeterministic)
Can we achieve more given more resources?
log n space
log2 n space
log3 n space
My work: hierarchy theorems for randomized algorithms
22 / 35
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems
Fix a model of computing(deterministic, randomized, nondeterministic)
Can we achieve more given more resources?
log n spacelog3 n space =
My work: hierarchy theorems for randomized algorithms
22 / 35
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems
Fix a model of computing(deterministic, randomized, nondeterministic)
Can we achieve more given more resources?
log n space
log2 n space
log3 n space
My work: hierarchy theorems for randomized algorithms
22 / 35
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems
Fix a model of computing(deterministic, randomized, nondeterministic)
Can we achieve more given more resources?
My work: hierarchy theorems for randomized algorithms
22 / 35
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems
Fix a model of computing(deterministic, randomized, nondeterministic)
Can we achieve more given more resources?
My work: hierarchy theorems for randomized algorithms
22 / 35
Two Big Questions Derandomization Hierarchy Theorems
Computational Complexity Theory
How much time, memory space, etc. are needed to solveproblems?
Is nondeterminism powerful? P vs. NP
Conjecture: P 6= NP
Techniques: hierarchy theorems, others
Is randomness powerful? P vs. BPP, L vs. BPL
Conjecture: P=BPP, L=BPL
My work: derandomization, hierarchy theorems
23 / 35
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Deterministic Algorithms
24 / 35
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Deterministic Algorithms
A2 A3 ...A1
All Algorithms time n
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Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Deterministic Algorithms
A2 A3 ...
x1
x2
x3
...
A1
All Algorithms
AllI
nput
s
time n
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Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Deterministic Algorithms
A2 A3 ...
A1(x1) A2(x1) A3(x1)
A1(x2) A2(x2) A3(x2) ...
A1(x3) A2(x3) A3(x3)
x1
x2
x3
...
A1
All Algorithms
AllI
nput
s
...
time n
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Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Deterministic Algorithms
A2 A3 ... D
A1(x1) A2(x1) A3(x1)
A1(x2) A2(x2) A3(x2) ...
A1(x3) A2(x3) A3(x3)
x1
x2
x3
...
A1
All Algorithms
AllI
nput
s
...
time ntime n2
24 / 35
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Deterministic Algorithms
A2 A3 ... D
A1(x1) A2(x1) A3(x1)
A1(x2) A2(x2) A3(x2) ...
A1(x3) A2(x3) A3(x3)
x1
x2
x3
...
A1
All Algorithms
AllI
nput
s
...
time ntime n2
24 / 35
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Deterministic Algorithms
A2 A3 ... D
A1(x1) A2(x1) A3(x1) ¬A1(x1)
A1(x2) A2(x2) A3(x2) ... ¬A2(x2)
A1(x3) A2(x3) A3(x3) ¬A3(x3)
...
x1
x2
x3
...
A1
All Algorithms
AllI
nput
s
...
time ntime n2
24 / 35
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Deterministic Algorithms
n time
n2 time
Poly Timen3 time
25 / 35
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Deterministic Algorithms
log n space
log2 n space
log3 n space
25 / 35
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Nondeterministic Algorithms?
26 / 35
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Nondeterministic Algorithms?
A2 A3 ... D
A1(x1) A2(x1) A3(x1) ¬A1(x1)
A1(x2) A2(x2) A3(x2) ... ¬A2(x2)
A1(x3) A2(x3) A3(x3) ¬A3(x3)
...
x1
x2
x3
...
A1
All Nondet. Algorithms
AllI
nput
s
...
time ntime n2
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Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Nondeterministic Algorithms
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Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Nondeterministic Algorithms
A1 Dtime ntime n2
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Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Nondeterministic Algorithms
A1 D
x1
0x1
00x1
...
0`−1x1
0`x1
0`−2x1
inpu
ts
` ≈ 2|x1|
time ntime n2
27 / 35
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Nondeterministic Algorithms
A1 D
A1(x1)
A1(0x1)A1(00x1)
...
A1(0`−1x1)
x1
0x1
00x1
...
0`−1x1
A1(0`x1)0`x1
A1(0`−2x1)0`−2x1
inpu
ts
` ≈ 2|x1|
time ntime n2
27 / 35
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Nondeterministic Algorithms
A1 D
A1(x1)
A1(0x1)A1(00x1)
...
A1(0`−1x1)
x1
0x1
00x1
...
0`−1x1
A1(0`x1) ¬A1(x1)0`x1
A1(0`−2x1)0`−2x1
inpu
ts
6=` ≈ 2|x1|
time ntime n2
27 / 35
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Nondeterministic Algorithms
A1 D
A1(x1) A1(0x1)
A1(0x1) A1(02x1)A1(00x1) A1(03x1)
... ...A1(0`−1x1) A1(0`x1)
x1
0x1
00x1
...
