A Cryptographic Proof of Regularity Lemmas: Simpler Unified Proofs and Refined Bounds

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1 A Cryptographic Proof of Reguarity Lemmas: Simper Unified Proofs and Refined Bounds Maciej Skórski IST Austria Abstract. In this work we present a short and unified proof for the Strong and Weak Reguarity Lemma, based on the cryptographic technique caed ow-compexity approximations. In short, both probems reduce to a task of finding constructivey an approximation for a certain target function under a cass of distinguishers (test functions), where distinguishers are combinations of simpe rectange-indicators. In our case these approximations can be earned by a simpe iterative procedure, which yieds a unified and simpe proof, achieving for any graph with density d and any approximation parameter ɛ the partition size a tower of 2 s of height O ( dɛ 2) for a variant of Strong Reguarity a power of 2 with exponent O ( dɛ 2) for Weak Reguarity The novety in our proof is as foows: (a) a simpe approach which yieds both strong and weaker variant, and (b) improvements for sparse graphs. At an abstract eve, our proof can be seen a refinement and simpification of the anaytic proof given by Lovasz and Szegedy. Keywords: reguarity emmas, boosting, ow-compexity approximations, convex optimization, computationa indistingusiabiity 1 Introduction Szemeredi s Reguarity Lemma was first used in his famous resut on arithmetic progressions in dense sets of integers [Sze75]. Since then, it has emerged as an important too in graph theory, with appications to extrema graph theory, property testing in computer science, combinatoria number theory, compexity theory and others. See for exampe [DLR95, FK99, HMT88] to mention ony few. Roughy speaking, the emma says that every graph can be partitioned into a finite number of parts such that the edges between these pairs behave randomy. There are two popuar forms of this resut, the origina resut referred to as the Strong Reguarity Lemma and the weaker version deveoped by Frieze and Kannan [FK99] for agorithmic appications. The purpose of this work is to give yet another proof of reguarity emmas, based on the cryptographic notion of computationa indistinguishabiity. Supported by the European Research Counci Consoidator Grant ( TOCNe).

2 2 We don t revisit appications as it woud be beyond the scope. For more about appications of reguarity emmas, we refer to surveys [KS96, RS, KR02] From now, G is a fixed graph with a vertex set V (G) = V and the edge set E(G) = E V 2. By a partition of V we understand every famiy of disjoint subsets that cover V. The rest of the paper is organized as foows: the remaining part of this section introduces necessary notions (Section 1.1), states reguarity emmas (Section 1.2), and summarizes our contribution (Section 1.3). In Section 2 we show how to obtain strong reguarity and in Section 3 we dea with weak reguarity. We concude our work in Section Preiminaries By the edge density of two vertex subsets we understand the fraction of pairs covered by graph edges. Definition 1 (Edge density). For two disjoint subsets T, S of a given graph G we define the edge density of the pair T, S as d G (T, S) = E G(T, S) T S We sighty abuse the notation denoting d G = d G (V, V ) for the graph density. (1) Sets Reguarity The notion of set irreguarity measures the difference between the number of actua edges and expected edges as if the graph was random. Note that for a random bipartite graph with a bipartirtion (T, S) we expect that for amost a subsets S, T roughy the same fraction of vertex pairs is covered by graph edges. The deviation is precisey measured as foows Definition 2 (Irreguarity [LS07, FL14]). The irreguarity of a pair (S, T ) of two vertex subsets is defined as irreg G (S, T ) = max S S,T T E(S, T ) d G (S, T ) S T If this quantity is a sma fraction of S T then the edge distribution is homogeneous or, if we want, random-ike. In turn, two vertex subsets are caed reguar if the density is amost preserved on their (sufficienty big) subsets 1 Definition 3 (Reguarity). A pair (S, T ) of two disjoint subsets of vertices is said to be ɛ-reguar, if d G (S, T ) d G (S, T ) ɛ for a S S, T T such that S ɛ S, T ɛ T. 1 The requirement of being sufficienty big is to make this notion equivaent with the irreguarity above.

