Topological lower bounds for computation trees

On topological lower bounds for algebraic computation trees

Andrei Gabrielov Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA agabriel@math.purdue.edu  and  Nicolai Vorobjov Department of Computer Science, University of Bath, Bath BA2 7AY, England, UK nnv@cs.bath.ac.uk
Abstract.

We prove that the height of any algebraic computation tree for deciding membership in a semialgebraic set is bounded from below by

where is the -th Betti number of with respect to “ordinary” (singular) homology, and are some (absolute) positive constants. This result complements the well known lower bound by Yao [7] for locally closed semialgebraic sets in terms of the total Borel-Moore Betti number.

We also prove that if is the projection map, then the height of any tree deciding membership in is bounded from below by

for some positive constants .

We illustrate these general results by examples of lower complexity bounds for some specific computational problems.

Communicated by Felipe Cucker
2010 Mathematics Subject Classification 68Q17, 14P25
Keywords Complexity lower bounds, algebraic computation trees, semialgebraic sets

1. Introduction

The algebraic computation tree is a standard sequential model of computation for deciding membership problems for semialgebraic sets. Among various general methods for obtaining lower complexity bounds for this model, one of the most efficient uses homotopy invariants, Euler characteristic and Betti numbers, as arguments for the bounding functions. The history of this approach started probably in mid 70s with the work of Dobkin and Lipton [3], and features prominent results such as Ben-Or’s [1] in 1983 and Yao’s [7] in 1997. The present paper is inspired by the latter. We will discuss the results of [7] in some detail.

We assume that the reader is familiar with the concept of the algebraic computation tree, so we give here just a brief formal description, closely following [2].

Definition 1.1.

An algebraic computation tree with input variables taking real values, is a tree having three types of vertices: computation (outdegree 1), branch (outdegree 3), and leaves (outdegree 0). To each vertex of a variable is assigned.

With each computation vertex an expression is associated, where , and are either real constants, or input variables, or variables associated with predecessor vertices of , or a combination of these.

At each branch vertex , the variable is assigned the value which is either a real constant, or an input variable, or a variable associated with a predecessor vertex. The three outgoing edges of correspond to signs , , .

With each leaf a basic semialgebraic set (called leaf set) is associated, defined by equations of the kind either , or , or , for all variables associated with predecessor branch vertices of along the branch leading from the root to . The sign of each is determined by the outgoing edge in the branch vertex. In addition, each leaf carries a label “Yes” or “No”. The tree tests membership in the union of all Yes leaf sets.

The semantics of this model of computation is straightforward. On an input , the input variables get the corresponding values , the arithmetic operations are executed in computation vertices and the real values are obtained by variables . At branch vertices the sign of the value of is determined, and the corresponding outgoing edge is chosen. As a result a certain branch ending up in a leaf is selected. The input belongs to the semialgebraic set assigned to . If is a Yes leaf, then is said to be accepted by the tree .

It can be assumed without loss of generality [2], that there are no divisions used in a tree.

We will be interested in lower bounds on the heights of algebraic computation trees testing membership in a given semialgebraic set . A detailed outline of the development of lower bounds that depend on topological characteristics of a set can be found in [7] (see also [2]). We mention here just two highlights.

The first most important achievement was the proof by Ben-Or [1] of the bound , where is the number of connected components of , and are some absolute positive constants. This general bound implies non-trivial, and sometimes tight, lower bounds for specific computational problems, such as Distinctness and Knapsack.

One of the most general results so far in this direction belongs to Yao [7]. Suppose a semialgebraic set is locally closed and bounded. Let be the total Betti number (the sum of all Betti numbers) of with respect to the Borel-Moore homology . Yao proved the lower bound

(1.1)

where are some absolute positive constants. From this he deduced a tight lower bound for -Distinctness problem, and other non-trivial lower bounds for specific problems.

The Borel-Moore homology is a very strong condition, which implies subadditivity of the total Betti number. Subadditivity is the property on which the whole of the Yao’s argument depends. It is natural to ask whether an analogous bound can be found for the usual, singular, homology theory, which is applicable to arbitrary (not necessarily locally closed) semialgebraic sets. Of course, in this case subadditivity is not necessarily true. Observe that for compact sets Borel-Moore Betti numbers coincide with singular Betti numbers, while for non-compact locally closed sets these two types of Betti numbers can be incomparable.