0`−1x1
A1(0`x1) ¬A1(x1)0`x1
A1(0`−2x1) A1(0`−1x1)0`−2x1
inpu
ts
6=
=========
=========
` ≈ 2|x1|
time ntime n2
27 / 35
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Nondeterministic Algorithms
A1 D
A1(x1) A1(0x1)
A1(0x1) A1(02x1)A1(00x1) A1(03x1)
... ...A1(0`−1x1) A1(0`x1)
x1
0x1
00x1
...
0`−1x1
A1(0`x1) ¬A1(x1)0`x1
A1(0`−2x1) A1(0`−1x1)0`−2x1
Assume A1 the same
inpu
ts
6=
=========
=========
as D on all inputs
` ≈ 2|x1|
time ntime n2
27 / 35
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Nondeterministic Algorithms
A1 D
A1(x1) A1(0x1)
A1(0x1) A1(02x1)A1(00x1) A1(03x1)
... ...A1(0`−1x1) A1(0`x1)
x1
0x1
00x1
...
0`−1x1
A1(0`x1) ¬A1(x1)0`x1
A1(0`−2x1) A1(0`−1x1)0`−2x1
Assume A1 the same
inpu
ts
6=
=========
=========
===
===
===
===
===
===
as D on all inputs
` ≈ 2|x1|
time ntime n2
27 / 35
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Nondeterministic Algorithms
¬A1(x1)
A1 D
... ...
x1
0x1
00x1
...
0`−1x1
¬A1(x1)0`x1
0`−2x1
Assume A1 the same
inpu
ts
6=
=========
=========
===
===
===
===
===
===
as D on all inputs
` ≈ 2|x1|
¬A1(x1)¬A1(x1) ¬A1(x1)
¬A1(x1)¬A1(x1)
¬A1(x1)¬A1(x1)¬A1(x1)
¬A1(x1)¬A1(x1)
time ntime n2
27 / 35
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Nondeterministic Algorithms
¬A1(x1)
A1 D
... ...
x1
0x1
00x1
...
0`−1x1
¬A1(x1)0`x1
0`−2x1
Assume A1 the same
inpu
ts
6=
=========
=========
===
===
===
===
===
===
as D on all inputs
` ≈ 2|x1|
¬A1(x1)¬A1(x1) ¬A1(x1)
¬A1(x1)¬A1(x1)
¬A1(x1)¬A1(x1)¬A1(x1)
¬A1(x1)¬A1(x1)
time ntime n2
27 / 35
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Nondeterministic Algorithms
n time
n2 time
Poly Timen3 time
28 / 35
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Nondeterministic Algorithms
log n space
log2 n space
log3 n space
28 / 35
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized Algorithms?
What if Pr[A1(x1) = “yes”] ≈ .5?
Then D does not have bounded error, not valid
29 / 35
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized Algorithms?
A2 A3 ... D
A1(x1) A2(x1) A3(x1) ¬A1(x1)
A1(x2) A2(x2) A3(x2) ... ¬A2(x2)
A1(x3) A2(x3) A3(x3) ¬A3(x3)
...
x1
x2
x3
...
A1
All Randomized Algorithms
AllI
nput
s
...
time ntime n2
What if Pr[A1(x1) = “yes”] ≈ .5?
Then D does not have bounded error, not valid
29 / 35
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized Algorithms?
A2 A3 ... D
A1(x1) A2(x1) A3(x1) ¬A1(x1)
A1(x2) A2(x2) A3(x2) ... ¬A2(x2)
A1(x3) A2(x3) A3(x3) ¬A3(x3)
...
x1
x2
x3
...
A1
All Randomized Algorithms
AllI
nput
s
...
time ntime n2
What if Pr[A1(x1) = “yes”] ≈ .5?
Then D does not have bounded error, not valid
29 / 35
Two Big Questions Derandomization Hierarchy Theorems
Randomized Algorithm
Random Strings
good
bad
01 10 10 1011 01 10
yes/no
Bounded error: Correct with probability > 99%
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Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized Algorithms?
A2 A3 ... D
A1(x1) A2(x1) A3(x1) ¬A1(x1)
A1(x2) A2(x2) A3(x2) ... ¬A2(x2)
A1(x3) A2(x3) A3(x3) ¬A3(x3)
...
x1
x2
x3
...
A1
All Randomized Algorithms
AllI
nput
s
...
time ntime n2
What if Pr[A1(x1) = “yes”] ≈ .5?
Then D does not have bounded error, not valid
31 / 35
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized Algorithms?
A2 A3 ... D
A1(x1) A2(x1) A3(x1) ¬A1(x1)
A1(x2) A2(x2) A3(x2) ... ¬A2(x2)
A1(x3) A2(x3) A3(x3) ¬A3(x3)
...
x1
x2
x3
...
