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 Is nondeterminism powerful? P vs. NP Conjecture: P6= NP Techniques: hierarchy theorems, others
Is randomness powerful? P vs. BPP, L vs. BPL Conjecture: P=BPP, L=BPL My work: derandomization, hierarchy theorems
Two Big Questions Derandomization Hierarchy Theorems
Computational Complexity Theory How much time, memory space, etc. are needed to solve problems?
2 / 35 P vs. NP Conjecture: P6= NP Techniques: hierarchy theorems, others
Is randomness powerful? P vs. BPP, L vs. BPL Conjecture: P=BPP, L=BPL My work: derandomization, hierarchy theorems
Two Big Questions Derandomization Hierarchy Theorems
Computational Complexity Theory How much time, memory space, etc. are needed to solve problems?
Is nondeterminism powerful?
2 / 35 Conjecture: P6= NP Techniques: hierarchy theorems, others
Is randomness powerful? P vs. BPP, L vs. BPL Conjecture: P=BPP, L=BPL My work: derandomization, hierarchy theorems
Two Big Questions Derandomization Hierarchy Theorems
Computational Complexity Theory How much time, memory space, etc. are needed to solve problems?
Is nondeterminism powerful? P vs. NP
2 / 35 Techniques: hierarchy theorems, others
Is randomness powerful? P vs. BPP, L vs. BPL Conjecture: P=BPP, L=BPL My work: derandomization, hierarchy theorems
Two Big Questions Derandomization Hierarchy Theorems
Computational Complexity Theory How much time, memory space, etc. are needed to solve problems?
Is nondeterminism powerful? P vs. NP Conjecture: P6= NP
2 / 35 Is randomness powerful? P vs. BPP, L vs. BPL Conjecture: P=BPP, L=BPL My work: derandomization, hierarchy theorems
Two Big Questions Derandomization Hierarchy Theorems
Computational Complexity Theory How much time, memory space, etc. are needed to solve problems?
Is nondeterminism powerful? P vs. NP Conjecture: P6= NP Techniques: hierarchy theorems, others
2 / 35 P vs. BPP, L vs. BPL Conjecture: P=BPP, L=BPL My work: derandomization, hierarchy theorems
Two Big Questions Derandomization Hierarchy Theorems
Computational Complexity Theory How much time, memory space, etc. are needed to solve problems?
Is nondeterminism powerful? P vs. NP Conjecture: P6= NP Techniques: hierarchy theorems, others
Is randomness powerful?
2 / 35 Conjecture: P=BPP, L=BPL My work: derandomization, hierarchy theorems
Two Big Questions Derandomization Hierarchy Theorems
Computational Complexity Theory How much time, memory space, etc. are needed to solve problems?
Is nondeterminism powerful? P vs. NP Conjecture: P6= NP Techniques: hierarchy theorems, others
Is randomness powerful? P vs. BPP, L vs. BPL
2 / 35 My work: derandomization, hierarchy theorems
Two Big Questions Derandomization Hierarchy Theorems
Computational Complexity Theory How much time, memory space, etc. are needed to solve problems?
Is nondeterminism powerful? P vs. NP Conjecture: P6= NP Techniques: hierarchy theorems, others
Is randomness powerful? P vs. BPP, L vs. BPL Conjecture: P=BPP, L=BPL
2 / 35 Two Big Questions Derandomization Hierarchy Theorems
Computational Complexity Theory How much time, memory space, etc. are needed to solve problems?