3 3 For competeness we mention that irreguarity and reguarity are pretty much equivaent (up to changing ɛ) Remark 1 (Irreguarity vs Reguarity). It it easy to see that irreg G (S, T ) ɛ S T is impied by ɛ-reguarity, and it impies ɛ 1 3 -reguarity. Partition Reguarity The next important objects are reguar partitions, for which amost a pairs of parts are reguar. Note that irreguar indexes are weighted by set sizes, to propery address partitions with parts of different size. Definition 4 (Reguar Partitions). A partition V 1,..., V k of the vertex set is said to be ɛ-reguar if there is a set I V V such that V i V j ɛ V 2 (i,j) I and for a (i, j) I the pair (V i, V j ) is ɛ-reguar. We say that a partition is equitabe (or simpy: is an equipartition) if any two parts differ in size by at most one. Note that for equitabe partitions the above conditions simpy means that a but ɛ-fraction of pairs are reguar. There is aso a notion of partition irreguarity based on sets irreguarity Definition 5 (Partition Irreguarity). The irreguarity of a partition V = {V 1,..., V k } is defined to be irreg(v) = i,j irreg G(V i, V j ). Remark 2 (Partition Irreguarity vs Partition Reguarity). Again it it easy to see that both notions are equivaent up to a change in ɛ. Concretey, ɛ-reguarity is impied by irreguarity smaer than ɛ 4 V 2 and impies ɛ-irreguarity [FL14]. The partition size in the Strong Reguarity Lemma grows as fast as powers of twos. For competeness, we state the definition of the tower function. Definition 6 (Power tower). For any n we denote 1.2 Reguarity Lemmas 2 T (n) = }{{} n times Summary of the state of the art Having introduced necessary notation, we are now in position to state reguarity emmas. There is a strong (origina) and weak variant of the reguarity emma (deveoped ater for agorithmic appications), which differ dramaticay in the partition size. The strong variant has a few sighty reaxed statements, which are more convenient for appications and simper to prove. These versions are equivaent up to a repacing ɛ by ɛ O(1). The state of the art is that the variant of Strong Reguarity Lemma (Theorem 2 beow) and the Weak Reguarity Lemma (Theorem 4 beow) are tight in genera, as shown recenty 2 in [FL14]. For the sake of the competeness we note that there are more works offering the proofs for Reguarity Lemmas, for exampe [Fri99] but they are not discussed here as they do not achieve optima bounds. 2 Worse bounds were known before for exampe [Gow97].

4 4 Strong Reguarity The origina variant of the Strong Reguarity Lemma simpy says that there is aways an equipartition such that amost every pair of parts is reguar, and the partition size is not dependent on the graph size. Theorem 1 (Strong Reguarity Lemma, origina variant 1). For any graph G there exists a partition V 1,..., V k of vertices such that for a up to ɛ-fraction of pairs (i, j) E(S, T ) d G (V i, V j ) S T ɛ V i V j for any S V i, T V j such that S ɛ V i, T ɛ V j. Moreover, the size of partition is at most a power of twos of height poy(1/ɛ). It has been observed that proofs are much easier when one works with the tota irreguarity, rather than separate bounds for each pair. The foowing version is equivaent (up to changing ɛ) Theorem 2 (Strong Reguarity Lemma, variant 2 [FL14]). For any graph G there exists a partition V 1,..., V k of the vertices such that irreg G (V i, V j ) ɛ V 2. (2) i<j Moreover, the partition size k is a power of twos of ength O(ɛ 2 ). The reguarity emma can be aso formuated as an approximation by a weighted graph. Theorem 3 (Strong Reguarity Lemma, variant 3 [LS07]). For any graph G there exists a partition V 1,..., V k of the vertices and rea numbers d i,j such that max E(S, T ) d i,j S T ɛ V 2, (3) S V i,t V j i<j and moreover the partition size k is at most a tower 3 of twos of height O(ɛ 2 ). Weak Reguarity Finay, we state the weaker version obtained by Frieze and Kannan Theorem 4 (Weak Reguarity Lemma). For any graph G there exists a partition V 1,..., V k of the vertices such that E(S V i, T V j ) d i,j S V i T V j i,j i,j ɛ V 2 (4) for a S, T. Moreover, the partition is generated 4 by O ( ɛ 2) subsets of V. In particuar, k is at most 2 O(ɛ 2). 3 The origina work [LS07] proves a bound being O(ɛ 2 ) iterations of the function s(1) = 1, s(k + 1) = 2 s(1)4...s(k) 4 starting at 1. It is easy to see that s(k) can be bounded by a tower of height k + O(1). 4 The generated partition arises as intersections of the generating sets with their compements.

