## Complexity Measures and Decision Tree Complexity: A Survey (2000)

Venue: | Theoretical Computer Science |

Citations: | 122 - 14 self |

### BibTeX

@ARTICLE{Buhrman00complexitymeasures,

author = {Harry Buhrman and Ronald De Wolf},

title = {Complexity Measures and Decision Tree Complexity: A Survey},

journal = {Theoretical Computer Science},

year = {2000},

volume = {288},

pages = {2002}

}

### Years of Citing Articles

### OpenURL

### Abstract

We discuss several complexity measures for Boolean functions: certificate complexity, sensitivity, block sensitivity, and the degree of a representing or approximating polynomial. We survey the relations and biggest gaps known between these measures, and show how they give bounds for the decision tree complexity of Boolean functions on deterministic, randomized, and quantum computers. 1 Introduction Computational Complexity is the subfield of Theoretical Computer Science that aims to understand "how much" computation is necessary and sufficient to perform certain computational tasks. For example, given a computational problem it tries to establish tight upper and lower bounds on the length of the computation (or on other resources, like space). Unfortunately, for many, practically relevant, computational problems no tight bounds are known. An illustrative example is the well known P versus NP problem: for all NP-complete problems the current upper and lower bounds lie exponentially ...

### Citations

1366 |
Quantum Computation and Quantum Information
- Nielsen, Chuang
- 2000
(Show Context)
Citation Context ... f0; 1g n . R 2 (f) denotes the complexity of the optimal randomized decision tree that computes f with bounded error. 2 3.3 Quantum We briefly sketch the framework of quantum computing, referring to =-=[30]-=- for more details. The classical unit of computation is a bit, which can take on the values 0 or 1. In the quantum case, the unit of computation is a quantum bit or qubit which is a linear combination... |

846 | A fast quantum mechanical algorithm for database search
- Grover
- 1996
(Show Context)
Citation Context ...ctor of 2: D(PARITY n ) = n and QE (PARITY n ) = dn=2e. The biggest gap we know between D(f) and Q 2 (f) is quadratic: D(OR n ) = n and Q 2 (OR n ) 2 \Theta( p n) by Grover's quantum search algorithm =-=[19]-=-. Also, R 2 (OR n ) 2 \Theta(n), deg(OR n ) = n, g deg(OR n ) 2 \Theta( p n). Open problem 5 What are the biggest gaps between the classical D(f), R 2 (f) and their quantum analogues QE (f), Q 2 (f)? ... |

508 | Concrete Mathematics - A Foundation for Computer Science - Graham, Knuth, et al. - 1989 |

429 |
zur Gathen and
- von
- 1991
(Show Context)
Citation Context ...e relationship between polynomials that represent symmetric functions, and single-variate polynomials that assume values 0 or 1 on f0; 1; : : : ; ng. Using this relationship, von zur Gathen and Roche =-=[17]-=- prove deg(f) = (1 \Gamma o(1))n for all symmetric f : Theorem 19 (von zur Gathen & Roche) If f is non-constant and sym20 metric, then deg(f) = n \Gamma O(n 0:548 ). If, furthermore, n + 1 is prime, t... |

359 | The complexity of Boolean functions
- Wegener
- 1987
(Show Context)
Citation Context ...classical, randomized, and quantum settings. We also identify some of the main remaining open questions. The complexity measures discussed here also have interesting relations with circuit complexity =-=[47,4,7]-=-, parallel computing [10,41,31,47], communication complexity [33,9], and the construction of oracles in computational complexity theory [6,43,15,16], which we will not discuss here. The paper is organ... |

266 | Quantum lower bounds by polynomials
- Beals, Buhrman, et al.
(Show Context)
Citation Context ...n p has degreesD(f ), and p(x) = 1 iff a 1-leaf is reached on input x, so p represents f . 2 14 Below we give some upper bounds on D(f) in terms of bs(f ), C(f ), deg(f ), and g deg(f ). Beals et.al. =-=[3] prove The-=-orem 11 D(f)sC (1) (f)bs(f). Proof The following describes an algorithm to compute f(x), querying at most C (1) (f)bs(f) variables of x (in the algorithm, by a "consistent" certificate C or ... |