In this paper we prove two main theorems. Firstly, we prove the lower bound

where is the -th Betti number of an arbitrary semialgebraic set with respect to singular homology, and are some absolute positive constants. Note that this bound depends on an individual Betti number rather than on the sum of Betti numbers. For Betti numbers of a small (fixed) index the bound turns into which is similar to Yao’s bound. The proof is based on a construction from [5] which transforms into a compact semialgebraic set having the same Betti numbers as up to a given index . We then prove that for any algebraic computation tree for there is an algebraic computation tree for having the height not exceeding, up to a multiplicative constant, times the height of . It remains to apply Yao’s bound to .

Our second main result is a lower bound in terms of Betti numbers of the projection of to a subspace, rather than Betti numbers of itself. Note that the topology of the image under a projection may be much more complex than the topology of the set being projected. We are not aware of previous lower bounds of this sort. The bound is

for some positive constants , which again should be applied for small (fixed) values of . The proof uses (implicitly) a spectral sequence associated with the projection map, which allows to bound from above Betti numbers of the projection of in terms of Betti numbers of fiber products by itself of the compactification of [6].

We illustrate these general results by examples of lower complexity bounds for some specific computational problems.

2. Topological tools

In this section we formulate the results from [4, 5, 6] which are used further in this paper.

In what follows, for a topological space , let be its -th Betti number with respect to the singular homology group with coefficients in some fixed Abelian group. By we denote the total Betti number of , i.e., the sum .

2.1. Upper bounds on Betti numbers

Consider a semialgebraic set , where is a Boolean combination of polynomial equations and inequalities of the kind or , and . Suppose that the number of different polynomials is and that their degrees do not exceed .

Proposition 2.1 ([5], Theorem 6.3).

The -th Betti number of satisfies

  1. ;

  2. .

Remark 2.2.

Unlike classical Petrovski-Oleinik-Thom-Milnor bound for basic semialgebraic sets, used in [7], the bounds in Proposition 2.1 are applicable to arbitrary semialgebraic sets defined by a quantifier-free formulae. They are slightly weaker than the classical bound by a multiplicative factor at the base of the exponent, namely in (1) and in (2).

Further, in the proof of Theorem 3.1, we will need the bound (1) from this proposition. We won’t need bound (2) as such but we shall use in an essential way the constructions from [5] used for proving this bound. We now proceed to describing this technique.

2.2. Approximation by monotone families

Definition 2.3.

Let be a compact semialgebraic set. Consider a semialgebraic family of compact subsets of , such that for all , if , then . Denote .

For each , let be a semialgebraic family of compact subsets of such that:

  • for all , if , then ;

  • ;

  • for all sufficiently small and for all , there exists an open in set such that .

We say that is represented by the families and in .

Consider the following two particular cases.

Case 1. Let a semialgebraic set be given as a disjoint union of basic semialgebraic sets (i.e., sets each defined by a conjunction of equations and strict inequalities). (Note that an algebraic computation tree represents the corresponding set in exactly this way.) Let and be some positive constants.

Suppose first that is bounded in , and take as a closed ball of a sufficiently large radius centered at 0. The set is the result of the replacement, independently in each basic set in the union, of all inequalities and by and respectively. The set is obtained by replacing, independently in each basic set, all expressions , and by , and , respectively. One can easily verify (see [5]) that the set , is represented by families and in .

Now suppose that is not necessarily bounded. In this case one can take the semialgebraic one-point (Alexandrov) compactification of as . Define sets and as in the bounded case, replacing equations and inequalities independently in each basic set, and then taking the conjunction of the resulting formula with . Again, is represented by and in .

Case 2. Let be the projection map, and be a semialgebraic set, given as a disjoint union of basic semialgebraic sets. The set is represented by families , in the compactification of as described in Case 1. One can easily verify (see [5]), that the projection is represented by families , (in the Alexandrov compactification of if necessary).

Returning to the general case, suppose that a semialgebraic set is represented by families and in .

For a sequence , where , introduce the compact set

Observe that in Case 2, we have the equality

(2.1)

In what follows, for two real numbers and we write to mean “ is sufficiently smaller than ” (see formal Definition 1.7 in [5]).

Proposition 2.4 ([5], Theorem 1.5).

For any , and

we have

  • for every , there is an epimorphism , in particular, ;

  • in Case 1, for every , the epimorphism is an isomorphism, in particular, . Moreover, if , then is homotopy equivalent to .

2.3. Betti numbers of projections

Definition 2.5.