Is nondeterminism powerful? P vs. NP Conjecture: P6= 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 Decision Problem: yes/no questions
Two Big Questions Derandomization Hierarchy Theorems
Time Complexity
4 / 35 Decision Problem: yes/no questions
Two Big Questions Derandomization Hierarchy Theorems
Time Complexity
Exp Time (EXP)
Poly Time (P)
4 / 35 Decision Problem: yes/no questions
Two Big Questions Derandomization Hierarchy Theorems
Time Complexity
Exp Time (EXP)
Poly Time (P) factoring, NP-complete sorting shortest path
4 / 35 Decision Problem: yes/no questions
Two Big Questions Derandomization Hierarchy Theorems
Time Complexity
Exp Time (EXP)
Poly Time (P)
sorting ? shortest path factoring, NP-complete
4 / 35 Two Big Questions Derandomization Hierarchy Theorems
Time Complexity
Exp Time (EXP)
Poly Time (P)
sorting ? shortest path factoring, NP-complete
Decision Problem: yes/no questions
4 / 35 Amount of working space memory needed
Two Big Questions Derandomization Hierarchy Theorems
Memory Space Complexity
5 / 35 Amount of working space memory needed
Two Big Questions Derandomization Hierarchy Theorems
Memory Space Complexity
Poly Space (PSPACE)
Log Space (L)
5 / 35 Two Big Questions Derandomization Hierarchy Theorems
Memory Space Complexity
Poly Space (PSPACE)
Log Space (L)
Amount of working space memory needed
5 / 35 Two Big Questions Derandomization Hierarchy Theorems
Memory Space Complexity
Poly Space (PSPACE)
Log Space (L)
factoring, arithmetic NP-complete
Amount of working space memory needed
5 / 35 Two Big Questions Derandomization Hierarchy Theorems
Memory Space Complexity
Poly Space (PSPACE)
Log Space (L)
? arithmetic factoring, NP-complete
Amount of working space memory needed
5 / 35 Bounded error: Correct with probability > 99%
Two Big Questions Derandomization Hierarchy Theorems
Randomized Algorithm
6 / 35 Bounded error: Correct with probability > 99%
Two Big Questions Derandomization Hierarchy Theorems
Randomized Algorithm
yes /no
11 01 10 01 10 10 10
6 / 35 Bounded error: Correct with probability > 99%
Two Big Questions Derandomization Hierarchy Theorems
Randomized Algorithm
Random Strings yes /no
bad 11 01 10 good 01 10 10 10
6 / 35 Two Big Questions Derandomization Hierarchy Theorems
Randomized Algorithm
Random Strings yes /no
bad 11 01 10 good 01 10 10 10
Bounded error: Correct with probability > 99%
6 / 35 p(x) = x3 · (3x − x2)2 − x4 · (2x3 + 5x) + x · (4x2 − x)3 Do all terms cancel?
Two Big Questions Derandomization Hierarchy Theorems
Polynomial Identity Testing
7 / 35 Do all terms cancel?
Two Big Questions Derandomization Hierarchy Theorems
Polynomial Identity Testing
p(x) = x3 · (3x − x2)2 − x4 · (2x3 + 5x) + x · (4x2 − x)3
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 5 30 15 p(x1, x2, x3, x4) = x4 · (x1 − x2 − x3) + (x4 + x3 − x1) · 20 20 15 (x3 − x2 + x1) − (x2 − x3 + x4) · (x2 + x1) Do all terms cancel?
Randomized Algorithm
Pick point (x1, ..., x4), each xi ∈R S d p non-zero, degree d ⇒ Pr[p(x1, ..., x4) = 0] ≤ |S|
Two Big Questions Derandomization Hierarchy Theorems
Multi-variate Polynomial Identity Testing
8 / 35 Do all terms cancel?
Randomized Algorithm
Pick point (x1, ..., x4), each xi ∈R S d p non-zero, degree d ⇒ Pr[p(x1, ..., x4) = 0] ≤ |S|
Two Big Questions Derandomization Hierarchy Theorems
Multi-variate Polynomial Identity Testing
5 30 15 p(x1, x2, x3, x4) = x4 · (x1 − x2 − x3) + (x4 + x3 − x1) · 20 20 15 (x3 − x2 + x1) − (x2 − x3 + x4) · (x2 + x1)
8 / 35 Randomized Algorithm
Pick point (x1, ..., x4), each xi ∈R S d p non-zero, degree d ⇒ Pr[p(x1, ..., x4) = 0] ≤ |S|
Two Big Questions Derandomization Hierarchy Theorems
Multi-variate Polynomial Identity Testing
5 30 15 p(x1, x2, x3, x4) = x4 · (x1 − x2 − x3) + (x4 + x3 − x1) · 20 20 15 (x3 − x2 + x1) − (x2 − x3 + x4) · (x2 + x1) Do all terms cancel?
8 / 35 Pick point (x1, ..., x4), each xi ∈R S d p non-zero, degree d ⇒ Pr[p(x1, ..., x4) = 0] ≤ |S|
Two Big Questions Derandomization Hierarchy Theorems
Multi-variate Polynomial Identity Testing
5 30 15 p(x1, x2, x3, x4) = x4 · (x1 − x2 − x3) + (x4 + x3 − x1) · 20 20 15 (x3 − x2 + x1) − (x2 − x3 + x4) · (x2 + x1) Do all terms cancel?
