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*Let <math>F_C = F_A \and F_B</math> be the Bloom filter formed by computing the bitwise AND of <math>F_A</math> and <math>F_B</math>. Argue that <math> ...

= Computing Graph Expansion = ...can be verified in poly-time. However, there is no efficient algorithm for computing the <math>\phi(G)</math> unless '''NP'''='''P'''. ...

...on algorithms for counting and sampling problems. Typical problems include computing marginal probabilities and expectations of random variables, the evaluation :何昆：Kun HE received his Ph.D. degree from the Institute of Computing Technology, Chinese Academy of Sciences in 2019. Currently he is a postdoct ...

* [https://arxiv.org/abs/1907.09415 Ronald de Wolf. Quantum Computing: Lecture Notes.] ...

The proposition can be formally verified by computing these expectations. Although these equations look formal, the intuitive int ...

Besides directly computing the probability <math>\Pr[X\ge t]</math>, we want to have some general way ...

...r <math>v</math> is adjacent to a vertex in <math>D</math>. The problem of computing minimum dominating set is NP-hard. ...

...d store the results in memory. Afterwards he can decrypt the ciphertext by computing ''D_K''(''C'') for each ''K''. Any matches between these two resu ...

...for deterministic communication protocols, this is the best we can get for computing EQ. :Any deterministic communication protocol computing EQ on two <math>n</math>-bit strings costs <math>n</math> bits of communica ...

|data17 = '''Probability and Computing''' (2E) <br> Michael Mitzenmacher and Eli Upfal <br> Cambridge University * '''[MU]''' ''Probability and Computing: Randomization and Probabilistic Techniques in Algorithms and Data Analysis ...

...r <math>v</math> is adjacent to a vertex in <math>D</math>. The problem of computing minimum dominating set is NP-hard. ...

...r <math>v</math> is adjacent to a vertex in <math>D</math>. The problem of computing minimum dominating set is NP-hard. ...

...r <math>v</math> is adjacent to a vertex in <math>D</math>. The problem of computing minimum dominating set is NP-hard. ...

...mal language] (not entirely the same thing as the c language). The task of computing <math>f(x)</math> is equivalent to that of determining whether <math>x\in L ...nomial size certificate, the algorithm <math>A</math> verifies (instead of computing) the <math>f(x)</math> for any positive instance <math>x</math> in polynomi ...

A naive way of checking the equality is first computing <math>AB</math> and then comparing the result with <math>C</math>. The (asy ...

;The Monte Carlo method for computing <math>\mathrm{e}</math> ...

...r <math>v</math> is adjacent to a vertex in <math>D</math>. The problem of computing minimum dominating set is NP-hard. ...

* [[Computer language]] for application of formal languages in computing ...

...e need the following theorem, which holds generally and is very useful for computing the AND of many events. ...

...r <math>v</math> is adjacent to a vertex in <math>D</math>. The problem of computing minimum dominating set is NP-hard. ...

...r <math>v</math> is adjacent to a vertex in <math>D</math>. The problem of computing minimum dominating set is NP-hard. ...

...ok.html|title = Proceedings of the Third Annual ACM Symposium on Theory of Computing|pages = 151–158}}</ref> ...ust 1% of the worst-case number of computation and that saves a ''lot'' of computing, but that is still <math>0.01 \times (2^n)</math> computations. And every e ...

|data17 = '''Probability and Computing''' (2E) <br> Michael Mitzenmacher and Eli Upfal <br> Cambridge University * '''[MU]''' ''Probability and Computing: Randomization and Probabilistic Techniques in Algorithms and Data Analysis ...

Computing the moment generating function <math>\mathbf{E}[e^{\lambda X}]</math>: We propose an extremely simple "randomized algorithm" for computing a <math>b\in\{-1,+1\}^m</math>: for each <math>i=1,2,\ldots, m</math>, let ...

...tudies of algorithms for elimination orderings in combinatorial scientific computing, our result is the first rigorous incorporation of randomized tools in this :Huacheng Yu (俞华程) is a postdoctoral researcher in the Theory of Computing group at Harvard University. He obtained his PhD from Stanford University i ...