5 5 1.3 Our contribution and reated works We present a simpe proof of both Reguarity Lemmas, using the cryptographic framework of ow compexity approximations. Our contribution is twofod: (a) conceptua, as we show how the Reguarity Lemmas can be written and easy proved using the notion of indistinguishabiity, and (b) technica, as we improve known bounds by a factor equa to the graph density. We eaborate more on out techniques and resuts beow. A Simper Proof. Our proof uses ony a naive optimization agorithm, avoiding combinatoric cacuations using energy arguments based on Cauchy-Schwarz inequaities, that appear in other proofs ike [FL14]. Quantitative Improvements For the Strong Reguarity Lemma we bound the partition size by a tower of twos of height O(ɛ 2 d G ) which is an improvement by a factor of d G over best resuts [FL14]. Simiary, for the Weak Reguarity Lemma we prove that the partition is an overay of O(ɛ 2 d G ) subsets (in particuar has up to 2 O(ɛ 2 d G ) members) which is again an improvement by a factor of d G comparing to best bounds [FL14]. Note that for constant densities d G, this matches both best upper and ower bounds [FL14]. Our improvements for smaer densities doesn t contradict the ower bounds as they depend on the density in a compicated and non-expicit way (and hence don t appy to a regimes of d G ). Reguarity Lemmas as Low Compexity Approximations We show that a variant of the Szemeredi Reguarity Lemma, equivaent to the most often used statement, can be written in the foowing form f F : E g(e)f(e) E h(e)f(e) ɛ (5) e X e X for some functions g, f and a cass of functions F on a finite set X, where h is efficient in terms of compexity. More precisey, the resut states that a given function f (in our case reated to the irreguarity of the graph) can be efficienty approximated under a certain cass of test functions (caed aso distinguishers). In cryptography resuts of this sort are known as ow compexity approximations and are a powerfu and eegant technique of proving compicated resuts [TTV09,VZ13,JP14]. The quantity in the absoute vaues in Equation (5) is referred to as the advantage of f in distinguishing g and h, so the statement simpy means that h is indistinguishabe from g for sma ɛ by a functions in F. Depending on the cass F it may be a good repacement for g in appications. In our case the cass of test functions changes depending on the probem. For weak reguarity we use rectange indicator functions, whereas for strong reguarity we consider combinations of rectange-indicator functions F = {f : f = ±1 T S } F = f : f = i,j ±1 Ti,j S i,j (for Weak Reguarity) (for Strong Reguarity)

