H. Buhrman and R. de Wolf. Complexity measures and decision tree complexity: a survey, Theoretical Comput. Sci. 288:21--43, 2002.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....used as the rst stage of various data mining algorithms. We discuss (subsection 2.3) the problem of selecting a random sample out of a dataset. We present a computational model for probabilistic sampling algorithms that compute approximate solutions. This model is based on the decision tree model [25] and relates the query complexity to the size of the sample. We start by providing some (mostly) textbook de nitions for self containment purposes. In data mining we are interested in vectors and their relationships under several distance measures. For two vectors, v = v 1 ; v n ) u ....

H. Buhrman and R. de Wolf. Complexity measures and decision tree complexity: A survey. available at: http://www.cwi.nl/ rdewolf, 1999.

....algorithm in order to compute f on a worstcase input. The terms D (f ) N (f ) R 0 (f ) and R can be defined analogous to communication complexity. For formal definitions of these quantities, the readers are referred to Saks and Wigderson [23] and the survey by Buhrman and de Wolf [8]. Clearly R (f ) A consequence of a result discovered independently by several authors [7, 12, 26] is that D . For two sided error #, Nisan [20] showed that D # (f) thus, in contrast to communication complexity, there is only a polynomial gap between these two ....

H. Buhrman and R. de Wolf. Complexity measures and decision tree complexity: A survey. Theoretical Computer Science, 288(1):21--43, 2002.

....which we de ne as L(f) minimal size of a tree computing f: In this paper, we de ne the size of a binary tree as the number of its internal nodes . We should also mention that several di erent complexity measures for boolean functions have been proposed in the literature; see e.g. [2] for a recent survey. The aim of this paper is dual: a better understanding of the limiting probability distribution on the space of boolean functions, and a study of the relationship between the probability of a given boolean function and its complexity. For the rst topic, we need to make ....

H. Buhrman and R. de Wolf. Complexity measures and decision tree complexity : a survey. Theoretical Computer Science., 288:21-43, 2002.

....gaps be tween the measures, including a total f for whi C(f) is superquadratic in QC (f) and a symmetric partial f for which QC (f) O (1) yet Q2 (f) f2 (n logn) 1. Background Most of what is known about the power of quantum computing can be cast in the query or decision tree model [1, 2, 3, 5, 6, 9, 10, 11, 18, 22, 23]. Here one counts only the number of queries to the input, not the number of computational steps. The appeal of this model lies in its extreme simplicityin contrast to (say) the Turing machine model, one feels the query model ought to be completely understandable. In spite of this, open ....

....by a quantum algorithm that outputs f (Y) with probability 1 if f (Y) k, and with probability at least 1 2 if f (Y) k. Then let Q0 (f) max Q0 (f) Q (f) If we require a single algorithm that succeeds with probability 1 for all Y, we obtain QE (f) or exact quantum query complexity. See [10] for detailed definitions and a survey of these measures. Q2 (f) R2 (f) Ro (f) D (f) n, that Q0 (f) R0 (f) and that QE (f) D (f) If f is partial (i.e. Dom (f) 0, 1 ) then Q2 (f) can be superpolynomially smaller than R2 (f) this is what makes Shor s period finding ....

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H. Buhrman and R. de Wolf. Complexity measures and decision tree complexity: a survey, Theoretical Cornput. Sci. 288:21-43, 2002.

....In [4] an n) lower bound for the problem of nding a marked element in an unordered database of size n has been given, matching the upper bound of Grover s algorithm [12] This lower bound relies on the notion of query magnitude. For other lower bound techniques for query complexity see e.g. [9]. The query magnitude technique is basically an adversary argument. An adversary is able to change the black box input without the query algorithm noticing that. We use only the following statement derived as Corollary 3.4 in [4] Fact 4 Let x = x 1 ; x n be an input given as an access ....

H. Buhrman, R. de Wolf. Complexity Measures and Decision Tree Complexity: 12 A Survey. To appear in Theoretical Computer Science, 2002.

.... gaps between the mea sures, including a total f for which (f) is superquadratic in Q (f) and a symmetric partial f for which QC (f) O (1) yet Q2 (f) f (n logn) 1 Background Most of what is known about the power of quantum computing can be cast in the query or decision tree model [1, 2, 3, 5, 6, 9, 10, 11, 17, 21, 22]. Here one counts only the number of queries to the input, not the number of computational steps. The appeal of this model lies in its extreme simplicity in contrast to (say) the Turing machine model, one feels the query model ought to be completely understandable. In spite of this, open ....