226 | The influence of variables on boolean functions - Kahn, Kalai, et al. - 1988 |

146 | On the degree of boolean functions as real polynomials - Nisan, Szegedy - 1994 |

113 | Oracle quantum computing - Berthiaume, Brassard - 1994 |

95 |
Upper and lower time bounds for parallel random access machines without simultaneous writes
- Cook, Dwork, et al.
- 1986
(Show Context)
Citation Context ...antum settings. We also identify some of the main remaining open questions. The complexity measures discussed here also have interesting relations with circuit complexity [47,4,7], parallel computing =-=[10,41,31,47]-=-, communication complexity [33,9], and the construction of oracles in computational complexity theory [6,43,15,16], which we will not discuss here. The paper is organized as follows. In Section 2 we i... |

94 | Complexity limitations on quantum computation
- FORTNOW, ROGERS
- 1999
(Show Context)
Citation Context ...also have interesting relations with circuit complexity [47,4,7], parallel computing [10,41,31,47], communication complexity [33,9], and the construction of oracles in computational complexity theory =-=[6,43,15,16]-=-, which we will not discuss here. The paper is organized as follows. In Section 2 we introduce some notation concerning Boolean functions and multivariate polynomials. In Section 3 we define the three... |

81 |
CREW PRAMs and decision trees
- Nisan
- 1991
(Show Context)
Citation Context ...antum settings. We also identify some of the main remaining open questions. The complexity measures discussed here also have interesting relations with circuit complexity [47,4,7], parallel computing =-=[10,41,31,47]-=-, communication complexity [33,9], and the construction of oracles in computational complexity theory [6,43,15,16], which we will not discuss here. The paper is organized as follows. In Section 2 we i... |

72 |
On the degree of polynomials that approximate symmetric Boolean functions
- PATURI
- 1992
(Show Context)
Citation Context ...= n \Gamma 3 for some specific f and n. Via Theorems 10 and 17, the above degree lower bounds give strong lower bounds on D(f) and QE (f ). For the case of approximate degrees of symmetric f , Paturi =-=[34]-=- gave the following tight characterization. Define \Gamma(f ) = minfj2k \Gamma n + 1j : f k 6= f k+1 g. Informally, this quantity measures the length of the interval around Hamming weight n=2 where f ... |

67 | The polynomial method in circuit complexity - Beigel - 1993 |

65 | The quantum query complexity of approximating the median and related statistics
- Nayak, Wu
- 1999
(Show Context)
Citation Context ...lynomial relations between classical and quantum complexities of [3]: Corollary 4 D(f) 2 O(QE (f) 4 ) and D(f) 2 O(Q 2 (f) 6 ). Some other quantum lower bounds via degree lower bounds may be found in =-=[3,1,29,14,8]-=-. The biggest gap that is known between D(f) and QE (f) is only a factor of 2: D(PARITY n ) = n and QE (PARITY n ) = dn=2e. The biggest gap we know between D(f) and Q 2 (f) is quadratic: D(OR n ) = n ... |

61 | Communication complexity lower bounds by polynomials
- Buhrman, Wolf
- 2001
(Show Context)
Citation Context ... the main remaining open questions. The complexity measures discussed here also have interesting relations with circuit complexity [47,4,7], parallel computing [10,41,31,47], communication complexity =-=[33,9]-=-, and the construction of oracles in computational complexity theory [6,43,15,16], which we will not discuss here. The paper is organized as follows. In Section 2 we introduce some notation concerning... |

56 |
Probabilistic Boolean decision trees and the complexity of evaluating game trees
- Saks, Wigderson
- 1986
(Show Context)
Citation Context ... R 2 (f) refers to the 2-sided error of the algorithm: it may err on 0-inputs as well as on 1-inputs. We will not discuss zero-error (Las Vegas) or one-sided error randomized decision trees here. See =-=[38,31,22,23,20,8]-=- for some results concerning such trees. 6 We formalize a query to an input x 2 f0; 1g n as a unitary transformation O which maps ji; b; zi to ji; b \Phi x i ; zi. Here ji; b; zi is some m-qubit basis... |