Randomized Algorithm
8 / 35 d p non-zero, degree d ⇒ Pr[p(x1, ..., x4) = 0] ≤ |S|
Two Big Questions Derandomization Hierarchy Theorems
Multi-variate Polynomial Identity Testing
5 30 15 p(x1, x2, x3, x4) = x4 · (x1 − x2 − x3) + (x4 + x3 − x1) · 20 20 15 (x3 − x2 + x1) − (x2 − x3 + x4) · (x2 + x1) Do all terms cancel?
Randomized Algorithm
Pick point (x1, ..., x4), each xi ∈R S
8 / 35 d ⇒ Pr[p(x1, ..., x4) = 0] ≤ |S|
Two Big Questions Derandomization Hierarchy Theorems
Multi-variate Polynomial Identity Testing
5 30 15 p(x1, x2, x3, x4) = x4 · (x1 − x2 − x3) + (x4 + x3 − x1) · 20 20 15 (x3 − x2 + x1) − (x2 − x3 + x4) · (x2 + x1) Do all terms cancel?
Randomized Algorithm
Pick point (x1, ..., x4), each xi ∈R S p non-zero, degree d
8 / 35 Two Big Questions Derandomization Hierarchy Theorems
Multi-variate Polynomial Identity Testing
5 30 15 p(x1, x2, x3, x4) = x4 · (x1 − x2 − x3) + (x4 + x3 − x1) · 20 20 15 (x3 − x2 + x1) − (x2 − x3 + x4) · (x2 + x1) Do all terms cancel?
Randomized Algorithm
Pick point (x1, ..., x4), each xi ∈R S d p non-zero, degree d ⇒ Pr[p(x1, ..., x4) = 0] ≤ |S|
8 / 35 BPP: Bounded-error Probabilistic Poly time , BPL: log space
P =? BPP
Two Big Questions Derandomization Hierarchy Theorems
Randomized Algorithm
Random Strings yes /no
p(x ; x ; x ; x ) bad 11 01 10 1 2 3 4 good 01 10 10 10 x1; :::; x4
9 / 35 , BPL: log space
P =? BPP
Two Big Questions Derandomization Hierarchy Theorems
Randomized Algorithm
Random Strings yes /no
p(x ; x ; x ; x ) bad 11 01 10 1 2 3 4 good 01 10 10 10 x1; :::; x4
BPP: Bounded-error Probabilistic Poly time
9 / 35 P =? BPP
Two Big Questions Derandomization Hierarchy Theorems
Randomized Algorithm
Random Strings yes /no
p(x ; x ; x ; x ) bad 11 01 10 1 2 3 4 good 01 10 10 10 x1; :::; x4
BPP: Bounded-error Probabilistic Poly time , BPL: log space
9 / 35 Two Big Questions Derandomization Hierarchy Theorems
Randomized Algorithm
Random Strings yes /no
p(x ; x ; x ; x ) bad 11 01 10 1 2 3 4 good 01 10 10 10 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 NP =? coNP
NP: Nondeterministic Polynomial time “NP-Complete” problems: 3-Coloring, TSP, Knapsack, ... P =? NP
Two Big Questions Derandomization Hierarchy Theorems
Nondeterministic Algorithm
12 / 35 NP =? coNP
“NP-Complete” problems: 3-Coloring, TSP, Knapsack, ... P =? NP
NP: Nondeterministic Polynomial time
Two Big Questions Derandomization Hierarchy Theorems
Nondeterministic Algorithm
yes /no
graph 11 01 10 coloring 01 10 10 10 Proof/Certificate
12 / 35 NP =? coNP
P =? NP
NP: Nondeterministic Polynomial time “NP-Complete” problems: 3-Coloring, TSP, Knapsack, ...