We propose an extremely simple "randomized algorithm" for computing a <math>b\in\{-1,+1\}^m</math>: for each <math>i=1,2,\ldots, m</math>, let ...

...> multiplications, which is too expensive. We describe an efficient way of computing the [http://en.wikipedia.org/wiki/Modular_exponentiation modular exponent] ...ed a number instead of a list of all colorings, we expect cleverer ways of computing this number than the brute force enumeration. ...

...he search for new Mersenne primes is the goal of the [[GIMPS]] distributed computing project. *[http://www.oddperfect.org OddPerfect.org] A projected distributed computing project to search for odd perfect numbers. ...

|data13 = Mitzenmacher and Upfal. <br>''Probability and Computing: Randomized Algorithms and Probabilistic Analysis''. <br> Cambridge Univ Pr ...

[[Decryption]] of a ciphertext <math>(R_n, L_n)</math> is accomplished by computing for <math>i=n,n-1,\ldots,1</math> ...

...]</math> by the definition of expectations. There is also a smarter way of computing <math>\mathbf{E}[X]</math>, by using indicators and the linearity of expec ...e need the following theorem, which holds generally and is very useful for computing the AND of many events. ...

|data13 = Mitzenmacher and Upfal. <br>''Probability and Computing: Randomized Algorithms and Probabilistic Analysis''. <br> Cambridge Univ Pr ...

Generally, the '''Principle of Inclusion-Exclusion''' states the rule for computing the union of <math>n</math> finite sets <math>A_1,A_2,\ldots,A_n</math>, su === Computing Möbius functions=== ...

Generally, the '''Principle of Inclusion-Exclusion''' states the rule for computing the union of <math>n</math> finite sets <math>A_1,A_2,\ldots,A_n</math>, su === Computing Möbius functions=== ...

Generally, the '''Principle of Inclusion-Exclusion''' states the rule for computing the union of <math>n</math> finite sets <math>A_1,A_2,\ldots,A_n</math>, su === Computing Möbius functions=== ...

Generally, the '''Principle of Inclusion-Exclusion''' states the rule for computing the union of <math>n</math> finite sets <math>A_1,A_2,\ldots,A_n</math>, su === Computing Möbius functions=== ...

Generally, the '''Principle of Inclusion-Exclusion''' states the rule for computing the union of <math>n</math> finite sets <math>A_1,A_2,\ldots,A_n</math>, su === Computing Möbius functions=== ...

Generally, the '''Principle of Inclusion-Exclusion''' states the rule for computing the union of <math>n</math> finite sets <math>A_1,A_2,\ldots,A_n</math>, su === Computing Möbius functions=== ...

...for deterministic communication protocols, this is the best we can get for computing EQ. :Any deterministic communication protocol computing EQ on two <math>n</math>-bit strings costs <math>n</math> bits of communica ...

...]</math> by the definition of expectations. There is also a smarter way of computing <math>\mathbf{E}[X]</math>, by using indicators and the linearity of expec ...e need the following theorem, which holds generally and is very useful for computing the AND of many events. ...

* '''Reading assignment''': Mitzenmacher and Upfal, ''Probability and Computing, second edition'', '''Chapter 17''' "Balanced Allocations and Cuckoo Hashin ...

|data13 = Mitzenmacher and Upfal. <br>''Probability and Computing: Randomized Algorithms and Probabilistic Analysis''. <br> Cambridge Univ Pr ...

...on algorithms for counting and sampling problems. Typical problems include computing marginal probabilities and expectations of random variables, the evaluation ...

...on algorithms for counting and sampling problems. Typical problems include computing marginal probabilities and expectations of random variables, the evaluation ...

== Computing graph expansion == ...can be verified in poly-time. However, there is no efficient algorithm for computing the <math>\phi(G)</math> unless '''NP'''='''P'''. ...

...ber of perfect matchings of <math>G</math>. This problem can be reduced to computing the '''permanent''' of a square matrix. ...his problem can be reduced to computing the permanent, thus the problem of computing the permanents is also '''#P-hard'''. ...

...for deterministic communication protocols, this is the best we can get for computing EQ. :Any deterministic communication protocol computing EQ on two <math>n</math>-bit strings costs <math>n</math> bits of communica ...

