Monotone expanders - constructions and applications

The main purpose of this work is to formally deﬁne monotone expanders and motivate their study with (known and new) connections to other graphs and to several computational and pseudorandomness problems. In particular we explain how monotone expanders of constant degree lead to:


Introduction
Expander graphs are families of highly connected sparse graphs.These combinatorial objects have found numerous applications in diverse areas of mathematics and computer science and have drawn an enormous amount of attention in past years (see [14] for a survey).Many of the applications require explicit constructions of such graphs, where 'explicit' means graphs that can be constructed by an algorithm that runs in polynomial time in the size of the graph.Some applications require an even stronger notion of explicitness in which there exists an algorithm that can compute the neighbor function of a vertex in time polynomial in the description of the vertex (and hence in time poly-logarithmic in the size of the graph).Such constructions are called 'strongly explicit'.
It is easy to show that a random sparse graph is an expander, but proving that some 'specific' graph is an expander is considerably harder.Today there are many known constructions of expander graphs ranging from group theoretic constructions (e.g.[19,22,23]) to purely combinatorial ones [27,2].
The question of whether restricted classes of graphs can be expanders of constant degree has also received attention, and there are many negative examples for natural classes, e.g.planar graphs (and graphs with other excluded minors) [18,3] and Cayley graphs of Abelian and near-Abelian groups [16,20].More relevant to us are the classes of k-page graphs and k-pushdown graphs (which we define later) that arise as computation graphs of Turing machines, for which the expansion question remained open and is key to resolving some complexity questions [12,28].
The focus of this paper are monotone graphs and their expansion properties.Monotone graphs are defined by monotone mappings.Throughout the paper the vertex set of graphs will be [n], the first n integers with their natural ordering.A partial monotone function f : [n] → [n] satisfies f (x) > f (y) for all pairs x > y for which the function is defined.Such a function defines a 1monotone graph whose edges are simply all (i, f (i)) for all i ∈ Dom(f ).Similarly, d (partial) monotone functions define a d-monotone graph.Note that such graphs have in (or out) degree at most d.The most natural way to think of these graphs is as bipartite directed graphs with n left vertices and n right vertices and edges going from left to right 1 .In this setting a graph is expanding if every set of left vertices of size t ≤ n/2 has at least (1 + α) • t neighbors for some constant α > 0. Monotonicity (with degree d) is equivalent to saying that the graph edges can be partitioned into d disjoint monotone matchings (that is, in each matching the edges do not cross).
Unlike the question in general graphs, there is no obvious way of even proving the existence of dmonotone expanders for small d.Attempting the probabilistic method one is faced with the choice of distribution on monotone mappings, and then with the analysis.One fundamental problem seems to be that there are only exp(O(n)) partial monotone mappings on [n], whereas there are exp(n log(n)) unrestricted mappings.While it is trivial to prove that O(1) random mappings create an expander with high probability, we have no analogous result for monotone mappings.To give an example of the difficulty of analyzing a random monotone construction consider the following natural way of sampling a monotone mapping: pick two sets S, T ⊂ [n] of size n/2 uniformly at random and map S onto T in a one-to-one way (mapping the i'th element of S to the i'th element of T ).It is easy to see that the set [1, t] will not expand w.h.p for t = Ω(n) since this set will have all of its neighbors in the interval [1, t + O( √ n)] with high probability.
Lets return to explicit constructions (which are anyway more interesting for applications).The first construction of monotone expanders was given in [11] in connection with dimension expanders (we elaborate on this connection below).This construction has logarithmic degree and is composed mainly of shift maps (that is, maps of the form f (x) = x + i).This construction was recently significantly improved by Bourgain [7] which achieves an optimal O(1)-monotone expanders.Independently of Bourgain, in this work, we construct O(log (c) n)-monotone expanders for every integer c (with log (c) (•) denotes the c-times iterated logarithm).Bourgain's construction is quite involved mathematically, using in particular the recent breakthrough on the "Tits Alternative" by Breuillard [8], and its analysis is even more involved, extending work on spectral gaps on the unitary group SU (2) by Bourgain and Gamburd [6] to the group SL 2 (R) (the reference [7] contains only sketches of the construction and analysis).In Section 6 we provide some more details on the workings behind this theorem.Theorem 1.1 (Bourgain [7]).There exists an explicit family of constant degree monotone expanders.
We will give our (suboptimal) construction here for two reasons.First, it is simple: it uses an iterated replacement product [27], observing that with appropriate ordering of vertices it preserves monotonicity of its components.Second, it shows that any (even nonconstructive) existence proof of O(1)-monotone expanders can be used as a base graph (in the style of [2]) to make our construction a fully explicit one of O(1)-monotone expanders.