Two Big Questions Derandomization Hierarchy Theorems
Nondeterministic Algorithm
yes Potential Proofs /no
graph bad 11 01 10 coloring 01 10 10 10 good Proof/Certificate
12 / 35 NP =? coNP
“NP-Complete” problems: 3-Coloring, TSP, Knapsack, ... P =? NP
Two Big Questions Derandomization Hierarchy Theorems
Nondeterministic Algorithm
yes Potential Proofs /no
graph bad 11 01 10 coloring 01 10 10 10 good Proof/Certificate
NP: Nondeterministic Polynomial time
12 / 35 NP =? coNP
P =? NP
Two Big Questions Derandomization Hierarchy Theorems
Nondeterministic Algorithm
yes Potential Proofs /no
graph bad 11 01 10 coloring 01 10 10 10 good Proof/Certificate
NP: Nondeterministic Polynomial time “NP-Complete” problems: 3-Coloring, TSP, Knapsack, ...
12 / 35 NP =? coNP
Two Big Questions Derandomization Hierarchy Theorems
Nondeterministic Algorithm
yes Potential Proofs /no
graph bad 11 01 10 coloring 01 10 10 10 good Proof/Certificate
NP: Nondeterministic Polynomial time “NP-Complete” problems: 3-Coloring, TSP, Knapsack, ... P =? NP
12 / 35 Two Big Questions Derandomization Hierarchy Theorems
Nondeterministic Algorithm
yes Potential Proofs /no
graph bad 11 01 10 coloring 01 10 10 10 good Proof/Certificate
NP: Nondeterministic Polynomial time “NP-Complete” problems: 3-Coloring, TSP, Knapsack, ... P =? NP NP =? coNP
12 / 35 P =? BPP Does randomness truly add power?
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?
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 solve problems?
Is nondeterminism powerful? P vs. NP Conjecture: P6= 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 Random Strings yes /no
p(x ; x ; x ; x ) bad 11 01 10 1 2 3 4 good 01 10 10 10 x1; :::; x4
Try all possible random bit strings – exponentially many
Two Big Questions Derandomization Hierarchy Theorems
Naive Derandomization
16 / 35 Try all possible random bit strings – exponentially many
Two Big Questions Derandomization Hierarchy Theorems
Naive Derandomization
Random Strings yes /no
p(x ; x ; x ; x ) bad 11 01 10 1 2 3 4 good 01 10 10 10 x1; :::; x4
16 / 35 – exponentially many
Two Big Questions Derandomization Hierarchy Theorems
Naive Derandomization
Random Strings yes /no
p(x ; x ; x ; x ) bad 11 01 10 1 2 3 4 good 01 10 10 10 x1; :::; x4
Try all possible random bit strings
16 / 35 Two Big Questions Derandomization Hierarchy Theorems
Naive Derandomization
Random Strings yes /no
p(x ; x ; x ; x ) bad 11 01 10 1 2 3 4 good 01 10 10 10 x1; :::; x4
Try all possible random bit strings – exponentially many
16 / 35 ⇒ O(log n) seed, exp stretch
Poly many strings to try
Two Big Questions Derandomization Hierarchy Theorems
Derandomization – the Standard PRG Approach
17 / 35 ⇒ O(log n) seed, exp stretch
Poly many strings to try
Two Big Questions Derandomization Hierarchy Theorems
Derandomization – the Standard PRG Approach
Random Strings yes /no
p(x ; x ; x ; x ) bad 11 01 10 1 2 3 4 good 01 10 10 10 x1; :::; x4
17 / 35 ⇒ O(log n) seed, exp stretch
Poly many strings to try
Two Big Questions Derandomization Hierarchy Theorems
Derandomization – the Standard PRG Approach
Random Strings yes /no pseudo–random
p(x ; x ; x ; x ) bad 11 01 10 1 2 3 4 good 01 10 10 10 x1; :::; x4
17 / 35 ⇒ O(log n) seed, exp stretch
Poly many strings to try
Two Big Questions Derandomization Hierarchy Theorems
Derandomization – the Standard PRG Approach
Random Strings yes /no pseudo–random
p(x ; x ; x ; x ) bad 11 01 10 1 2 3 4 x ; :::; x good 01 10 10 10 1 4 PRG
17 / 35 ⇒ O(log n) seed, exp stretch
Two Big Questions Derandomization Hierarchy Theorems
Derandomization – the Standard PRG Approach
Random Strings yes /no pseudo–random
p(x ; x ; x ; x ) bad 11 01 10 1 2 3 4 x ; :::; x good 01 10 10 10 1 4 PRG
Poly many strings to try
17 / 35 Two Big Questions Derandomization Hierarchy Theorems
Derandomization – the Standard PRG Approach
Random Strings yes /no pseudo–random
p(x ; x ; x ; x ) bad 11 01 10 1 2 3 4 x ; :::; x good 01 10 10 10 1 4 PRG
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 yes /no pseudo–random
p(x ; x ; x ; x ) bad 11 01 10 1 2 3 4 x ; :::; x good 01 10 10 10 1 4 hard PRG function h
Poly many strings to try ⇒ O(log n) seed, exp stretch
17 / 35 Seed length n, poly stretch
Two Big Questions Derandomization Hierarchy Theorems
A New Approach – Typically-Correct Derandomization
18 / 35 Seed length n, poly stretch
Two Big Questions Derandomization Hierarchy Theorems
A New Approach – Typically-Correct Derandomization
Random Strings yes /no pseudo–random
p(x ; x ; x ; x ) bad 11 01 10 1 2 3 4 x ; :::; x good 01 10 10 10 1 4 PRG