A1
All Randomized Algorithms
AllI
nput
s
...
time ntime n2
What if Pr[A1(x1) = “yes”] ≈ .5?
Then D does not have bounded error, not valid
31 / 35
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized Algorithms?
A1 D
A1(x1) A1(0x1)
A1(0x1) A1(02x1)A1(00x1) A1(03x1)
... ...A1(0`−1x1) A1(0`x1)
x1
0x1
00x1
...
0`−1x1
A1(0`x1) ¬A1(x1)0`x1
A1(0`−2x1) A1(0`−1x1)0`−2x1
inpu
ts
6=
=========
=========
` ≈ 2|x1|
time ntime n2
Make sure D has bounded error – 1 bit of advice
32 / 35
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized Algorithms?
A1 D
A1(x1) A1(0x1)
A1(0x1) A1(02x1)A1(00x1) A1(03x1)
... ...A1(0`−1x1) A1(0`x1)
x1
0x1
00x1
...
0`−1x1
A1(0`x1) ¬A1(x1)0`x1
A1(0`−2x1) A1(0`−1x1)0`−2x1
inpu
ts
6=
=========
=========
` ≈ 2|x1|
time ntime n2
Make sure D has bounded error – 1 bit of advice
32 / 35
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized Algorithms?
A1 D
A1(x1) A1(0x1)
A1(0x1) A1(02x1)A1(00x1) A1(03x1)
... ...A1(0`−1x1) A1(0`x1)
x1
0x1
00x1
...
0`−1x1
A1(0`x1) ¬A1(x1)0`x1
A1(0`−2x1) A1(0`−1x1)0`−2x1
inpu
ts
6=
=========
=========
` ≈ 2|x1|
time ntime n2
Make sure D has bounded error
– 1 bit of advice
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Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized Algorithms?
A1 D
A1(x1) A1(0x1)
A1(0x1) A1(02x1)A1(00x1) A1(03x1)
... ...A1(0`−1x1) A1(0`x1)
x1
0x1
00x1
...
0`−1x1
A1(0`x1) ¬A1(x1)0`x1
A1(0`−2x1) A1(0`−1x1)0`−2x1
inpu
ts
6=
=========
=========
` ≈ 2|x1|
time ntime n2
Make sure D has bounded error – 1 bit of advice
32 / 35
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized Algorithms?
Yes, for algorithms with 1 bit of advice!
My work [ Kinne, Van Melkebeek ]Memory Space hierarchies: randomized, quantum, ...
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Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized Algorithms?
Yes, for algorithms with 1 bit of advice!
My work [ Kinne, Van Melkebeek ]Memory Space hierarchies: randomized, quantum, ...
33 / 35
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized Algorithms?
Yes, for algorithms with 1 bit of advice!
n time
n2 time
Poly Timen3 time
My work [ Kinne, Van Melkebeek ]Memory Space hierarchies: randomized, quantum, ...
33 / 35
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized Algorithms?
Yes, for algorithms with 1 bit of advice!
log n space
log2 n space
log3 n space
My work [ Kinne, Van Melkebeek ]Memory Space hierarchies: randomized, quantum, ...
33 / 35
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized Algorithms?
Yes, for algorithms with 1 bit of advice!
log n space
log2 n space
log3 n space
My work [ Kinne, Van Melkebeek ]Memory Space hierarchies: randomized, quantum, ...
33 / 35
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized Algorithms?
Yes, for algorithms with 1 bit of advice!
log n space
log2 n space
log3 n space
My work [ Kinne, Van Melkebeek ]
Memory Space hierarchies: randomized, quantum, ...
33 / 35
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized Algorithms?
Yes, for algorithms with 1 bit of advice!
log n space
log2 n space
log3 n space
My work [ Kinne, Van Melkebeek ]Memory Space hierarchies: randomized, quantum, ...
33 / 35
Two Big Questions Derandomization Hierarchy Theorems
Computational Complexity Theory
How much time, memory space, etc. are needed to solveproblems?
Is nondeterminism powerful? P vs. NP
Conjecture: P 6= NP
Techniques: hierarchy theorems, others
Is randomness powerful? P vs. BPP, L vs. BPL
Conjecture: P=BPP, L=BPL
My work: derandomization, hierarchy theorems
34 / 35
Two Big Questions Derandomization Hierarchy Theorems
The End, Thank You!
Slides available at:http://www.kinnejeff.com/GoSycamores/
(or E-mail me)
More on my research (slides, papers, etc.) at:http://www.kinnejeff.com/
35 / 35
Two Big Questions Derandomization Hierarchy Theorems
The End, Thank You!
Slides available at:http://www.kinnejeff.com/GoSycamores/
(or E-mail me)
More on my research (slides, papers, etc.) at:http://www.kinnejeff.com/
35 / 35