Monotone expanders and dimension expanders
Monotone graphs, and the question of their expansion for small d arose implicitly in the paper of Dvir and Shpilka [11].They reduce the problem of constructing degree d dimension expanders (proposed in [4]) to the construction of O(d)-monotone expanders, and moreover give explicit construction of O(log n)-monotone expanders of size n, leaving possible improvements as an open question.Since this connection was never given explicitly, we sketch it below.
A degree d dimension expander is a set of d linear mappings T 1 , . . ., T d : F n → F n such that for every linear subspace V ⊂ F n with dim(V ) ≤ n/2 we have dim Here, F is a field, possible finite, and α is a positive constant independent of n.The probabilistic method shows that families of constant degree dimension expanders exist.An explicit construction over fields of characteristic zero was given in [21]. 2n [11] an approach toward constructing dimension expanders over arbitrary fields was given.This approach involved applying a monotone expander graph on the coordinates (that is, permute the coordinates using d monotone mappings).Theorem 1.2 ([11]).If there exists an explicit construction of d-monotone expander graphs than there is an explicit construction of degree d dimension expanders over any field F.
Proof.Let e 1 , . . ., e n ∈ F n denote the standard basis vectors.For each monotone partial function f : [n] → [n] in the expander we define a linear map L f : F n → F n as follows: L f (e i ) = e f (i) if f (i) is defined and L f (e i ) = 0 otherwise.To see why these linear maps are a dimension expander consider a vector space V ⊂ F n of dimension t ≤ n/2.For a non-zero vector v ∈ F n denote by where, for a set S, f (S) is defined as the set of all images of f on elements of S on which f is defined (or the empty set if f is not defined on S).So, using expansion we get that π giving the required bound on the dimension.This theorem combined with Bourgain's construction gives explicit constant degree dimension expanders over any field.
Corollary 1.3.Over any field F, there exists an explicit family of constant degree dimension expanders.

Monotone graphs and multi-pushdown graphs
The second contribution of this paper is the observation of a simple connection between d-monotone graphs and d-pushdown graphs: if the former are expanding then so are the latter (for the same d).This is of consequence since the question of whether O(1)-pushdown graphs have small separators (and thus cannot be expanders) is intimately related (and in some cases equivalent) to several questions in Turing machine complexity.With this observation, Bourgain's result simultaneously proves some conjectures and refutes others in one shot.The fundamental connection to complexity arises from the basic fact [15] that computation graphs of certain Turing machines are O(1)-pushdown graphs.We describe this connection and its consequences below.
A d-pushdown graph [25,24] is a graph on an ordered set of vertices such that, if we order the vertices along the spine of a book, the edges can be drawn on d-pages of the book such that in each page, the edges do not touch (even not at a vertex).These graphs come up naturally as the computation graphs of Turing machines with d tapes and are also a natural subclass of d-page graphs [9].For a formal definition of d-pushdown graphs see Section 5.
Let us recall the definition of a separator: A separator S is a subset of the vertices of a graph G on n vertices such that the vertices outside S can be partitioned into two disjoint sets A and B each of size at most 2n/3 and such that there are no edges between A and B. Observe that a graph with sub-linear size separator is not an expander.The planar separator theorem [17] says that a planar graph has a O( √ n) size separator.Since d-pushdown graphs are generalizations of planar graph it is natural to question whether such graphs also have o(n) separators.This question (conjectured to be true in [25]) turns out to be related to the time needed to simulate a deterministic TM by a non deterministic one3 .A 'segregator' theorem (proving the existence of a weaker object than a separator) is the combinatorial 'heart' of the celebrated separation [24] of non deterministic linear time from deterministic linear time (see also [28] for more recent results in this spirit).
In [12] it was shown that there exist 3-pushdown graphs which are 'almost' expanders, in the sense that every separator must be of size Ω(n/ log (c) (n)) for any constant c.In fact, their construction is very similar to our construction of degree log (c) (n) monotone expanders, and is probably one of the earliest occurrence of what is now referred to as an iterative 'Zig-Zag' construction.In this paper we observe that d-monotone expanders can be transformed easily into d-pushdown expanders with d equal to the degree of the graph.Combining this observation with Bourgain's result we get that there are (explicit) d-pushdown graphs which are expanders, settling an old open problem.We prove the following theorem in Section 5.

