Complexity Measures and Decision Tree Complexity A Survey a; a;b; Harry Buhrman Ronald de Wolf a CWI PO Box Amsterdam The Netherlands fbuhrmanrdewolfgcwinl b University of Amsterdam Abstract We discuss several complexity measures for Bo olean functions certicate complex ity sensitivity blo ck sensitivity and the degree of a representing or approximating p olynomial We survey the relations and biggest gaps known b etween these mea sures and show how they give b ounds for the decision tree complexity of Bo olean functions on deterministic randomized and quantum computers Intro duction Computational Complexity is the subeld of Theoretical Computer Science that aims to understand how much computation is necessary and sucient to p erform certain computational tasks For example given a computational problem it tries to establish tight upp er and lower b ounds on the length of the computation or on other resources like space Unfortunately for many practically relevant computational problems no tight b ounds are known An illustrative example is the well known P versus NP problem for all NPcomplete problems the current upp er and lower b ounds lie exp onentially far apart That is the b est known algorithms for these com putational problems need exp onential time in the size of the input but the b est lower b ounds are of a linear nature One of the general approaches towards solving a hard problem mathematical or otherwise is to set the goals a little bit lower and try to tackle a simpler Partially supp orted by the EU fth framework pro ject QAIP IST Preprint submitted to Elsevier Preprint Novemb er problem rst The hop e is that understanding of the simpler problem will lead to a b etter understanding of the original more dicult problem This approach has b een taken with resp ect to Computational Complexity simpler and more limited mo dels of computation have b een studied Perhaps the simplest mo del of computation is the decision tree The goal here is to n compute a Bo olean function f f g f g using queries to the input In the most simple form a query asks for the value of the bit x and the answer is i this value The queries may b e more complicated In this survey we will only deal with this simple typ e of query The algorithm is adaptive that is the k th query may dep end on the answers of the k previous queries The algorithm can therefore b e describ ed by a binary tree whence its name decision tree For a Bo olean function f we dene its deterministic decision tree complexity D f as the minimum numb er of queries that an optimal deterministic al gorithm for f needs to make on any input This measure corresp onds to the depth of the tree that an optimal algorithm induces Once the computational p ower of decision trees is b etter understo o d one can extend this notion to more p owerful mo dels of query algorithms This results in randomized and even quantum decision trees In order to get a handle on the computational p ower of decision trees whether deterministic randomized or quantum other measures of the complexity of Bo olean functions have b een dened and studied Some prime examples are certicate complexity sensitivity block sensitivity the degree of a representing polynomial and the degree of an approximating polynomial We survey the known relations and biggest gaps b etween these complexity measures and show how they apply to decision tree complexity giving pro ofs of some of the central results The main results say that all of these complexity measures with the p ossible exception of sensitivity are p olynomially related to each other and to the decision tree complexities in each of the classical randomized and quantum settings We also identify some of the main remaining op en questions The complexity measures discussed here also have interesting relations with circuit complexity parallel computing communication complexity and the construction of oracles in computational complexity theory which we will not discuss here The pap er is organized as follows In Section we intro duce some notation concerning Bo olean functions and multivariate p olynomials In Section we dene the three main variants of decision trees that we discuss determin istic decision trees randomized decision trees and quantum decision trees In Section we intro duce certicate complexity sensitivity blo ck sensitivity and the degree of a representing or approximating p olynomial We survey the main relations and known upp er and lower b ounds b etween these measures In Section we show how the complexity measures of Section imply upp er and lower b ounds on deterministic randomized and quantum decision tree complexity This section gives b ounds that apply to all Bo olean functions Finally in Section we examine some sp ecial sub classes of Bo olean functions and tighten the general b ounds of Section for those sp ecial cases Bo olean Functions and Polynomials Boolean functions n A Bo olean function is a function f f g f g Note that f is total n ie dened on all nbit inputs For an input x f g we use x to denote i its ith bit so x x x We use jxj to denote the Hamming weight of x n S its numb er of s If S is a set of indices of variables then we use x to fig denote the input obtained by ipping the S variables in x We abbreviate x i fg to x For example if x then x and x We call f symmetric if f x only dep ends on jxj Some common symmetric functions that we will refer to are OR x i jxj n AND x i jxj n n PARITY x i jxj is o dd n MAJ x i jxj n n We call f monotone increasing if f x cannot decrease if we set more vari ables of x to A function that we will refer to sometimes is the address k function This is a function on n k variables where the rst k bits k of the input provide an index in the last bits The value of the indexed variable is the output of the function Wegener gives a monotone version of the address function Multilinear polynomials If S is a set of indices of variables then the monomial X is the pro duct of S variables X x The degree of this monomial is the cardinality of S A S iS i n multilinear polynomial on n variables is a function p R C which can b e P c X for some complex numb ers c We call c the written as px S S S S S n coecient of the monomial X in p The degree of p is the degree of its largest S monomial deg p maxfjS j j c g Note that if we restrict attention to S n k the Bo olean domain f g then x x for all k so considering only i i multilinear p olynomials is no restriction when dealing with Bo olean inputs The next lemma implies that if multilinear p olynomials p and q are equal on all Bo olean inputs then they are identical n Lemma Let p q R R be multilinear polynomials of degree at most d n If px q x for al l x f g with jxj d then p q Pro of Dene r x px q x Supp ose r is not identically zero Let V b e a minimaldegree term in r with nonzero co ecient c and x b e the input where x i x o ccurs in V Then jxj d and hence px q x j j However since all monomials in r except for V evaluate to on x we have r x c px q x which is a contradiction It follows that r is identically zero and p q 2 Below we sketch the metho d of symmetrization due to Minsky and Pap ert n see also Section Let p R R b e a p olynomial If is some p ermutation and x x x then x x x Let S b e the n n n sy m set of all n p ermutations The symmetrization p of p averages over all p ermutations of the input and is dened as P p x S sy m n p x n sy m Note that p is a p olynomial of degree at most the degree of p Symmetrizing sy m may actually lower the degree if p x x then p The following lemma allows us to reduce an nvariate p olynomial to a singlevariate one n Lemma Minsky Pap ert If p R R is a multilinear polynomial then there exists a singlevariate polynomial q R R of degree at most the sy m n degree of p such that p x q jxj for al l x f g sy m Pro of Let d b e the degree of p which is at most the degree of p Let V j n denote the sum of all pro ducts of j dierent variables so V x x n j sy m V x x x x x x etc Since p is symmetrical it is easily n n shown by induction that it can b e written as sy m p x c c V c V c V d d jxj with c R Note that V assumes value jxjjxj jxj jxj j i j j j on x which is a p olynomial of degree j of jxj Therefore the singlevariate p olynomial q dened by jxj jxj jxj c c q jxj c c d d satises the lemma 2 Decision Tree Complexity on Various Machine Mo dels Below we dene decision tree complexity for three dierent kinds of machine mo dels deterministic randomized and quantum Deterministic A deterministic decision tree is a ro oted ordered binary tree T Each internal no de of T is lab eled with a variable x and each leaf is lab eled with a value i n or Given an input x f g the tree is evaluated as follows Start at the ro ot If this is a leaf then stop Otherwise query the variable x that lab els i the ro ot If x then recursively evaluate the left subtree if x then i i recursively evaluate the right subtree The output of the tree is the value or of the leaf that is reached eventually Note that an input x deterministically determines the leaf and thus the output that the pro cedure ends up in n We say a decision tree computes f if its output equals f x for all x f g Clearly there are many dierent decision trees that compute the same f The complexity of such a tree is its depth ie the numb er of queries made on the worstcase input We dene