....by a quantum algorithm that outputs f (Y) with probability 1 if f (Y) k, and with probability at least 1 2 if f (Y) k k. Then let Q0 (f) max(Q0 (f) Q (f) If we require a sin gle algorithm that succeeds with probability 1 for all Y, we obtain QE (f) or exact quantum query complexity. See [10] for detailed definitions and a survey of these measures. It is immediate that Q2 (f) R2 (f) R0 (f) D (f) n, that (0 (f) R0 (f) and that (E (f) D (f) If f is partial (i.e. Dom (f) 0, 1 ) then Q2 (f) can be superpolynomially smaller than R2 (f) this is what makes Shor s ....

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H. Buhrman and R. de Wolf. Complexity mea- sures and decision tree complexity: a survey, to appear in Theoretical Comput. Sci.

....for instance# ( # n) if the number of bits is restricted to (say) O (log n) To our knowledge, this would be the first example in quantum computing of a provable tradeo# between query complexity and space complexity. The general power of the polynomial method remains open. Buhrman and de Wolf [8] raised the question of whether approximate degree as a polynomial yields an optimal lower bound on quantum query complexity for any function. They were discussing total Boolean functions, but the question is open for non total functions also, if the polynomial p must satisfy p (X) 0, 1] ....

H. Buhrman and R. de Wolf. Complexity measures and decision tree complexity: a survey. To appear in Theoretical Computer Science, 2002.

....Key words. algorithm, Boolean function, truth table, query complexity, quantum computation. AMS subject classifications. 68Q10, 68Q17, 68Q25, 68W01, 81P68. 1. Introduction. The query complexity of Boolean functions, also called blackbox or decision tree complexity, has been well studied for years [5, 7, 16]. Counting how many queries are needed to evaluate a function is easier than counting how many computational steps are needed; thus, nontrivial lower bounds are more readily shown for the former measure than for the latter. Also, query complexity has proved to be a powerful tool for studying the ....

.... for all X , p(X) f(X) Degree was introduced to query complexity by Nisan and Szegedy [17] who observed the relationship deg(f) D(f ) Later Beals et al. 5] related degree to quantum query complexity by showing that deg(f) 2QE (f ) The following lemma, adapted from Lemma 4 of [7], is easily seen to yield an O(n3 ) O(N log N) dynamic programming algorithm for deg(f ) Say that a function obeys the parity property if the number of inputs X with odd parity for which f(X) 1 equals the number of inputs X with even parity for which f(X) 1. Lemma 3.1 (Shi and Yao) ....

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H. Buhrman and R. de Wolf, Complexity measures and decision tree complexity: a survey, to appear in Theoretical Comp. Sci.

....to our basic proof needed for this generalization (Remark 4) 3. Preliminaries In this section we compile together some de nitions and previously known results needed for our proof. 3.1. Quantum search vs. quantum communication For a precise de nition of a quantum decision tree see e.g. BW00] Given a Boolean function g(x 1 ; x n ) we will denote by QDT (g) the minimal number of queries needed to compute g by a quantum decision tree with error at most 1 3 at any input x 2 f0; 1g . Let us denote by f g : P( n] P( n] f0; 1g the predicate f g (x; y) g(x y) where ....

H. Buhrman and R. de Wolf. Complexity measures and decision tree complexity: a survey. Manuscript to appear in Theoretical Computer Science, 2000.

....algorithm or a streaming algorithm [17, 1, 13] then it admits lossy compression (in a technically precise sense) These results are described in Section 7. In Section 8 we discuss related prior work and in particular the relationship of our model to the areas of Boolean decision tree complexity [6], PAC and statistical learning Theory [27, 20, 29] statistical decision theory [4] statistical estimation theory [28] and statistical sequential analysis [26] 2. PRELIMINARIES In this section we introduce a notion of approximation for functions f : A n B, where A and B are arbitrary ....