18 / 35 Seed length n, poly stretch
Two Big Questions Derandomization Hierarchy Theorems
A New Approach – Typically-Correct Derandomization
Random Strings yes /no pseudo–random
p(x ; x ; x ; x ) bad 11 01 10 1 2 3 4 x ; :::; x good 01 10 10 10 1 4 PRG
11 01 10
18 / 35 Two Big Questions Derandomization Hierarchy Theorems
A New Approach – Typically-Correct Derandomization
Random Strings yes /no pseudo–random
p(x ; x ; x ; x ) bad 11 01 10 1 2 3 4 x ; :::; x good 01 10 10 10 1 4 PRG
11 01 10
Seed length n, poly stretch
18 / 35 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
Two Big Questions Derandomization Hierarchy Theorems
Standard Use of PRG’s vs. Typ-Correct
19 / 35 • 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
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]
19 / 35 • 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
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
19 / 35 • Need exponential stretch • Need only poly stretch • Conditional results • Unconditional results: fast parallel time, streaming, communication protocols
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
19 / 35 • Conditional results • Unconditional results: fast parallel time, streaming, communication protocols
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
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 solve problems?
Is nondeterminism powerful? P vs. NP Conjecture: P6= 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 Fix a model of computing (deterministic, randomized, nondeterministic) Can we achieve more given more resources?
My work: hierarchy theorems for randomized algorithms
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems
22 / 35 (deterministic, randomized, nondeterministic) Can we achieve more given more resources?
My work: hierarchy theorems for randomized algorithms
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems
Fix a model of computing
22 / 35 Can we achieve more given more resources?
My work: hierarchy theorems for randomized algorithms
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems
Fix a model of computing (deterministic, randomized, nondeterministic)
22 / 35 My work: hierarchy theorems for randomized algorithms
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems
Fix a model of computing (deterministic, randomized, nondeterministic) Can we achieve more given more resources?
22 / 35 My work: hierarchy theorems for randomized algorithms
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
22 / 35 My work: hierarchy theorems for randomized algorithms
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems
Fix a model of computing (deterministic, randomized, nondeterministic) Can we achieve more given more resources?
n2 time
n time
22 / 35 My work: hierarchy theorems for randomized algorithms
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems
Fix a model of computing (deterministic, randomized, nondeterministic) Can we achieve more given more resources?
n3 time n2 time
n time
22 / 35 My work: hierarchy theorems for randomized algorithms
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems
Fix a model of computing (deterministic, randomized, nondeterministic) Can we achieve more given more resources?
Poly Time n3 time n2 time
n time
22 / 35 My work: hierarchy theorems for randomized algorithms
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems
Fix a model of computing (deterministic, randomized, nondeterministic) Can we achieve more given more resources?
Poly time = n time
22 / 35 My work: hierarchy theorems for randomized algorithms
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems
Fix a model of computing (deterministic, randomized, nondeterministic) Can we achieve more given more resources?
log3 n space log2 n space
log n space
22 / 35 My work: hierarchy theorems for randomized algorithms
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems
Fix a model of computing (deterministic, randomized, nondeterministic) Can we achieve more given more resources?
log3 n space = log n space
22 / 35 My work: hierarchy theorems for randomized algorithms
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems
Fix a model of computing (deterministic, randomized, nondeterministic) Can we achieve more given more resources?
log3 n space log2 n space
log n space
22 / 35 My work: hierarchy theorems for randomized algorithms
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems
Fix a model of computing (deterministic, randomized, nondeterministic) Can we achieve more given more resources?
