Ultra-efficient algorithms for testing well-parenthesised expressions by Tatiana Starikovskaya
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The document discusses the development of ultra-efficient algorithms for testing well-parenthesised expressions, focusing on property testing and streaming algorithms for string processing. It outlines the challenges of processing input streams while ensuring correctness with limited reading capacity, and introduces a new property tester for well-balanced strings of parentheses. The take-home message emphasizes the potential of these algorithms in streaming and property testing settings, requiring minimal space and read capacity for effective solutions.
![Well-parenthesised expressions
Dm = well-balanced strings on parentheses of m types
Task: develop a property tester that decides whether
the input is in Dm
()([]())[]([]) ()(([][)()((([]
1. It accepts all inputs that are in Dm with
probability at least 2/3
2. It rejects all inputs that are ε-far from Dm
with probability at least 2/3
Time = number of read characters!](https://image.slidesharecdn.com/tatiana-wimlds-171127140851/85/Ultra-efficient-algorithms-for-testing-well-parenthesised-expressions-by-Tatiana-Starikovskaya-7-320.jpg)
![What do we know
()(()())()
()(()()(((
()([]())([])
()(([)()(([]
const.m =1 Alon et al.’01
m ≥ 2
Parnas et al.’03c n1/11 < T < C n2/3
c n1/5 < T < C n2/5+δ
NEW!
Dm = well-balanced strings on parentheses of m types](https://image.slidesharecdn.com/tatiana-wimlds-171127140851/85/Ultra-efficient-algorithms-for-testing-well-parenthesised-expressions-by-Tatiana-Starikovskaya-9-320.jpg)
![New tester for Dm-membership
Dm = well-balanced strings on parentheses of m types
Hmmm… does not look like a simple property to test!
Let’s start with a property tester for strawberries
()({()})([]){((([]())([])([{}]())}([])))
red
sweet
yellow seeds
simple
properties,
easy to test!
?](https://image.slidesharecdn.com/tatiana-wimlds-171127140851/85/Ultra-efficient-algorithms-for-testing-well-parenthesised-expressions-by-Tatiana-Starikovskaya-10-320.jpg)
![New tester for Dm-membership
Dm = well-balanced strings on parentheses of m types
If we replace all opening parentheses with (, and all closing
parentheses with ), the resulting string must be in D1
And we know how to test in O(1) time [Alon et al.’01]!
Not sufficient: becomes
()({()})([]){((([]())([])([{}]()))([]))}
()((()))(())((((()())(())((())()))(())))
()({{)}) ()((()))](https://image.slidesharecdn.com/tatiana-wimlds-171127140851/85/Ultra-efficient-algorithms-for-testing-well-parenthesised-expressions-by-Tatiana-Starikovskaya-11-320.jpg)
![New tester for Dm-membership
Dm = well-balanced strings on parentheses of m types
Each block is Dm-consistent = is a substring of a string in Dm
We test that the blocks are Dm-consistent by running our
Dm-test in a recursive fashion
()({()})([ ]){((([]() )([])([{}] ()))([]))}
()({()})([ ]){((([]() )([])([{}] ()))([]))}
b = n4/5 b = n4/5 b = n4/5 b = n4/5](https://image.slidesharecdn.com/tatiana-wimlds-171127140851/85/Ultra-efficient-algorithms-for-testing-well-parenthesised-expressions-by-Tatiana-Starikovskaya-12-320.jpg)
![New tester for Dm-membership
Dm = well-balanced strings on parentheses of m types
We have checked that the string is good locally, but can we
guarantee that it is good globally?