Theorem 1.4. If there exists (explicit) d-monotone expanders than there exists (explicit) d-pushdown expanders.
In [28] the assumption that the class of d-pushdown graphs have sub-linear separators was used to derive several complexity theoretic results.In particular, it was shown in [28] that this assumption implies the separation for all time bounds t.The existence of expanding d-pushdown graphs shows that this approach to prove such a separation can not succeed.
Lower bounds on simulation time.It was shown in [12] that the question of whether dpushdown graphs are separable or not (i.e if these graphs have sub-linear size separators) is equivalent to a question regarding the simulation time of certain Turing machines.More formally, let t(n) denote the time it takes a 1-tape online non-deterministic TM to simulate a two tape real-time non-deterministic TM 4 .One can easily show that t(n) = O(n 2 ).The question of whether or not t(n) = o(n 2 ) was shown in [12] to be equivalent to whether or not d-pushdown graphs have small (sub-linear) separators.Combining Bourgain's result with Theorem 1.4 above we have the following corollary.

Organization
In Section 2 we give formal definitions and notation that will be used later on.In Section 3 we give the analysis of the replacement product for monotone graphs.In Section 4 we describe our iterative construction of near-constant degree monotone expanders.In Section 5 we explain the connection between monotone and multi-pushdown graphs.Section 6 contains a brief outline of Bourgain's construction of monotone expanders.We conclude in Section 7 with some open problems and directions for future research.
Let V be a finite totally ordered set.We denote by Part[V ] the set of partial functions f : is monotone if, whenever f is defined at two distinct points x < y ∈ V , we have f (x) < f (y) (in this paper we use 'monotone' to mean 'strictly increasing').Notice that if f is monotone it is also injective and so there exists a (partial are monotone then we can define their composition f • g which is also monotone (the composition is defined on those points x where g(x) ̸ = ⊥ and f (g(x)) ̸ = ⊥).
An ordered directed graph is a pair G = (V, (f 1 , . . ., f d )) with V a finite totally ordered set and f i ∈ Part[V ] where we think of f i (x) as mapping x to its i'th neighbor (if one exists).In the following we will refer to these simply as 'graphs'.We will call d the degree of the graph.A graph neighbor functions and their d/2 inverses).For a graph G = (V, (f 1 , . . ., f d )) and a subset S ⊂ V we denote the boundary of S in G as (we omit the subscript G if it is clear from the context).The edge-expansion of G is denoted by Another notion of expansion is vertex-expansion.
Since G is monotone, the in-degree of a vertex is at most d and so there have to be at least h(G) • |S| neighbors of S that are not in S.
We will require the following simple claim.Proof.The inequality follows from the definition of h(G) and from the fact that, in an inverse-closed graph, the number of edges from any set S to its complement is equal to the number of edges in the opposite direction.

The replacement product of monotone graphs
Let V 1 , V 2 be two finite totally ordered sets.The reverse lexicographical ordering The functions t 1 , . . ., t d are all equal to the same function where the set V 2 is identified in some arbitrary one-to-one way with [D] (again, if f a 2 (a 1 ) = ⊥ then t(a 1 , a 2 ) = ⊥).It is clear from the definitions that, if G 1 and G 2 are monotone, then so is their product G 1 • G 2 .The reason for including many parallel edges in the definition will make sense later when we argue about the expansion properties of the replacement product of two expanders (this is necessary to avoid small cuts).
The proof of the following lemma, bounding the edge-expansion of the replacement product, is essentially the same as the proof for undirected graphs appearing in [2].Since our definitions are more involved we retrace the argument below.