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 solve problems?
Is nondeterminism powerful? P vs. NP Conjecture: P6= 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
n All Algorithms time
A1 A2 A3 ...
24 / 35 Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Deterministic Algorithms
n All Algorithms time
A1 A2 A3 ...
x1
x2
x3
All Inputs ...
24 / 35 Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Deterministic Algorithms
n All Algorithms time
A1 A2 A3 ...
x1 A1(x1) A2(x1) A3(x1)
x2 A1(x2) A2(x2) A3(x2) ...
x3 A1(x3) A2(x3) A3(x3)
All Inputs
......
24 / 35 Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Deterministic Algorithms
2 n n All Algorithms time time
A1 A2 A3 ... D
x1 A1(x1) A2(x1) A3(x1)
x2 A1(x2) A2(x2) A3(x2) ...
x3 A1(x3) A2(x3) A3(x3)
All Inputs
......
24 / 35 Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Deterministic Algorithms
2 n n All Algorithms time time
A1 A2 A3 ... D
x1 A1(x1) A2(x1) A3(x1)
x2 A1(x2) A2(x2) A3(x2) ...
x3 A1(x3) A2(x3) A3(x3)
All Inputs
......
24 / 35 Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Deterministic Algorithms
2 n n All Algorithms time time
A1 A2 A3 ... D
x1 A1(x1) A2(x1) A3(x1) ¬A1(x1)
x2 A1(x2) A2(x2) A3(x2) ... ¬A2(x2)
x3 A1(x3) A2(x3) A3(x3) ¬A3(x3)
All Inputs
...
......
24 / 35 Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Deterministic Algorithms
Poly Time n3 time n2 time
n time
25 / 35 Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Deterministic Algorithms
log3 n space log2 n space
log 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?
2 n n All Nondet. Algorithms time time
A1 A2 A3 ... D
x1 A1(x1) A2(x1) A3(x1) ¬A1(x1)
x2 A1(x2) A2(x2) A3(x2) ... ¬A2(x2)
x3 A1(x3) A2(x3) A3(x3) ¬A3(x3)
All Inputs
...
......
26 / 35 Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Nondeterministic Algorithms
27 / 35 Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Nondeterministic Algorithms
2 n n A1 time D time
27 / 35 Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Nondeterministic Algorithms
2 n n A1 time D time
x1
0x1
00x1
...
inputs
`−2 0 x1 `−1 0 x1 ` 0 x1
` ≈ 2|x1|
27 / 35 Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Nondeterministic Algorithms
2 n n A1 time D time
x1 A1(x1)
0x1 A1(0x1)
00x1 A1(00x1)
...
...
inputs
`−2 `−2 0 x1 A1(0 x1) `−1 `−1 0 x1 A1(0 x1) ` ` 0 x1 A1(0 x1)
` ≈ 2|x1|
27 / 35 Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Nondeterministic Algorithms
2 n n A1 time D time
x1 A1(x1)
0x1 A1(0x1)
00x1 A1(00x1)
...
...
inputs
`−2 `−2 0 x1 A1(0 x1) `−1 `−1 0 x1 A1(0 x1) ` ` 0 x1 A1(0 x1) ¬A1(x1) ` ≈ 2|x1| 6=
27 / 35 Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Nondeterministic Algorithms
2 n n A1 time D time
x1 A1(x1) A1(0x1) === 2 0x1 A1(0x1) A1(0 x1) === 3 00x1 A1(00x1) A1(0 x1)
...
...... ===
inputs
`−2 `−2 === `−1 0 x1 A1(0 x1) A1(0 x1) `−1 `−1 === ` 0 x1 A1(0 x1) A1(0 x1) ` ` === 0 x1 A1(0 x1) ¬A1(x1) ` ≈ 2|x1| 6=
27 / 35 Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Nondeterministic Algorithms
2 n n A1 time D time
x1 A1(x1) A1(0x1) === 2 0x1 A1(0x1) A1(0 x1) === 3 00x1 A1(00x1) A1(0 x1) Assume A1 the same
...