()({()})([ ]){((([]() )([])([{}] ()))([]))}
()({()})([ ]){((([]() )([])([{}] ()))([]))}
b = n4/5 b = n4/5 b = n4/5 b = n4/5](https://image.slidesharecdn.com/tatiana-wimlds-171127140851/85/Ultra-efficient-algorithms-for-testing-well-parenthesised-expressions-by-Tatiana-Starikovskaya-13-320.jpg)
![New tester for Dm-membership
Dm = well-balanced strings on parentheses of m types
Approximate matching graph: nodes = blocks, edge (B1,B2) =
many excess parentheses in block B1 must be matched with excess
parentheses in block B2
()({()})([ ]){((([]() )([])([{}] ()))([]))}
()({()})([ ]){((([]() )([])([{}] ()))([]))}
b = n4/5 b = n4/5 b = n4/5 b = n4/5](https://image.slidesharecdn.com/tatiana-wimlds-171127140851/85/Ultra-efficient-algorithms-for-testing-well-parenthesised-expressions-by-Tatiana-Starikovskaya-14-320.jpg)
![New tester for Dm-membership
Dm = well-balanced strings on parentheses of m types
()({()})([ ]){((([]() )([])([{}] ()))([]))}
()({()})([ ]){((([]() )([])([{}] ()))([]))}
1. Build an approximate matching graph
2. Run a recursive inter-block matching procedure
b = n4/5 b = n4/5 b = n4/5 b = n4/5](https://image.slidesharecdn.com/tatiana-wimlds-171127140851/85/Ultra-efficient-algorithms-for-testing-well-parenthesised-expressions-by-Tatiana-Starikovskaya-15-320.jpg)
![()({()})([ ]){((([]() )([])([{}] ())}([])))
1. Build an approximate matching graph
2. Run a recursive inter-block matching procedure
]){((([]() ))((((()() (())((((()()))))
S S w/o types D1
{e1(S) = 2
e0(S) = 4
e1(S) - excess closing parentheses
e0(S) - excess opening parentheses
T1, T2, …, Tn/b - blocks of the input
Parentheses in Ti that must be matched with parentheses in Tj
min(e0(Ti), e1(Ti+1Ti+2…Tj)) - e1(Ti+1Ti+2…Tj-1)](https://image.slidesharecdn.com/tatiana-wimlds-171127140851/85/Ultra-efficient-algorithms-for-testing-well-parenthesised-expressions-by-Tatiana-Starikovskaya-16-320.jpg)
![()({()})([ ]){((([]() )([])([{}] ())}([])))
1. Build an approximate matching graph
2. Run a recursive inter-block matching procedure
]){((([]() ))((((()() (())((((()()))))
S S w/o types D1
{e1(S) = 2
e0(S) = 4
Observation e1(S) = max{S’ - prefix of S} (n1(S’) - n0(S’))
n1(S’) = |closing parentheses in S’|
n0(S’) = |opening parentheses in S’|
Lemma By querying x2/Δ2 positions of a string S of length x,
we can compute a Δ-additive approximation of n1(S’) for any
substring S’ of S correctly w.h.p.](https://image.slidesharecdn.com/tatiana-wimlds-171127140851/85/Ultra-efficient-algorithms-for-testing-well-parenthesised-expressions-by-Tatiana-Starikovskaya-17-320.jpg)
![()({()})([ ]){((([]() )([])([{}] ())}([])))
1. Build an approximate matching graph
2. Run a recursive inter-block matching procedure
Lemma By querying x2/Δ2 positions of a string S of length x,
we can compute a Δ-additive approximation of n1(S’) for any
substring S’ of S correctly w.h.p.
Proof
Query x2/Δ2 positions of S uniformly at random
If |S’| ≤ Δ, output Δ
Otherwise, |S’| = yΔ, where y > 1
S’ contains ~yx/Δ of the queried positions](https://image.slidesharecdn.com/tatiana-wimlds-171127140851/85/Ultra-efficient-algorithms-for-testing-well-parenthesised-expressions-by-Tatiana-Starikovskaya-18-320.jpg)
![()({()})([ ]){((([]() )([])([{}] ())}([])))
1. Build an approximate matching graph
2. Run a recursive inter-block matching procedure
Lemma By querying x2/Δ2 positions of a string S of length x,
we can compute a Δ-additive approximation of n1(S’) for any
substring S’ of S correctly w.h.p.
Proof (cont.)