6 6 The proof is in both cases very simpe and can be viewed as a specia case of the genera subgradient descent agorithm we known in convex optimization 5. The agorithm is given beow in pseudocode (see Agorithm 1) Agorithm 1: Low Compexity Approximations Input : target function g to approximate, cass of test functions F, a starting point h 0, accuracy parameter ɛ, stepsize t Output: function h of ow compexity w.r.t F and indistinguishabe from g (with respect to test functions F) 1.1 n whie can distinguish h n and g by some f F with advantage ɛ do 1.3 n n h n h n 1 t f A simiar resut has been shown by Trevisan et a. [TTV09] with respect to the weak reguarity emma. It turns out that the weak reguarity emma can be directy transated to a form of Equation (5). The case of the Strong Reguarity Lemma is however a bit different, because the standard statement doesn t admit a direct transation to Equation (5) so we need first to reduce the Reguarity Lemma to a sighty reaxed form simiar 6 to Theorem 2 and prove the reaxed statement by ow compexity approximation toos. Aso, the same cass of functions appear in the anaytic proof in [LS07] but in a different approximation technique. Abstracting the concept of pseudo-reguarity In the Weak Reguarity Lemma, we measure the irreguarity of the partition as average difference between the actua number of edges and the expected number of edges across the pairs of parts of the partition. Therefore, the Weak Reguarity Lemma is obtained from the bound E(T i, S j ) d i,j T i S j V 2 i,j i,j (where T i, S j are subsets of V i and V j respectivey; note that i,j E(T i, S j ) = E(T, S)). In turn, to prove the Strong Reguarity Lemma, we measure the av- 5 If we consider the mapping h max f E g(e)f(e) E h(e)f(e) then its subgradient equas f for some f F. Then the update is h := h t f precisey as in e X e X the proof of Section The reaxed form we use is except that we aow any numbers d i,j in pace of densities d G(V i, V j).

7 7 erage absoute difference between the actua number of edges and the expected number of edges. To prove our resut we introduce the foowing condition (for some constants d i,j ) E(T i,j, S i,j ) d i,j T i,j S i,j V 2. i,j (S i,j, T i,j being subsets of V i and V j respectivey), and refer to this property as pseudo-reguarity 7. This condition extends sighty the notion of irreguarity, where the true densities of pairs (V i, V j ) appear in pace of d i,j. Note that pseudo-reguarity can be understood as approximating the graph by a weighted graph, where we contro the absoute deviation of number of edges across pairs of partition parts. The approach with unrestricted constants is much easier to prove and is more fexibe. In fact, the idea of reaxing restrictions on densities (equivaenty: considering a weighted graph) goes back to [FK99]. The concept of pseudoreguarity is what aows us to connect the approximation emma with the Strong Reguarity Lemma. 1.4 Proof techniques The key ingredient of our proof is a descent agorithm, which transated back to the partition anguage is simiar to the popuar technique of proving reguarity emmas. As ong as the current partition fais to satisfy the desired property, the agorithm uses sets being counterexampes to refine the partition. Moreover, we show that a certain quantity, caed the energy function, decreases with every step by a constant (depending on ɛ). From this one concudes that the process of refining the partition hats after a number of step (the bound depends on concrete energy estimates). Our proof is different with respect to the energy function, as we use simpy the eucidean distance (second norm) between the candidate soution and the target. This aows us to decrease the number of rounds by the initia distance, which in our case equas d G, as we start from f = 1 E (where E is the edge set) and g = 0. An overview of the proof (of the Strong Reguarity Lemma) is iustrated in Figure 1. Indistinguishabiity A tower of height ɛ 2 d G (decent agorithm, Section 2.1) Pseudo-reguarity (impies Theorem 3) ɛ repaced by ɛ 0.25 (Section 2.2) Reguarity A constant oss (Section 2.3) Equiparition (impies Theorem 1) Fig. 1. An overview of our proof of the Strong Reguarity Lemma. 7 This property was aso impicity used in [LS07]