53 |
On recognizing graph properties from adjacency matrices
- Rivest, Vuillemin
- 1976
(Show Context)
Citation Context ...ture is still open; see [27] for an overview. The conjecture has been proved for graphs where the number of vertices is a prime power [25], but the best known general bound is D(P ) 2 \Omega\Gamma N) =-=[35,25,26]-=-. This bound also follows from a degree-bound by Dodis and Khanna [11, Theorem 2]: Theorem 22 (Dodis & Khanna) If P is a non-constant monotone graph property, then deg(P ) 2 \Omega\Gamma N). Corollary... |

48 | A limit on the speed of quantum computation in determining parity
- Farhi, Goldstone, et al.
- 1998
(Show Context)
Citation Context ... deg(f) and hence deg(f)s2QE (f ). 2 By a similar proof: Theorem 18 g deg(f)s2 Q 2 (f). Both theorems are tight for f = PARITY n : here we have deg(f) = g deg(f) = n [28] and QE (f) = Q 2 (f) = dn=2e =-=[3,13]-=-. No f is known with QE (f) ? deg(f) or Q 2 (f) ? g deg(f ), so the following question presents itself: Open problem 4 Are QE (f) 2 O(deg(f)) and Q 2 (f) 2 O( g deg(f))? Note that the degree lower bou... |

47 | An oracle builder’s toolkit
- Fenner, Fortnow, et al.
- 1993
(Show Context)
Citation Context ...also have interesting relations with circuit complexity [47,4,7], parallel computing [10,41,31,47], communication complexity [33,9], and the construction of oracles in computational complexity theory =-=[6,43,15,16]-=-, which we will not discuss here. The paper is organized as follows. In Section 2 we introduce some notation concerning Boolean functions and multivariate polynomials. In Section 3 we define the three... |

37 | On rank vs. communication complexity
- Nisan, Wigderson
- 1995
(Show Context)
Citation Context ... the main remaining open questions. The complexity measures discussed here also have interesting relations with circuit complexity [47,4,7], parallel computing [10,41,31,47], communication complexity =-=[33,9]-=-, and the construction of oracles in computational complexity theory [6,43,15,16], which we will not discuss here. The paper is organized as follows. In Section 2 we introduce some notation concerning... |

34 |
Lower Bounds for Probabilistic Linear Decision Trees
- Snir
- 1983
(Show Context)
Citation Context ...omplete binary AND-OR-tree of depth k. For instance, for k = 2 we have f(x) = (x 1sx 2 )s(x 3sx 4 ). It is easy to see that deg(f) = n and hence D(f) = n. There is a simple randomized algorithm for f =-=[42,38]-=-: randomly choose one of the two subtrees of the root and recursively compute the value of that subtree; if its value is 0 then output 0, otherwise compute the other subtree and output its value. It c... |

33 |
Schwankung von polynomen zwischen gitterpunkten, Mathematische Zeitschrift 86:41–44
- Ehlich, Zeller
- 1964
(Show Context)
Citation Context ...n and Szegedy's proof that deg(f) can be no more than quadratically smaller than bs(f) [32]. This shows that the gap of the last example is close to optimal. The proof uses the following theorem from =-=[12,36]-=-: Theorem 3 (Ehlich & Zeller; Rivlin & Cheney) Let p : R ! R be a polynomial such that b 1sp(i)sb 2 for every integer 0sisn, and its derivative has jp 0 (x)jsc for some real 0sxsn. Then deg(p)sq cn=(c... |

32 |
Bounds for small-error and zero-error quantum algorithms
- Zalka
- 1999
(Show Context)
Citation Context ... R 2 (f) refers to the 2-sided error of the algorithm: it may err on 0-inputs as well as on 1-inputs. We will not discuss zero-error (Las Vegas) or one-sided error randomized decision trees here. See =-=[38,31,22,23,20,8]-=- for some results concerning such trees. 6 We formalize a query to an input x 2 f0; 1g n as a unitary transformation O which maps ji; b; zi to ji; b \Phi x i ; zi. Here ji; b; zi is some m-qubit basis... |