Lemma 3.1 (Replacement product of two monotone expanders). Let
Then H is a monotone graph (w.r.t reverse lexicographical ordering) and Proof.As was observed before, if G 1 and G 2 are monotone then so is G 1 •G 2 .We are thus concerned only with bounding the edge expansion.

Let us denote n
and let I ′′ = V 1 \ I ′ .We partition the set S into two parts, S ′ = ∪ x∈I ′ S x and S ′′ = S \ S ′ .We separate the analysis into two cases.The first is when |S ′ | ≥ 1  10 δ 1 |S|.In this case we will get expansion using the mappings that act on the disjoint copies of V 2 .For every x ∈ I ′ we have (using Claim 2.2) that (we abuse notation and treat S x as a subset of V 2 ).Therefore, using the bound on |S ′ |, we get that We now turn to the case when |S ′ | < 1 10 δ 1 |S|.In this case we have |S ′′ | ≥ (1 − 1 10 δ 1 ) • |S|.We will get expansion using the d identical copies of the map t(x, y) = (f y (x), y).Since for all x ∈ I ′′ we have In particular, since Therefore, by Claim 2.2, we have Consider the corresponding d • M edges in H.Of these d • M edges at most 1 4 δ 1 dD • |I ′′ | come from outside S ′′ and so there are at least 1  4 δ 1 dD • |I ′′ | edges from S ′′ .Among these edges, at most d • |S ′ | can land in S ′ (here we rely on the fact that G 1 is monotone and so the mappings acting on the copies of V 1 are injective).We can bound this number of edges by We can therefore conclude that the number of edges from S ′′ to the complement of S is at least As

An iterative construction with near optimal degree
In this section we describe a simple iterative construction of monotone expanders with almost constant degree.We give our construction in two parts: The first (Section 4.1) gives a construction of a monotone expander with poly-logarithmic degree and the second (Section 4.2) shows how to iterate the construction, using the tools from Section 3, and reduce the degree to log (c) (n) for any constant c.

The base graph
In this section we describe a construction of a monotone expander graph with poly-logarithmic degree which will be the basis of the full construction.The graph we will use is similar to the one used in [11].However, since we are shooting for a strongly explicit graph, we will refrain from using the base graph from [30] (which is not strongly explicit) and will use the expanders of [1,26] instead.The degree of these graphs is poly-logarithmic (instead of logarithmic in [30]) but this difference will 'disappear' in the recursion done in the next section.

Theorem 4.1 ([1, 26] Expanders for Z n
).There exists a constant h 0 > 0 and a strongly explicit family of graphs 1) , h(G) ≥ h 0 and such that for each i there exists an integer a i such that We now turn the above graph into a monotone expander in a similar manner to [11].In fact, the proof of the following theorem implies that any Cayley expander for Z n with degree d gives a monotone expander with degree 2d.

Theorem 4.2 (Monotone expander with poly-log degree). There exists a constant h 1 > 0 and a strongly explicit family of monotone, inverse-closed graphs
with d = log(n) O (1) and h(G) ≥ h 1 .
Proof.W.l.o.g suppose n is even (otherwise we can run the construction on n − 1 vertices and then connect a single vertex arbitrarily).Let m = n/2 and let A m = ({0, 1, . . ., m − 1}, (f 1 , . . ., f d ′ )) be the graph given by Theorem 4.1.Let a 1 , . . ., a d ′ be integers such that, for 0 ≤ x ≤ m − 1, We define the graph B n on vertex set {0, 1, . . ., n − 1} to have d = 2d ′ neighbor functions.The first d ′ functions, call them g 1 , . . ., g d ′ will be defined as Since we can assume w.l.o.g that the integers a i are smaller than m we have that the mappings g i , i ∈ [d ′ ] are indeed monotone.We now add to B n another d ′ mappings g d ′ +1 , . . ., g 2d ′ which will all be the same and will be defined as It is clear that the graphs B n are monotone, have degree log(n) O (1) and are strongly explicit.We can also assume w.l.o.g that B n is inverse closed simply by adding all the inverses (this will decrease the edge expansion by at most half).We now bound the edge expansion of B n .
Let S ⊂ {0, 1, . . ., n − 1} be such that |S| ≤ n/2.We will show that the number of edges leaving S is at least 1  16 • h 0 • d • |S|.Define S 1 = S ∩ {0, 1, . . ., m − 1} and S 2 = S \ S 1 .We separate the analysis into two cases based on the size of the set to the outside of S 1 and so we are done.
We now deal with the case when |I| < h 0 8 • |S|.In this case we must have for otherwise |I| can be bounded from below by We now use the expansion of A m to claim that there are at least edges from S to its complement in this case.This concludes the proof of the theorem.