8 8 The proof of the Weak Reguarity Lemma is even simper and consists of ony first step (with the cass of test functions changed accordingy). 1.5 Organization In Section 2 we prove a variant of the Strong Reguarity Lemma, in Section 3 we prove the Weak Reguarity Lemma and concude the work in Section 4. 2 Strong Reguarity Lemma 2.1 Obtaining a partition with sma pseudo-irreguarity The key ingredient is the foowing approximation resut, proved by the technique sketched in Agorithm 1. Theorem 5 (Simuating against stepwise functions). For any rea function g on V 2 and any ɛ > 0, there exists a partition V 1,..., V k and a piece-wise function h constant on rectanges V i V j such that h and g are ɛ-indistinguishabe by functions piecewise constant on subrectanges of V i V j where i j F k = f = a i,j 1 Si,j T i,j : a i,j = ±1, S i,j V i, T i,j V j, (6) i j where indistinguishabiity means f F k : h(e)f(e) g(e)f(e) ɛ V 2, (7) e V 2 e V 2 and moreover k is not bigger than O(dɛ 2 ) iterations of the function k k 2 k+1 at k = 1, where d = 1 V 2 e V 2 g(e)2. In particuar, k is at most a tower of 2 s of height O ( dɛ 2). Remark 3 (Symmetrizing cass F). Note that ordering pairs (i j) in the definition of cass F is crucia to obtain the compexity being a power of 2. Otherwise, we woud obtain a (much worse) tower of 4 s of the same height. Remark 4. It is easy to see that the function is a power-tower of twos of height O(d G δ 2 ) (a forma proof can obtained by induction as in [FL14]. As a coroary we obtain the foowing statement which is precisey the variant stated in Theorem 3. Coroary 1 (Reguarity Lemma in terms of pseudo-reguarity (variant 3)). For any graph G there is a partition of vertices V such that the absoute pseudo-irreguarity is at most ɛ V 2, that is for some numbers d i,j we have max E(T, S) d i,j T S ɛ V 2 (8) S V i,t V j (i,j):i j k and moreover, the number of partition parts is is a power-tower of twos of height O(d G δ 2 ).

9 Proof (Proof of Coroary 1). It suffices to appy Theorem 5 to g = 1 E and h = 0. We have then e g(e)t(e) = i j a i,je(s i,j, T i,j ) and e h(e)t(e) = i j a i,jd i,j S i,j T i,j. The absoute vaues in Equation (8) are achieved by fitting signs of the coefficients a i,j = ±1. Proof (of Theorem 5). Suppose we have a function h on a partition V 1,..., V k which is δ V -indistingushabe from g by a function f piecewise constant on squares T i S j, that is (g(e) h(e))f(e) δ V 2 (9) e 9 Consider now h = h + t f and note that (h (e) g(e)) 2 = (h(e) h(e)) 2 2t e e e (g(e) h(e))f(e) + t 2 e f(e) 2. Setting t = δ in the above equation, by Equation (9) we obtain (h (e) g(e)) 2 (h(e) h(e)) 2 δ 2 V 2, e e which means that by repacing h by h we decrease the distance to g by δ 2 V 2. Since our first choice for h is the zero function, the initia distance was equa to e g(e)2 = d V 2 and the oop ends after at most O(dδ 2 ). Regarding the compexity of h = h+t i j a i,j1 Si,j T i,j note that when adding step functions 1 Si,j T i,j, any fixed partition member V i is intersected by at most k + 1 sets of the form S i,j or T i,j (because we consider ony ordered pairs i j!). Therefore, the function h is piecewise constant on the partition V generated by V and sets S i,j, T i,j which has at most k 2 k+1 members. 2.2 Sma pseudo-irreguarity impies reguarity In this section we show that pseudo-reguarity impies reguarity in the sense of Definition 3. Proposition 1. Suppose that for a partition V 1,..., V k of V there exist numbers d i,j such that E(S i,j, T i,j ) d i,j T i,j S i,j ɛ 4 V 2 (10) i,j k for a disjoint rectanges T i,j S i,j V i V j. Then the partition is 2ɛ-reguar. Proof. Rewrite Equation (10) as i,j k S i,j T i,j V 2 d G (S i,j, T i,j ) d i,j ɛ 4