...for deterministic communication protocols, this is the best we can get for computing EQ. :Any deterministic communication protocol computing EQ on two <math>n</math>-bit strings costs <math>n</math> bits of communica ...

...for deterministic communication protocols, this is the best we can get for computing EQ. :Any deterministic communication protocol computing EQ on two <math>n</math>-bit strings costs <math>n</math> bits of communica ...

...for deterministic communication protocols, this is the best we can get for computing EQ. :Any deterministic communication protocol computing EQ on two <math>n</math>-bit strings costs <math>n</math> bits of communica ...

...for deterministic communication protocols, this is the best we can get for computing EQ. :Any deterministic communication protocol computing EQ on two <math>n</math>-bit strings costs <math>n</math> bits of communica ...

...for deterministic communication protocols, this is the best we can get for computing EQ. :Any deterministic communication protocol computing EQ on two <math>n</math>-bit strings costs <math>n</math> bits of communica ...

...for deterministic communication protocols, this is the best we can get for computing EQ. :Any deterministic communication protocol computing EQ on two <math>n</math>-bit strings costs <math>n</math> bits of communica ...

Computing the moment generating function <math>\mathbf{E}[e^{\lambda X}]</math>: We propose an extremely simple "randomized algorithm" for computing a <math>b\in\{-1,+1\}^m</math>: for each <math>i=1,2,\ldots, m</math>, let ...

Computing the moment generating function <math>\mathbf{E}[e^{\lambda X}]</math>: The problem raises from parallel computing. Consider that we have <math>N</math> processors, connected by a communicat ...

Computing the moment generating function <math>\mathbf{E}[e^{\lambda X}]</math>: ...

== Computing graph expansion == ...can be verified in poly-time. However, there is no efficient algorithm for computing the <math>\phi(G)</math> unless '''NP'''='''P'''. ...

We propose an extremely simple "randomized algorithm" for computing a <math>b\in\{-1,+1\}^m</math>: for each <math>i=1,2,\ldots, m</math>, let The problem raises from parallel computing. Consider that we have <math>N</math> processors, connected by a communicat ...

...> multiplications, which is too expensive. We describe an efficient way of computing the [http://en.wikipedia.org/wiki/Modular_exponentiation modular exponent] ...

...for deterministic communication protocols, this is the best we can get for computing EQ. :Any deterministic communication protocol computing EQ on two <math>n</math>-bit strings costs <math>n</math> bits of communica ...

|caption = ''Probability and Computing: Randomized Algorithms and Probabilistic Analysis'', Mitzenmacher and Upfal ...

...d the analysis of random graphs by [Janson, Combinatorics, Probability and Computing `94]. This leads to a routine that in quadratic time, sparsifies a graph do ...

The proposition can be formally verified by computing these expectations. Although these equations look formal, the intuitive int ...

== Computing graph expansion == ...can be verified in poly-time. However, there is no efficient algorithm for computing the <math>\phi(G)</math> unless '''NP'''='''P'''. ...

...]</math> by the definition of expectations. There is also a smarter way of computing <math>\mathbf{E}[X]</math>, by using indicators and the linearity of expec ...e need the following theorem, which holds generally and is very useful for computing the AND of many events. ...

The proposition can be formally verified by computing these expectations. Although these equations look formal, the intuitive int ...

The proposition can be formally verified by computing these expectations. Although these equations look formal, the intuitive int ...

The proposition can be formally verified by computing these expectations. Although these equations look formal, the intuitive int ...

The proposition can be formally verified by computing these expectations. Although these equations look formal, the intuitive int ...

The proposition can be formally verified by computing these expectations. Although these equations look formal, the intuitive int ...

*Let <math>F_C = F_A \land F_B</math> be the Bloom filter formed by computing the bitwise AND of <math>F_A</math> and <math>F_B</math>. Argue that <math> ...

...y | first4=BP | year=2007 | title=Numerical Recipes: The Art of Scientific Computing | edition=3rd | publisher=Cambridge University Press | publication-place=N ...

...mal language] (not entirely the same thing as the c language). The task of computing <math>f(x)</math> is equivalent to that of determining whether <math>x\in L ...nomial size certificate, the algorithm <math>A</math> verifies (instead of computing) the <math>f(x)</math> for any positive instance <math>x</math> in polynomi ...