The iterative construction
We now combine Lemma 3.1 with Theorem 4.2 to give a construction of monotone expanders with degree close to constant.We denote by log (c) (n) the log function iterated c-times.
Theorem 4.3 (Monotone expanders with degree log (c) (n)).For every c > 0 there exists a strongly explicit family of monotone graphs ) .
Proof.We describe an algorithm for producing M k in c iterations.We start with a graph G 1 that has degree log(k) O (1) and at each iteration decrease the degree to the logarithm of the previous degree using the replacement product.
Set G 1 to be B k -the graph given by Theorem 4.2 on input k.Let D 1 be the degree of G 1 , so D 1 = log(k) O (1) .Let B D 1 be given again by Theorem 4.2, this time on D 1 vertices and with degree D 2 = log(D 1 ) O (1) .We can take the replacement product From Lemma 3.1 we see that G 2 is a monotone graph on k • D 1 vertices with degree 2D 2 = (log log(k)) O (1) and where h 1 > 0 is the constant given by Theorem 4.3.
We can continue in this manner c-times.At the j'th iteration we are given a graph G j with degree D j = poly(log (j) (k)) and we take its replacement product with B D j .If we denote by h j the edges expansion at the j'th iteration we see that and so, after c iterations we will have expansion at least Ω(exp exp(c)) −1 .
Tracing the growth of the vertex set size we conclude that the size of the final graph is some explicit function of k of the form ≈ k • poly(log(k)) • poly(log log(k)) • • • poly(log (c) (k)).We can choose a starting integer k ′ , slightly smaller than k, so that the final graph will have size Θ(k).
The part of the theorem claiming strong explicitness can be easily verified by induction on the number of iterations (since the base graph is strongly explicit).
From this last proof we see that, if had an existence proof of constant degree monotone expanders, we could transform it into an explicit construction in the following manner: start with two iterations of the replacement products as we did above, then, do an exhaustive search for a constant degree monotone expander on poly(log log(n)) vertices (this can be done in polynomial time) and use it in the last replacement product.This approach (introduced in [2] to give a simple explicit construction of expander graphs) can be summarized in the next corollary.as was required.We will rely on the fact that the identity mapping belongs to the graph Assume w.l.o.g that k 1 ≥ k 2 .Notice that the neighbors of S 1 belong to the set [n + 1, 2n] and that the neighbors of S 2 are in [n].This means that |Γ(S)| ≥ max{2k 1 , k + αk 2 }, since k 2 expands (it is smaller than n/2) and k 1 is copied by the identity map.If k 2 ≥ k/4 then k + αk 2 ≥ (1 + α/4)k and if k 2 < k/4 than 2k 1 ≥ (1 + α/4)k (recall that α is at most 1).This completes the proof.
Combining Claim 5.3 with Bourgain's construction of constant degree monotone expanders [7] we get the following corollary.

Corollary 5.4.
There exists an integer d 0 and a family of n-vertex undirected graphs G n , with n going to infinity such that G n is a d 0 -pushdown graph and µ(G n ) ≥ α 0 for α 0 > 0 independent of n.Furthermore, the family G n is explicit (can be computed in polynomial time).