32 | Topological approach to evasiveness
- Kahn, Saks, et al.
(Show Context)
Citation Context ... non-constant monotone graph properties P are evasive. This conjecture is still open; see [27] for an overview. The conjecture has been proved for graphs where the number of vertices is a prime power =-=[25]-=-, but the best known general bound is D(P ) 2 \Omega\Gamma N) [35,25,26]. This bound also follows from a degree-bound by Dodis and Khanna [11, Theorem 2]: Theorem 22 (Dodis & Khanna) If P is a non-con... |

30 |
The average sensitivity of bounded-depth circuits
- Boppana
- 1997
(Show Context)
Citation Context ...classical, randomized, and quantum settings. We also identify some of the main remaining open questions. The complexity measures discussed here also have interesting relations with circuit complexity =-=[47,4,7]-=-, parallel computing [10,41,31,47], communication complexity [33,9], and the construction of oracles in computational complexity theory [6,43,15,16], which we will not discuss here. The paper is organ... |

27 |
HEMACHANDRA: One-way functions, robustness and the non-isomorphism of NP-complete sets
- HARTMANIS, LANE
- 1987
(Show Context)
Citation Context ...n and D(f) = n. Open problem 2 Is D(f) 2 O(bs(f) 2 )? Of course, Theorem 11 also holds with C (0) instead of C (1) . Since bs(f)smaxfC (0) (f); C (1) (f)g, we also obtain the following result, due to =-=[6,21,43]-=-. Corollary 2 D(f)sC (0) (f)C (1) (f). Now we will show that D(f) is upper bounded by deg(f) 4 and g deg(f) 6 . The first result is due to Nisan and Smolensky, below we give their (previously unpublis... |

26 | Quantum oracle interrogation: Getting all information for almost half the price
- Dam
- 1998
(Show Context)
Citation Context ...imply strong lower bounds on the quantum decision tree complexities of almost all f . In particular, Theorem 8 implies that Q 2 (f)sn=4 \Gamma O( p n log n) for almost all 19 f . In contrast, Van Dam =-=[45]-=- has shown that Q 2 (f)sn=2 + p n for all f . Combining Theorems 17 and 18 with Theorems 12 and 13 we obtain the polynomial relations between classical and quantum complexities of [3]: Corollary 4 D(f... |

23 | An Ω(n 4/3 ) lower bound on the randomized complexity of graph properties
- Hajnal
- 1991
(Show Context)
Citation Context ... R 2 (f) refers to the 2-sided error of the algorithm: it may err on 0-inputs as well as on 1-inputs. We will not discuss zero-error (Las Vegas) or one-sided error randomized decision trees here. See =-=[38,31,22,23,20,8]-=- for some results concerning such trees. 6 We formalize a query to an input x 2 f0; 1g n as a unitary transformation O which maps ji; b; zi to ji; b \Phi x i ; zi. Here ji; b; zi is some m-qubit basis... |

23 |
Query complexity, or why is it difficult to separate NP A ∩ co − NP A from P A by a random oracle
- Tardos
- 1989
(Show Context)
Citation Context ...also have interesting relations with circuit complexity [47,4,7], parallel computing [10,41,31,47], communication complexity [33,9], and the construction of oracles in computational complexity theory =-=[6,43,15,16]-=-, which we will not discuss here. The paper is organized as follows. In Section 2 we introduce some notation concerning Boolean functions and multivariate polynomials. In Section 3 we define the three... |

22 | An introduction to quantum complexity theory - Cleve - 1999 |

20 |
A comparison of uniform approximations on an interval and a finite subset thereof
- Rivlin, Cheney
- 1966
(Show Context)
Citation Context ...n and Szegedy's proof that deg(f) can be no more than quadratically smaller than bs(f) [32]. This shows that the gap of the last example is close to optimal. The proof uses the following theorem from =-=[12,36]-=-: Theorem 3 (Ehlich & Zeller; Rivlin & Cheney) Let p : R ! R be a polynomial such that b 1sp(i)sb 2 for every integer 0sisn, and its derivative has jp 0 (x)jsc for some real 0sxsn. Then deg(p)sq cn=(c... |