...... ===
inputs as D on all inputs
`−2 `−2 === `−1 0 x1 A1(0 x1) A1(0 x1) `−1 `−1 === ` 0 x1 A1(0 x1) A1(0 x1) ` ` === 0 x1 A1(0 x1) ¬A1(x1) ` ≈ 2|x1| 6=
27 / 35 Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Nondeterministic Algorithms
2 n n A1 time D time
x1 A1(x1) === A1(0x1) === 2 0x1 A1(0x1) === A1(0 x1) === 3 00x1 A1(00x1) === A1(0 x1) Assume A1 the same
...
...... ===
inputs as D on all inputs
`−2 `−2 === `−1 0 x1 A1(0 x1) === A1(0 x1) `−1 `−1 === ` 0 x1 A1(0 x1) === A1(0 x1) ` ` === 0 x1 A1(0 x1) === ¬A1(x1) ` ≈ 2|x1| 6=
27 / 35 Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Nondeterministic Algorithms
2 n n A1 time D time
x1 ¬A1(x1) === ¬A1(x1) === 0x1 ¬A1(x1) === ¬A1(x1) === 00x1 ¬A1(x1) === ¬A1(x1) Assume A1 the same
...
...... ===
inputs as D on all inputs
`−2 === 0 x1 ¬A1(x1) === ¬A1(x1) `−1 === 0 x1 ¬A1(x1) === ¬A1(x1) ` === 0 x1 ¬A1(x1) === ¬A1(x1) ` ≈ 2|x1| 6=
27 / 35 Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Nondeterministic Algorithms
2 n n A1 time D time
x1 ¬A1(x1) === ¬A1(x1) === 0x1 ¬A1(x1) === ¬A1(x1) === 00x1 ¬A1(x1) === ¬A1(x1) Assume A1 the same
...
...... ===
inputs as D on all inputs
`−2 === 0 x1 ¬A1(x1) === ¬A1(x1) `−1 === 0 x1 ¬A1(x1) === ¬A1(x1) ` === 0 x1 ¬A1(x1) === ¬A1(x1) ` ≈ 2|x1| 6=
27 / 35 Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Nondeterministic Algorithms
Poly Time n3 time n2 time
n time
28 / 35 Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Nondeterministic Algorithms
log3 n space log2 n space
log n space
28 / 35 What if Pr[A1(x1) = “yes”] ≈ .5? Then D does not have bounded error, not valid
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized Algorithms?
29 / 35 What if Pr[A1(x1) = “yes”] ≈ .5? Then D does not have bounded error, not valid
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized Algorithms?
2 n n All Randomized Algorithms time time
A1 A2 A3 ... D
x1 A1(x1) A2(x1) A3(x1) ¬A1(x1)
x2 A1(x2) A2(x2) A3(x2) ... ¬A2(x2)
x3 A1(x3) A2(x3) A3(x3) ¬A3(x3)
All Inputs
...
......
29 / 35 Then D does not have bounded error, not valid
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized Algorithms?
2 n n All Randomized Algorithms time time
A1 A2 A3 ... D
x1 A1(x1) A2(x1) A3(x1) ¬A1(x1)
x2 A1(x2) A2(x2) A3(x2) ... ¬A2(x2)
x3 A1(x3) A2(x3) A3(x3) ¬A3(x3)
All Inputs
...
......
What if Pr[A1(x1) = “yes”] ≈ .5?
29 / 35 Two Big Questions Derandomization Hierarchy Theorems
Randomized Algorithm
Random Strings yes /no
bad 11 01 10 good 01 10 10 10
Bounded error: Correct with probability > 99%
30 / 35 Then D does not have bounded error, not valid
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized Algorithms?
2 n n All Randomized Algorithms time time
A1 A2 A3 ... D
x1 A1(x1) A2(x1) A3(x1) ¬A1(x1)
x2 A1(x2) A2(x2) A3(x2) ... ¬A2(x2)
x3 A1(x3) A2(x3) A3(x3) ¬A3(x3)
All Inputs
...
......
What if Pr[A1(x1) = “yes”] ≈ .5?
31 / 35 Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized Algorithms?
2 n n All Randomized Algorithms time time
A1 A2 A3 ... D
x1 A1(x1) A2(x1) A3(x1) ¬A1(x1)
x2 A1(x2) A2(x2) A3(x2) ... ¬A2(x2)
x3 A1(x3) A2(x3) A3(x3) ¬A3(x3)
All Inputs
...
......