Bourgain's monotone expanders
We give here a brief outline of Bourgain's construction of monotone expanders.Let SL 2 (R) denote the group of 2 × 2 real matrices with determinant 1.This group acts6 on the projective line (the inclusion of ∞ allows for division by zero).One can verify that this is indeed a group action.Furthermore, the derivative of these mappings is given by and so, these are all monotone increasing functions on any interval not containing −d/c.Bourgain's construction proceeds to find (explicitly) a set of finite size in SL 2 (R) (in fact, with rational entries) that, roughly speaking, satisfies the following two conditions: 1.The matrices in the set are all close to the identity matrix and so their associated mappings do not move any point in P 1 by much.
2. They satisfy a so-called 'restricted spectral gap' property.This property ensures that applying their associated maps on 'nice' subsets in P 1 will expand their measure by a constant factor.
Combining these two properties and limiting our attention to a specific interval (which is possible due to Property 1.) one gets the following 'continuous' analog of monotone expanders.

Theorem 6.1 ([7]
).There exists α > 0 and an explicit finite family Ψ of smooth increasing maps ψ : The final step in the construction is to discretize the interval [0, 1] into n intervals of size 1/n and to associate to each smooth mapping a discrete mapping from [n] to [n] in a natural way so that expansion in measure is transformed into expansion in set size.The property of 'strong' monotonicity (no collisions) is obtained by observing that, since the derivatives of the smooth mappings ψ used in the construction are bounded from below, one does not have more than a constant number of edges mapping to a single vertex.This allows one to 'break up' each weakly monotone mapping into a constant number of strongly monotone (partial) mappings, while preserving expansion.

Conclusions and Open Problems
This paper wishes to popularize some facts we find remarkable: explicit constant degree expanders exist, whose graph structure is extremely restricted: the vertices are ordered, and edges decompose into few monotone maps, or few legal parenthesis sequences.In particular, these graphs are "nearly planar" in a precise sense.It would be interesting to explore the limitations of the expansion of these graphs.For example, can a constant-degree monotone graph be a lossless expander?a Ramanujan graph?Unique neighbor expander?Consequence of these constructions are new explicit pseudorandom objects (dimension expanders) as well as a better understanding of the computation graphs of various Turing machines.We believe that more applications of these expanders will be found.Specifically, algorithms on restricted models, especially operating with limited access to input, may use be able to utilize these new expanders.
Another remarkable feature of these expanders is that while they have an explicit construction, no known direct (and simpler) existence proof is known.Few pseudorandom structures are known which have an explicit construction but whose existence does not follow from some application of the probabilistic method.We find the question of providing new existence proof of constant-degree monotone expanders very appealing mathematically.Here we also motivate it by showing how any such proof (even highly nonconstructive) directly implies a simple, strongly explicit construction.This reduction uses the Zig-Zag graph product adapted to monotone graphs, further revealing the generality of this paradigm.
In a different direction, while constant degree monotone expanders lead to constant degree dimension expanders via the reduction of Dvir-Shpilka [11], in some sense this construction is not very natural.A possibly more natural construction was conjectured by Wigderson in [29], and we conclude this section with a formal statement of it.Recall the definition of dimension expanders from the introduction 1.1.Conjecture 7.1.[29] Let G be a finite group, S a symmetric set of generators, and assume that the associated Cayley graph Cay(G; S) is an expander -formally the (normalized) 2nd largest eigenvalue of its adjacency matrix is 1/2 (say).Let F be any field of characteristic 0 or positive characteristic relatively prime to |G|.Let ρ be any irreducible representation of G over F, and denote its dimension by n.Assume S = {g 1 , g 2 , . . ., g k } and define T i = ρ(g i ) for all i ∈ [k].Then T 1 , T 2 , . . ., T k : F n → F n form a dimension expander.
We note that the paper of Lubotzky and Zelmanov [21] mentioned above constructs constant degree dimension expanders over fields of characteristic 0 precisely by proving the conjecture above for such fields (and then using any group with a constant number of expanding generators and high dimensional irreducible representations, e.g.SL 2 (p)).The same paper gives purely algebraic motivations for resolving this conjecture for finite fields.
in the following way: with each matrix ordered according to reverse lexicographical ordering and with 2d neighbor functions s 1 , . . ., s d , t 1 , . . ., t d