...for deterministic communication protocols, this is the best we can get for computing EQ. :Any deterministic communication protocol computing EQ on two <math>n</math>-bit strings costs <math>n</math> bits of communica ...

...ath>G(n,p)</math> is connected, then we can compare the probabilities, but computing the probability that a random graph is connected is a very non-trivial task ...show that with coupling, we can compare two distributions without actually computing the probabilities. ...

...solve problems of other areas such as [[engineering]], [[physics]], and [[computing]]. ...

...for deterministic communication protocols, this is the best we can get for computing EQ. :Any deterministic communication protocol computing EQ on two <math>n</math>-bit strings costs <math>n</math> bits of communica ...

...for deterministic communication protocols, this is the best we can get for computing EQ. :Any deterministic communication protocol computing EQ on two <math>n</math>-bit strings costs <math>n</math> bits of communica ...

#* See Chapter 7.1.1 of Probability and Computing for an analysis of the random walk algorithm for 2SAT ...

...for deterministic communication protocols, this is the best we can get for computing EQ. :Any deterministic communication protocol computing EQ on two <math>n</math>-bit strings costs <math>n</math> bits of communica ...

...y when dividing two numbers in [[integer]] or fixed-point arithmetic; when computing mathematical functions such as [[square root]]s, [[logarithm]]s, and [[sine ...0.01 = 217.84, then rounding ''y'' to the integer ''q'' = 218, and finally computing ''z'' = ''q''×''m'' = 218×0.01 = 2.18. ...

...]</math> by the definition of expectations. There is also a smarter way of computing <math>\mathbf{E}[X]</math>, by using indicators and the linearity of expec ...

He also invented an original method for computing the probable size of a polymer in good solution. He invented the [[Flory-Hu ...

The problem raises from parallel computing. Consider that we have <math>N</math> processors, connected by a communicat ...

We propose an extremely simple "randomized algorithm" for computing a <math>b\in\{-1,+1\}^m</math>: for each <math>i=1,2,\ldots, m</math>, let The problem raises from parallel computing. Consider that we have <math>N</math> processors, connected by a communicat ...

...bit numbers is done <math>\bmod 2^u</math>, and have a one-line C-code for computing the hash function: ...

...bit numbers is done <math>\bmod 2^u</math>, and have a one-line C-code for computing the hash function: ...

...bit numbers is done <math>\bmod 2^u</math>, and have a one-line C-code for computing the hash function: ...

...ath>G(n,p)</math> is connected, then we can compare the probabilities, but computing the probability that a random graph is connected is a very non-trivial task ...show that with coupling, we can compare two distributions without actually computing the probabilities. ...

...can be verified in poly-time. However, there is no efficient algorithm for computing the <math>\phi(G)</math> unless '''NP'''='''P'''. ...

The [[Automated Readability Index]] was designed for [[real-time computing]] of readability for the electric [[typewriter]].<ref>{{cite journal ...

...bit numbers is done <math>\bmod 2^u</math>, and have a one-line C-code for computing the hash function: ...

Computing the moment generating function <math>\mathbf{E}[e^{\lambda X}]</math>: ...

Computing the moment generating function <math>\mathbf{E}[e^{\lambda X}]</math>: ...

Computing the moment generating function <math>\mathbf{E}[e^{\lambda X}]</math>: ...

Computing the moment generating function <math>\mathbf{E}[e^{\lambda X}]</math>: ...

Computing the moment generating function <math>\mathbf{E}[e^{\lambda X}]</math>: ...

Generally, the '''Principle of Inclusion-Exclusion''' states the rule for computing the union of <math>n</math> finite sets <math>A_1,A_2,\ldots,A_n</math>, su ...

Generally, the '''Principle of Inclusion-Exclusion''' states the rule for computing the union of <math>n</math> finite sets <math>A_1,A_2,\ldots,A_n</math>, su ...

Generally, the '''Principle of Inclusion-Exclusion''' states the rule for computing the union of <math>n</math> finite sets <math>A_1,A_2,\ldots,A_n</math>, su ...