....trees, which are suitable for exact computation of Boolean functions. Boolean query complexity was extensively studied, and is known to be related to various Boolean function properties, such as sensitivity, certificate complexity, and degree of representing and approximating polynomials (see [6] for a survey) In this paper we explored only the generalization of block sensitivity and its connection to query complexity of general functions. Similar generalizations are possible also for the other properties; however, as mentioned in Section 3, not all the connections between these ....

H. Buhrman and R. de Wolf. Complexity measures and decision tree complexity: A survey, 1999. Available at http://www.cwi.nl/rdewolf.

....or a streaming algorithm [HRR99, AMS99, FKSV99] then it admits lossy compression (in a technically precise sense) These results are described in Section 7. In Section 8 we discuss related prior work and in particular the relationship of our model to the areas of Boolean decision tree complexity [Bd99] PAC and statistical learning Theory [Val84, KV94, Vap98] statistical decision theory [Ber85] statistical estimation theory [Van68] and statistical sequential analysis [Sie85] Section 9 concludes with some open problems. 2 Preliminaries In this section we introduce a notion of approximation ....

....of the computer science literature on the query complexity of sampling algorithms concentrated on the variant of the model in which the functions are Boolean (i.e. A = B = f0; 1g) and the computation is required to be exact. This variant is captured by the famous Boolean decision tree model (see [Bd99] for a recent survey about this area) The query complexity of deterministic, randomized, and non deterministic computations of Boolean functions is well understood, and is known to be related to various function properties, like sensitivity and block sensitivity, certi cate complexity, and ....

H. Buhrman and R. de Wolf. Complexity measures and decision tree complexity: A survey, 1999. Available at http://www.cwi.nl/rdewolf.

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H. Buhrman and R. de Wolf. Complexity measures and decision tree complexity: A survey. Theoretical Computer Science, 288(1):21-43, 2002.

....worst case or average case input. We assume familiarity with classical computation and briefly sketch the definition of quantum query algorithms. For a general introduction to quantum computing, see the book of Nielsen and Chuang [20] For more details about (quantum) query complexity we refer to [10]. 2 An m qubit state is a 2 m dimensional unit vector of complex numbers, written P x2f0;1g m ff x jxi. A T query quantum algorithm corresponds to a unitary transformation A = U T OU T Gamma1 O : U 1 OU 0 : Here the U j are unitary transformations on some m qubits. These U j are ....

H. Buhrman and R. de Wolf. Complexity measures and decision tree complexity: A survey. To appear in Theoretical Computer Science. Available at http://www.cwi.nl/~rdewolf, 1999.

....deterministic and quantum algorithms that compute some f : f0; 1g n f0; 1g exactly. 1 Let deg(f) denote the degree of the multilinear polynomial that represents f . The following relations are known (see [BBC 98] the last inequality is due to Nisan and Smolensky unpublished, but see [BW99b] deg(f) 2 Q q (f) D q (f) O(deg(f) 4 ) Thus deg(f ) Q q (f) and D q (f) are all polynomially related for all total f (the situation is very different for partial f [DJ92, Sim97] A function is known with a near quadratic gap between D q (f) and deg(f) NS94] but no function is known ....

....and the monomial X S is the product Pi i2S x i of all variables in S. The degree of p is the degree of a largest monomial with non zero coefficient. It is well known that every total Boolean function f has a unique polynomial p such that p(x) f(x) for all x 2 f0; 1g n (see for instance [NS94, BW99b] Let deg(f) be the degree of this polynomial. For example, OR(x 1 ; x 2 ) x 1 x 2 Gamma x 1 x 2 , which has degree 2. We introduce the notion of a non deterministic polynomial for f . This is a polynomial p such that p(x) 6= 0 iff f(x) 1. Let the non deterministic degree of f , denoted ....

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H. Buhrman and R. de Wolf. Complexity measures and decision tree complexity: A survey. Submitted. Available at http://www.cwi.nl/~rdewolf, 1999.

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H. Buhrman and R. de Wolf. Complexity measures and decision tree complexity: a survey, Theoretical Comput. Sci. 288:21--43, 2002.

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H. Buhrman and R. de Wolf. Complexity measures and decision tree complexity: A survey. Theoretical Computer Science, 288(1):21-43, 2002.

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Buhrman, H. and de Wolf, R. (1999). Complexity measures and decision tree complexity: a survey. To appear in Theoretical Computer Science.

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