10 10 In particuar, we get i,j k V i V j V 2 d G (S i, T j ) d i,j ɛ 2 (11) when S i,j, T i,j ɛ V for a i. Let S i,j, T i,j (both bigger than ɛ V ) maximize d G (S i,j, T i,j ) d i,j. By the Markov inequaity (appied to the probabiity weights p i,j = Vi Vj V ), there exists an exceptiona set I {1..k} 2 such that 2 and (i,j) I V i V j ɛ V 2. (i, j) I : d G (S i,j, T i,j) d i,j ɛ By the choice of the pairs (S i,j, T i,j ) this impies d G(S i,j, T i,j ) d i,j ɛ for every pair S i,j V i, T i,j V j (provided that (i, j) I. In particuar, this is true with S i,j = V i and T i,j = V j which gives d G (V i, V j ) d i,j ɛ. By the triange inequaity we have d G (S i,j, T i,j ) d G V i V j 2ɛ for (i, j) I which finishes the proof. 2.3 Enforcing equipartition To compete the ast step of the proof we have to prove the foowing Lemma 1. For any ɛ-reguar partition V there exists a O(ɛ)-reguar equipartition W of size W = O ( ɛ 1 V ). The key observation is the foowing usefu fact, which simpy states that reguarity is preserved under refinements. A simpe proof is given in Appendix A. Lemma 2 (Reguarity preserved under refinements). For any graph G, if (S, T ) is ɛ-reguar and S S, T T, then (S, T ) is 2ɛ-reguar. Consider now a coarser partition {V i,i } i,i such that for every i the set V i is partitioned into k(i) k ɛ parts V i,i where i = 1,..., k(i) which are a, up to one, of equa size V V i,i =, i = 1,..., k(i) 1 V V i,i <, i = k(i) Let V = V k(i). In other words, the set V combines a residua parts into i one component. We partition W again into equa (except one) parts V 1,..., V r

11 11 so that V i = V r < V, i = 1,..., r 1 V Therefore, the famiy i=1,...,k i =1,...,k(i) 1 {V i,i } i,i i=1,...,r is a partition of V that has members, 1 of them being of size being a remainder of size smaer than to be of size at east V ( 1) V V {V i } (12) V and one. It foows that the ast term has, that is between V and V ( 1). Now by moving up to one eement from each of the other 1 components to the remaining component we arrive at an equipartition W 1,..., W where a members are of equa size up to one eement, that is W i W j 1 (13) Note that we moved from sets V i to V at most k V = O(ɛ V ) vertices, which by Equation (12) beong to at most O(ɛ) parts W j. Therefore Caim (Partition W i is a refinement of V i up to a sma fraction of members). For a up to a O(ɛ)-fraction of pairs (i, j) {1,..., } 2, the sets W i, W j are subsets of some pair V i, V j. Let I W be the set of a pairs (i, j) such that the pair (W i, W j ) is not ɛ-reguar, and et I V be the set of pairs (i, j) such that (V i, V j ) is not ɛ-reguar. W i W j ɛ V 2 + W i W j (14) (i,j) I W (i,j):w i V i,w j V j V i V j (15) (i,j) I V O ( ɛ V 2), (16) where the first ine foows by the ast caim and the fact that W i are disjoint, the second ine foows by the reguarity of the partition V i. Now Equation (13) impies I W = O(ɛ 2 ). 3 Weak Reguarity Lemma Theorem 6 (Simuating against rectange-indicator functions). For any function g : V 2 [ 1, 1], and any ɛ > 0, there exists a partition V 1,..., V k and