Xi = 1 iff the i-th queried position is a closing parenthesis
E[(Δ2/x) ⋅ Σ Xi] = (Δ2/x)⋅ n1(S’) (yx/Δ) / yΔ = n1(S’)
By additive Chernoff bound,
P[|(Δ2/x) ⋅ Σ Xi - n1(S’)| > Δ] < 2e-2](https://image.slidesharecdn.com/tatiana-wimlds-171127140851/85/Ultra-efficient-algorithms-for-testing-well-parenthesised-expressions-by-Tatiana-Starikovskaya-19-320.jpg)
![New tester for Dm-membership
1. Build an approximate matching graph
2. Run a recursive inter-block matching procedure
If we replace all opening parentheses with (, and all
closing parentheses with ), the resulting string ∈ D1
Test that the blocks are Dm-consistent by running
the test in a recursive fashion
Complexity: O(n2/5)
()({()})([ ]){((([]() )([])([{}] ())}([])))
()({()})([ ]){((([]() )([])([{}] ())}([])))
O(1)
O(√b)
b = n4/5
b = n4/5 b = n4/5 b = n4/5
O(n2/b2)](https://image.slidesharecdn.com/tatiana-wimlds-171127140851/85/Ultra-efficient-algorithms-for-testing-well-parenthesised-expressions-by-Tatiana-Starikovskaya-20-320.jpg)
Ultra-efficient algorithms for testing well-parenthesised expressions by Tatiana Starikovskaya
- 1. Ultra-efficient algorithms for testing well-parenthesised expressions Tatiana Starikovskaya (ENS Paris) Joint work with Eldar Fisher (Technion) and Frédéric Magniez (Paris-Diderot) WiMLDS Paris, November 24, 2017
- 2. Pattern matching: you use it every time you search for something More general: algorithms on strings (= sequences of characters) My research area
- 3. My research area Applications • Bioinformatics • Information Retrieval • … Classical approaches • We can read the whole input • We can afford to store linear-space data structures In the Big Data world, we must do better!
- 4. My research area Streaming algorithms We receive the input as a stream, and must process it on-the-fly, without storing it Property testing algorithms We must decide if the input has a property P, but we can read only a small part of the input ? ? ? We need efficient algorithms for string processing!
- 5. Property testers Wait a second! How can we make the decision not reading the whole input? Well, in general, we cannot… For example, we cannot say if the input is well-parenthesised by reading just a small fraction of it ? ? ? Task: We must decide if the input has a property P, but we can read only a small part of the input Objective: Save time ()(()())() ()(()()(() queried parentheses are identical ? ? ? ? ? ?
- 6. Property testers We must 1. accept, if the input has the property P 2. reject, if the input is far from having the property 3. accept or reject otherwise Far = we must fix at least εn characters of the input so that the property is satisfied The output must be correct probability at least 2/3 ? ? ? Task: We must decide if the input has a property P, but we can read only a small part of the input Objective: Save time ()(()())() ()(()()((( ()(()()(() ε = 0.2, n = 10, εn = 2 ?
- 7. Well-parenthesised expressions Dm = well-balanced strings on parentheses of m types Task: develop a property tester that decides whether the input is in Dm ()([]())[]([]) ()(([][)()((([] 1. It accepts all inputs that are in Dm with probability at least 2/3 2. It rejects all inputs that are ε-far from Dm with probability at least 2/3 Time = number of read characters!
- 8. Simplicity: simplest context-free language Universality: any context-free language can be expressed through it (Chomsky-Schützenberger theorem) Practicality: processing of semi-structured documents • Visibly pushdown languages • Nested strings Why is it interesting? Dm = well-balanced strings on parentheses of m types
- 9. What do we know ()(()())() ()(()()((( ()([]())([]) ()(([)()(([] const.m =1 Alon et al.’01 m ≥ 2 Parnas et al.’03c n1/11 < T < C n2/3 c n1/5 < T < C n2/5+δ NEW! Dm = well-balanced strings on parentheses of m types
- 10. New tester for Dm-membership Dm = well-balanced strings on parentheses of m types Hmmm… does not look like a simple property to test! Let’s start with a property tester for strawberries ()({()})([]){((([]())([])([{}]())}([]))) red sweet yellow seeds simple properties, easy to test! ?