What if Pr[A1(x1) = “yes”] ≈ .5? Then D does not have bounded error, not valid
31 / 35 2 n n A1 time D time
x1 A1(x1) A1(0x1) === 2 0x1 A1(0x1) A1(0 x1) === 3 00x1 A1(00x1) A1(0 x1)
...
...... ===
inputs
`−2 `−2 === `−1 0 x1 A1(0 x1) A1(0 x1) `−1 `−1 === ` 0 x1 A1(0 x1) A1(0 x1) ` ` === 0 x1 A1(0 x1) ¬A1(x1) ` ≈ 2|x1| 6=
Make sure D has bounded error – 1 bit of advice
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized Algorithms?
32 / 35 Make sure D has bounded error – 1 bit of advice
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized Algorithms?
2 n n A1 time D time
x1 A1(x1) A1(0x1) === 2 0x1 A1(0x1) A1(0 x1) === 3 00x1 A1(00x1) A1(0 x1)
...
...... ===
inputs
`−2 `−2 === `−1 0 x1 A1(0 x1) A1(0 x1) `−1 `−1 === ` 0 x1 A1(0 x1) A1(0 x1) ` ` === 0 x1 A1(0 x1) ¬A1(x1) ` ≈ 2|x1| 6=
32 / 35 – 1 bit of advice
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized Algorithms?
2 n n A1 time D time
x1 A1(x1) A1(0x1) === 2 0x1 A1(0x1) A1(0 x1) === 3 00x1 A1(00x1) A1(0 x1)
...
...... ===
inputs
`−2 `−2 === `−1 0 x1 A1(0 x1) A1(0 x1) `−1 `−1 === ` 0 x1 A1(0 x1) A1(0 x1) ` ` === 0 x1 A1(0 x1) ¬A1(x1) ` ≈ 2|x1| 6=
Make sure D has bounded error
32 / 35 Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized Algorithms?
2 n n A1 time D time
x1 A1(x1) A1(0x1) === 2 0x1 A1(0x1) A1(0 x1) === 3 00x1 A1(00x1) A1(0 x1)
...
...... ===
inputs
`−2 `−2 === `−1 0 x1 A1(0 x1) A1(0 x1) `−1 `−1 === ` 0 x1 A1(0 x1) A1(0 x1) ` ` === 0 x1 A1(0 x1) ¬A1(x1) ` ≈ 2|x1| 6=
Make sure D has bounded error – 1 bit of advice
32 / 35 Yes, for algorithms with 1 bit of advice!
My work [ Kinne, Van Melkebeek ] Memory Space hierarchies: randomized, quantum, ...
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized Algorithms?
33 / 35 My work [ Kinne, Van Melkebeek ] Memory Space hierarchies: randomized, quantum, ...
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized Algorithms?
Yes, for algorithms with 1 bit of advice!
33 / 35 My work [ Kinne, Van Melkebeek ] Memory Space hierarchies: randomized, quantum, ...
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized Algorithms?
Yes, for algorithms with 1 bit of advice!
Poly Time n3 time n2 time
n time
33 / 35 My work [ Kinne, Van Melkebeek ] Memory Space hierarchies: randomized, quantum, ...
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized Algorithms?
Yes, for algorithms with 1 bit of advice!
log3 n space log2 n space
log n space
33 / 35 My work [ Kinne, Van Melkebeek ] Memory Space hierarchies: randomized, quantum, ...
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized Algorithms?
Yes, for algorithms with 1 bit of advice!
log3 n space log2 n space
log n space
33 / 35 Memory Space hierarchies: randomized, quantum, ...
Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized Algorithms?
Yes, for algorithms with 1 bit of advice!
log3 n space log2 n space
log n space
My work [ Kinne, Van Melkebeek ]
33 / 35 Two Big Questions Derandomization Hierarchy Theorems
Hierarchy Theorems for Randomized Algorithms?
Yes, for algorithms with 1 bit of advice!
log3 n space log2 n space
log 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 solve problems?
Is nondeterminism powerful? P vs. NP Conjecture: P6= 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 Slides available at: http://www.kinnejeff.com/GoSycamores/ (or E-mail me)
More on my research (slides, papers, etc.) at: http://www.kinnejeff.com/
Two Big Questions Derandomization Hierarchy Theorems
The End, Thank You!
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