19 | Lower bounds of quantum black-box complexity and degree of approximating polynomials by influence of Boolean variables
- Shi
- 2000
(Show Context)
Citation Context ...while whether bs(f) can be upper bounded by a polynomial in s(f ). It may well be true that bs(f) 2 O(s(f) 2 ). 3 There has also been some work on average (block) sensitivity [5] and its applications =-=[7,40,2]-=-. In particular, Shi [40] has shown that the average sensitivity of a total function f is a lower bound on its approximate degree g deg(f ). 9 Open problem 1 Is bs(f) 2 O(s(f) k ) for some k? We proce... |

18 | On read-once threshold formulae and their randomized decision tree complexity
- Heiman, Newman, et al.
- 1993
(Show Context)
Citation Context |

17 | Characterization of non-deterministic quantum query and quantum communication complexity
- Wolf
- 2000
(Show Context)
Citation Context ...e that there may be many different minimaldegree polynomials that approximate f , whereas there is only one polynomial that represents f . 5 Also non-deterministic polynomials for f have been studied =-=[49]-=-, but we will not cover that notion in this survey. 13 By the same technique as Theorem 4, Nisan and Szegedy [32] showed Theorem 7 (Nisan & Szegedy) bs(f)s6 g deg(f) 2 . The approximate degree of f ca... |

16 |
Sensitivity vs. block sensitivity of Boolean functions
- Rubinstein
- 1995
(Show Context)
Citation Context ... x will have to contain at least one variable of each sensitive block). Hence: Proposition 1 s(f)sbs(f)sC(f). The biggest gap known between s(f) and bs(f) is quadratic and was exhibited by Rubinstein =-=[37]-=-: Example 1 Let n = 4k 2 . Divide the n variables in p n disjoint blocks of p n variables: the first block B 1 contains x 1 ; : : : ; x p n , the second block B 2 contains x p n+1 ; : : : ; x 2 p n , ... |

15 |
On the Monte Carlo decision tree complexity of read-once formulae
- Santha
- 1991
(Show Context)
Citation Context ...cted number of queries O(n ff ), where ff = log((1 + p 33)=4)s0:7537 : : : . Saks and Wigderson [38] showed that this is asymptotically optimal for zero-error algorithms for this function, and Santha =-=[39]-=- proved the same for bounded-error algorithms. Thus we have D(f) = n = \Theta(R 2 (f) 1:3::: ). Open problem 3 What is the biggest gap between D(f) and R 2 (f)? 5.3 Quantum As in the classical case, d... |

14 |
Lecture notes on evasiveness of graph properties
- Lovasz, Young
- 1994
(Show Context)
Citation Context ...orst case. The evasiveness conjecture (also sometimes called Aanderaa-Karp-Rosenberg conjecture) says that all non-constant monotone graph properties P are evasive. This conjecture is still open; see =-=[27]-=- for an overview. The conjecture has been proved for graphs where the number of vertices is a prime power [25], but the best known general bound is D(P ) 2 \Omega\Gamma N) [35,25,26]. This bound also ... |

13 |
How many functions can be distinguished with k quantum queries?, Physical Review A 60
- Farhi, Goldstone, et al.
- 1999
(Show Context)
Citation Context ...lynomial relations between classical and quantum complexities of [3]: Corollary 4 D(f) 2 O(QE (f) 4 ) and D(f) 2 O(Q 2 (f) 6 ). Some other quantum lower bounds via degree lower bounds may be found in =-=[3,1,29,14,8]-=-. The biggest gap that is known between D(f) and QE (f) is only a factor of 2: D(PARITY n ) = n and QE (PARITY n ) = dn=2e. The biggest gap we know between D(f) and Q 2 (f) is quadratic: D(OR n ) = n ... |

13 |
The critical complexity of graph properties
- TURÁN
- 1984
(Show Context)
Citation Context ...n R 2 (f) and Q 2 (f ). For Q 2 (f) these bounds are in fact tight (a matching upper bound was shown in [3]), but for R 2 (f) a stronger bound can be obtained from Theorem 16 and the following result =-=[44]-=-: Proposition 2 (Tur'an) If f is non-constant and symmetric, then s(f)sd n+1 2 e. Proof Let k be such that f k 6= f k+1 , and jxj = k. Without loss of generality assume ksb(n \Gamma 1)=2c (otherwise g... |