For two maps and , the fibered product of and is defined as

Proposition 2.6 ([6], Theorem 1).

Let be a closed surjective semialgebraic map (in particular, can be the projection map to a subspace, with a compact ). Then

where

3. General lower bounds

We start with a theorem which immediately follows from an upper bound on the total Betti number of an arbitrary semialgebraic set in Proposition 2.1.

Theorem 3.1.

Let be the height of an algebraic computation tree testing membership in a semi-algebraic set . Then

where is the total Betti number of .

Proof.

Since in each computation vertex at most one multiplication can be performed, every polynomial occurring in the disjunctive normal form defining has a degree at most . The number of polynomials in the conjunction defining the set attached to a Yes leaf is at most , while the number of Yes leaves does not exceed . It follows that the total number of polynomials defining is at most . Then, according to Proposition 2.1, (1), . Taking logarithms we get the result. ∎

Remark 3.2.

The bound in the theorem is significantly weaker than Yao’s bound (1.1). However, as explained in the introduction, it is applicable to any semialgebraic set, not necessarily a locally closed one. The upper bound on the total Betti number, used in the proof, is applicable to arbitrary semialgebraic set, unlike classical Petrovski-Oleinik-Thom-Milnor bounds employed in [7].

Theorem 3.3.

Let be the height of an algebraic computation tree testing membership in a semi-algebraic set . Then

where is the -th Betti number of , and are some positive constants.

Lemma 3.4.

Let be algebraic computation trees testing membership in semialgebraic sets and respectively, and having heights and respectively. Then there is a tree testing membership in , and a tree testing membership in , both having heights at most .

Proof.

To construct , attach a copy of to each No leaf of the tree . For , attach a copy of to each Yes leaf of the tree . ∎

Lemma 3.5.

Let be a tree for , having height . Then for any there exists a tree for whose height does not exceed for some positive constant .

Proof.

The plan of the proof is as follows. The construction of consists of two stages. On the first stage we perform the construction for and arbitrary , and get the tree . The height of is not larger than times the height of for a constant . On the second stage we construct for an arbitrary by induction. On the base step, construct the tree . Suppose we constructed the tree . The tree is obtained from by attaching to each No leaf of the latter, a copy of the tree , considering the leaf as the root of .

Figure 1. Tree .

Now we proceed to a more detailed proof.

Let . The root of the tree is a branch vertex with the polynomial attached. The child of , corresponding to , is a No leaf. Take the other two children as roots of two copies of the tree . The construction of now continues identically for both copies, by induction, as follows. In , let be the closest branch vertex to its root, and let be the polynomial attached to (if such branch vertex does not exist, then the construction of is completed). Then the neighbourhood of in looks like the tree on Figure 1. Here and are polynomials attached to children of . Replace this neighbourhood by the tree on Figure 2. Notice that the leaves of , are labelled again by while one of the leaves is a No leaf. Attach to each leaf of , labelled by , the subtree of rooted at (unless is a leaf of ). Denote the resulting tree by . This completes the base of the induction.

Figure 2. Tree .

On the next induction step perform the same replacement operation, as on the base step, for each subtree of rooted at a leaf of which is not a No leaf. If for a leaf of no such subtree exists, i.e., vertex is a leaf of , then this vertex is taken as a leaf of , it is a Yes leaf if and only if is a Yes leaf in . Denote the results of replacements again by , and the resulting tree again by .

Further induction steps are performed in the same fashion, by applying the replacement operation, described at the base step, to subtrees of rooted at leaves of the trees obtained on the previous induction step. The construction of is completed when all leaves of trees become leaves of .

Note that the height of is not larger than times the height of for a constant .

Now we prove by induction on the construction that is a tree testing membership in (recall the notation from Case 1, Section 2.2). Observe that either or is present in the definition of any Yes leaf set. Assume, as before, that in the vertex is the closest branch vertex to the root, and is the polynomial attached to . Observe that each leaf set of , in particular each Yes leaf set, is of the kind either , or , or . In the tree , on the base step of the construction of , the leaf will be replaced by two leaves, and , the leaf – by two leaves, and , while the leaf – by two leaves, and . It follows that if and only if is a subset of the set tested by , and similar for sets and . Proceeding by induction, we conclude that is the set tested by .