12 12 a piece-wise function h constant on squares V i V j such that f and g are ɛ- indistinguishabe by indicators of rectanges V i V j where i j F = {f = ±1 S T : S V i, T V j }, (17) that is f F : h(e)f(e) g(e)f(e) ɛ V 2. (18) e V 2 e V 2 Moreover, k is not bigger than 2 O(d Gɛ 2). In fact, the partition is an overay of O(d G ɛ 2 ) subsets of vertices. By appying this resut to the function 1 E on V 2 (being 1 for pairs e = (v 1, v 2 ) which are connected and 0 otherwise) we reprove Theorem 4 Coroary 2 (Deriving Weak Reguarity Lemma). The Weak Reguarity Lemma hods with k = O(d G ɛ 2 ). This resut, without the factor d G was proved in [TTV09]. We skip the proof of Theorem 6 as it merey repeats the argument from Theorem 5, noticing ony that the cacuation of k is different because the cass F is now simper. Note aso that for this resut the cass F doesn t change with every round. 4 Concusion We have shown that both: weak and strong reguarity emmas can be written as indistinguishabiity statements, where the edge indicator function is approximated by a combination of rectange-indicator functions. This extends the resut of Trevisan at a. for weak reguarity to the case of Strong Reguarity Lemma. Moreover, due to a different anaysis of the underying descent agorithm, our proof achieves quantitative improvements graphs with ow edge densities. References DLR95. FK99. FL14. Fri99. Richard A. Duke, Hanno Lefmann, and Vojtech Rd, A fast approximation agorithm for computing the frequencies of subgraphs in a given graph., SIAM J. Comput. 24 (1995), no. 3, Aan M. Frieze and Ravi Kannan, Quick approximation to matrices and appications., Combinatorica 19 (1999), no. 2, Jacob Fox and Lászó Mikós Lovász, A tight ower bound for szemerédi s reguarity emma, CoRR abs/ (2014). Kannan Ravi Frieze, Aan, A simpe agorithm for constructing szemerdi s reguarity partition., The Eectronic Journa of Combinatorics [eectronic ony] 6 (1999), no. 1, Research paper R17, 7 p. Research paper R17, 7 p. (eng).

13 13 Gow97. W.T. Gowers, Lower bounds of tower type for szemerédi s uniformity emma, Geometric & Functiona Anaysis GAFA 7 (1997), no. 2, HMT88. András Hajna, Wofgang Maass, and György Turán, On the communication compexity of graph properties, Proceedings of the Twentieth Annua ACM Symposium on Theory of Computing (New York, NY, USA), STOC 88, ACM, 1988, pp JP14. Dimitar Jetchev and Krzysztof Pietrzak, How to fake auxiiary input, TCC 2014, San Diego, CA, USA, February 24-26, Proceedings (Yehuda Linde, ed.), Lecture Notes in Computer Science, vo. 8349, Springer, 2014, pp KR02. Y. Kohayakawa and V. Rd, Szemeredi s reguarity emma and quasirandomness, KS96. Jnos Koms and Miks Simonovits, Szemerdi s reguarity emma and its appications in graph theory, LS07. Lászó Lovász and Baázs Szegedy, Szemerédi s emma for the anayst, Geom. Funct. Ana. 17 (2007), no. 1, MR MR (2008a:05129) RS. Vojtch Rd and Mathias Schacht, Reguarity emmas for graphs. Sze75. E. Szemerdi, On sets of integers containing no k eements in arithmetic progression, TTV09. Luca Trevisan, Madhur Tusiani, and Sai Vadhan, Reguarity, boosting, and efficienty simuating every high-entropy distribution, Proceedings of the th Annua IEEE Conference on Computationa Compexity (Washington, DC, USA), CCC 09, IEEE Computer Society, 2009, pp VZ13. Sai Vadhan and CoinJia Zheng, A uniform min-max theorem with appications in cryptography, Advances in Cryptoogy CRYPTO 2013 (Ran Canetti and JuanA. Garay, eds.), Lecture Notes in Computer Science, vo. 8042, Springer Berin Heideberg, 2013, pp (Engish). A Proof of Lemma 2 Proof. Let d be the edge density of the pair (T, S) and d be the edge density of the pair (T, S ). Denote ɛ = irreg G (T, S). For any two subsets T T, S S, which are aso subsets of T and S respectivey, by the definition of d we have which transates to E(T, S ) T S d ɛ. Therefore, by Equation (19) and the triange inequaity d d ɛ. (19) E(T, S ) d T S E(T, S ) d T S + ɛ T S. (20) Since the definition of d appied to T T, S S impies from Equation (20) we concude that which finishes the proof. E(T, S ) d T S ɛ T S, E(T, S ) d T S 2ɛ T S,