12 | A.: A note on quantum black-box complexity of almost all Boolean functions
- Ambainis
- 1999
(Show Context)
Citation Context ...ere are two ways to view a randomized decision tree. Firstly, we can add (possibly biased) coin flips as internal nodes to the tree. That is, the tree may contain internal nodes labeled by a bias p 2 =-=[0; 1]-=-, and when the evaluation procedure reaches such a node, it will flip a coin with bias p and will go to the left child on outcome `heads' and to the right child on `tails'. Now an input x no longer de... |

12 | Randomized vs. deterministic decision tree complexity for read-once boolean functions
- Heiman, Wigderson
- 1991
(Show Context)
Citation Context |

11 |
bounds on the complexity of graph properties
- Lower
- 1988
(Show Context)
Citation Context ...ture is still open; see [27] for an overview. The conjecture has been proved for graphs where the number of vertices is a prime power [25], but the best known general bound is D(P ) 2 \Omega\Gamma N) =-=[35,25,26]-=-. This bound also follows from a degree-bound by Dodis and Khanna [11, Theorem 2]: Theorem 22 (Dodis & Khanna) If P is a non-constant monotone graph property, then deg(P ) 2 \Omega\Gamma N). Corollary... |

9 | Space-Time Tradeoffs for Graph Properties - Dodis - 1998 |

6 | Average-case quantum query complexity
- Ambainis, Wolf
(Show Context)
Citation Context ...while whether bs(f) can be upper bounded by a polynomial in s(f ). It may well be true that bs(f) 2 O(s(f) 2 ). 3 There has also been some work on average (block) sensitivity [5] and its applications =-=[7,40,2]-=-. In particular, Shi [40] has shown that the average sensitivity of a total function f is a lower bound on its approximate degree g deg(f ). 9 Open problem 1 Is bs(f) 2 O(s(f) k ) for some k? We proce... |

6 |
Sensitivity vs. Block Sensitivity (an average-case study), Information processing letters
- Bernasconi
- 1996
(Show Context)
Citation Context ...as been open for quite a while whether bs(f) can be upper bounded by a polynomial in s(f ). It may well be true that bs(f) 2 O(s(f) 2 ). 3 There has also been some work on average (block) sensitivity =-=[5]-=- and its applications [7,40,2]. In particular, Shi [40] has shown that the average sensitivity of a total function f is a lower bound on its approximate degree g deg(f ). 9 Open problem 1 Is bs(f) 2 O... |

6 |
Generic oracles and oracle classes (extended abstract
- Blum, Impagliazzo
- 1987
(Show Context)
Citation Context |

5 |
A note on the relations between critical and sensitive complexity, EIK
- Wegener, Zádori
- 1989
(Show Context)
Citation Context ...ve a quadratic gap between s(f) and bs(f). Since bs(f)sC(f), this is also a quadratic gap between s(f) and C(f) (Wegener and Z'adori give a different function with a smaller gap between s(f) and C(f) =-=[48]-=-). It has been open for quite a while whether bs(f) can be upper bounded by a polynomial in s(f ). It may well be true that bs(f) 2 O(s(f) 2 ). 3 There has also been some work on average (block) sensi... |

4 |
The critical complexity of all (monotone) boolean functions
- Wegener
- 1985
(Show Context)
Citation Context ...tion". This is a function on n = k + 2 k variables, where the first k bits of the input provide an index in the last 2 k bits. The value of the indexed variable is the output of the function. Weg=-=ener [46]-=- gives a monotone version of the address function. 2.2 Multilinear polynomials If S is a set of (indices of) variables, then the monomial X S is the product of variables X S = \Pi i2S x i . The degree... |

3 |
A tight\Omega\Gamma245 log n)-bound on the time for parallel RAM's to compute nondegenerated boolean functions
- Simon
- 1983
(Show Context)
Citation Context ...lock sensitivity of f is bs(f) = max x bs x (f). (If f is constant, we define s(f) = bs(f) = 0.) Note that sensitivity is just block sensitivity with the size of the blocks B i restricted to 1. Simon =-=[Sim83]-=- gave a general lower bound on s(f ): Theorem 1 (Simon) If f depends on all n variables, then s(f)s1 2 log n \Gamma 1 2 log log n + 1 2 . 2 There has also been some work on average (block) sensitivity... |