Now construct for arbitrary by induction. On the base step, start with the path of computation vertices at the end of which the polynomial is computed. Continue with the tree . The result of these two steps is the tree . Suppose we constructed the tree for . The tree is obtained from by attaching to each No leaf of the latter, the tree , considering the leaf as the root of . By Lemma 3.4, the result is indeed .

Obviously the height of does not exceed for a constant . ∎

Proof of Theorem 3.3.

By Lemma 3.5, for a constant . By (1.1), since is compact,

for some positive constants . Hence, for the -th Betti number,

It follows, by Proposition 2.4, that

Hence, the theorem. ∎

4. Projections

Theorem 4.1.

Let be the height of an algebraic computation tree testing membership in a semi-algebraic set . Let be the projection map. Then

(4.1)

for some positive constants .

Let

Lemma 4.2.

Let be a tree for , having height . Then there exists a tree for whose height does not exceed for some positive constant .

Proof.

Lemma 3.5 implies that there is a tree for having the height not exceeding . The problem of membership in has input variables

Construct the tree inductively, starting with a copy of with input variables . Then, using Lemma 3.4, attach to each Yes leaf another copy of with input variables , and so on. The height of the resulting tree is at most times the height of the tree , as required. ∎

Proof of Theorem 4.1.

According to Proposition 2.6,

(4.2)

Let , and be the height of a tree for .

Since is compact, by (1.1), we have

for some positive constants , thus, replacing by a larger number according to Lemma 4.2, and using , we get

for some positive constants . But

so using (4.2) we have

Hence,

for some positive constants .

According to (2.1), , while, by Proposition 2.4,

It follows that

5. Applications

In this section we apply the general bounds from Theorems 3.3 and 4.1 to examples of specific computational problems. These problems admit obvious variations.

“Parity of integers”

This is the following computational problem.

Let be a positive integer. For given real numbers such that for all , decide whether the following property is true: either all are integer or exactly two of them are not integer.

Observe that complexity of this problem has an upper bound : the computation tree for each checks whether it coincides with one of the numbers using binary search.

To obtain a lower bound, consider the integer lattice in and let be the union of all open 2-dimensional squares and all vertices. Then the problem is equivalent to deciding membership in . Observe that is not locally closed. It is homotopy equivalent to a 2-plane with punctured points, so . By Theorem 3.3, the height of any algebraic computation tree testing membership in is .

“Crossing number”

Let be a smooth connected bounded semialgebraic curve in . Then is a (smooth) embedding of either the circle or the interval into (for and a circle this is a knot). The total Betti number of is at most 2.

Observe that the image under the projection of onto a generic 2-dimensional linear subspace has only double points as singular points.

The crossing number of is the maximal number of singular points of the image of the projection over all generic 2-dimensional linear subspaces.

Theorem 5.1.

The complexity of membership in is at least for some positive constants .

Proof.

Let be the image of under the projection to the plane on which the crossing number is realized. Then is less by 2 (if is an embedding of ), or otherwise by 1, than the number of connected components of the complement to in the plane. By Alexander duality, the number of connected components is the same as , hence the lower bound follows from Theorem 4.1. ∎

Acknowledgements

Andrei Gabrielov was partially supported by NSF grant DMS-1161629.

References

  • [1] M. Ben-Or, Lower bounds for algebraic computation trees, in Proceedings of 15th Annual ACM Symposium on Theory of Computing 1983, 80-86.
  • [2] P. Bürgisser, F. Cucker, Variations by complexity theorists on three themes of Euler, Bézout, Betti, and Poincaré, in Complexity of Computations and Proofs (Jan Krajicek ed.), Quaderini di Matematica 13 (2005), 73-152.
  • [3] D. Dobkin and R.J. Lipton, On the complexity of computations under varying sets of primitives, J. Comput. Syst. Sci. 18 (1979), 86-91.
  • [4] A. Gabrielov, N. Vorobjov, Betti numbers of semialgebraic sets defined by quantifier-free formulae, Discrete Comput. Geom. 33 (2005), 395-401.
  • [5] A. Gabrielov, N. Vorobjov, Approximation of definable sets by compact families, and upper bounds on homotopy and homology, J. London Math. Soc. 80 (2009), 35-54.
  • [6] A. Gabrielov, N. Vorobjov, T. Zell, Betti numbers of semialgebraic and sub-Pfaffian sets, J. London Math. Soc. 69, part 1 (2004), 27-43.
  • [7] A.C.C. Yao, Decision tree complexity and Betti numbers, J. Comput. Syst. Sci. 55 (1997), 36-43.
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