- 11. New tester for Dm-membership Dm = well-balanced strings on parentheses of m types If we replace all opening parentheses with (, and all closing parentheses with ), the resulting string must be in D1 And we know how to test in O(1) time [Alon et al.’01]! Not sufficient: becomes ()({()})([]){((([]())([])([{}]()))([]))} ()((()))(())((((()())(())((())()))(()))) ()({{)}) ()((()))
- 12. New tester for Dm-membership Dm = well-balanced strings on parentheses of m types Each block is Dm-consistent = is a substring of a string in Dm We test that the blocks are Dm-consistent by running our Dm-test in a recursive fashion ()({()})([ ]){((([]() )([])([{}] ()))([]))} ()({()})([ ]){((([]() )([])([{}] ()))([]))} b = n4/5 b = n4/5 b = n4/5 b = n4/5
- 13. New tester for Dm-membership Dm = well-balanced strings on parentheses of m types We have checked that the string is good locally, but can we guarantee that it is good globally? ()({()})([ ]){((([]() )([])([{}] ()))([]))} ()({()})([ ]){((([]() )([])([{}] ()))([]))} b = n4/5 b = n4/5 b = n4/5 b = n4/5
- 14. New tester for Dm-membership Dm = well-balanced strings on parentheses of m types Approximate matching graph: nodes = blocks, edge (B1,B2) = many excess parentheses in block B1 must be matched with excess parentheses in block B2 ()({()})([ ]){((([]() )([])([{}] ()))([]))} ()({()})([ ]){((([]() )([])([{}] ()))([]))} b = n4/5 b = n4/5 b = n4/5 b = n4/5
- 15. New tester for Dm-membership Dm = well-balanced strings on parentheses of m types ()({()})([ ]){((([]() )([])([{}] ()))([]))} ()({()})([ ]){((([]() )([])([{}] ()))([]))} 1. Build an approximate matching graph 2. Run a recursive inter-block matching procedure b = n4/5 b = n4/5 b = n4/5 b = n4/5
- 16. ()({()})([ ]){((([]() )([])([{}] ())}([]))) 1. Build an approximate matching graph 2. Run a recursive inter-block matching procedure ]){((([]() ))((((()() (())((((()())))) S S w/o types D1 {e1(S) = 2 e0(S) = 4 e1(S) - excess closing parentheses e0(S) - excess opening parentheses T1, T2, …, Tn/b - blocks of the input Parentheses in Ti that must be matched with parentheses in Tj min(e0(Ti), e1(Ti+1Ti+2…Tj)) - e1(Ti+1Ti+2…Tj-1)
- 17. ()({()})([ ]){((([]() )([])([{}] ())}([]))) 1. Build an approximate matching graph 2. Run a recursive inter-block matching procedure ]){((([]() ))((((()() (())((((()())))) S S w/o types D1 {e1(S) = 2 e0(S) = 4 Observation e1(S) = max{S’ - prefix of S} (n1(S’) - n0(S’)) n1(S’) = |closing parentheses in S’| n0(S’) = |opening parentheses in S’| Lemma By querying x2/Δ2 positions of a string S of length x, we can compute a Δ-additive approximation of n1(S’) for any substring S’ of S correctly w.h.p.
- 18. ()({()})([ ]){((([]() )([])([{}] ())}([]))) 1. Build an approximate matching graph 2. Run a recursive inter-block matching procedure Lemma By querying x2/Δ2 positions of a string S of length x, we can compute a Δ-additive approximation of n1(S’) for any substring S’ of S correctly w.h.p. Proof Query x2/Δ2 positions of S uniformly at random If |S’| ≤ Δ, output Δ Otherwise, |S’| = yΔ, where y > 1 S’ contains ~yx/Δ of the queried positions
- 19. ()({()})([ ]){((([]() )([])([{}] ())}([]))) 1. Build an approximate matching graph 2. Run a recursive inter-block matching procedure Lemma By querying x2/Δ2 positions of a string S of length x, we can compute a Δ-additive approximation of n1(S’) for any substring S’ of S correctly w.h.p. Proof (cont.) Xi = 1 iff the i-th queried position is a closing parenthesis E[(Δ2/x) ⋅ Σ Xi] = (Δ2/x)⋅ n1(S’) (yx/Δ) / yΔ = n1(S’) By additive Chernoff bound, P[|(Δ2/x) ⋅ Σ Xi - n1(S’)| > Δ] < 2e-2
- 20. New tester for Dm-membership 1. Build an approximate matching graph 2. Run a recursive inter-block matching procedure If we replace all opening parentheses with (, and all closing parentheses with ), the resulting string ∈ D1 Test that the blocks are Dm-consistent by running the test in a recursive fashion Complexity: O(n2/5) ()({()})([ ]){((([]() )([])([{}] ())}([]))) ()({()})([ ]){((([]() )([])([{}] ())}([]))) O(1) O(√b) b = n4/5 b = n4/5 b = n4/5 b = n4/5 O(n2/b2)
- 21. Take-home message • Streaming or property testing settings • We have new, ultra-efficient algorithms for string processing • It is enough to use a polylog space or to read a constant number of data items in the input to solve a problem with good guarantees Questions? Comments?