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A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag product Avraham Ben-Aroya Avraham Ben-Aroya Amnon Ta-Shma Amnon Ta-Shma Tel-Aviv University Tel-Aviv University

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Expander graphs - Sparse graphs with strong connectivity - Fundamental objects in Math and CS Applications in: Communication networks Communication networks Derandomization Derandomization Error correcting codes Error correcting codes PCPs PCPs Proof complexity Proof complexity …

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Properties of expanders Many pseudorandom properties: - No small cuts - Every not-too-large set expands - Random walks mix fast - Spectral gap - … Challenge: Explicit constructions of good expanders Explicit constructions of good expanders

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Spectral gap 00…1/D0 01/D…01/D … 1/D0…1/D0 01/D…00 Eigenvector basis v 1,…,v n, eigenvalues 1 =1 … n Eigenvector basis v 1,…,v n, eigenvalues 1 =1 … n Denote 2 (G)=max{| 2 |,| n |}. Denote 2 (G)=max{| 2 |,| n |}. Smaller 2 (G) better expansion Smaller 2 (G) better expansion 1 = n D-regular graphoperator spectrum

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What is the optimal spectral gap? [AlonBoppana] : [AlonBoppana] : Any D-regular graph satisfies Any D-regular graph satisfies 2 (G) 2 (D-1) /D – o(1) 2D -1/2 2 (G) 2 (D-1) /D – o(1) 2D -1/2 o(1) 0 as the number of vertices grows [Friedman] : [Friedman] : Random D-regular graphs satisfy Random D-regular graphs satisfy 2 (G) 2 (D-1) /D + 2 (G) 2 (D-1) /D + A graph is Ramanujan if 2 (G) 2 (D-1) /D A graph is Ramanujan if 2 (G) 2 (D-1) /D

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Expander constructions ConstructionType 2 (G) 2 (G) [ LubotzkyPhilipsSarnak86, Margulis88,Morgenstern94 ] AlgebraicRamanujan 2 (G) 2 (D-1) /D 2 (G) 2 (D-1) /D 2D -1/2 2D -1/2 [ReingoldVadhanWigderson00]Combinatorial O(D -1/3 )

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Why do we look for combinatorial constructions? A fundamental object in CS deserves a combinatorial proof [RVW00] A fundamental object in CS deserves a combinatorial proof [RVW00] Central component in: Central component in: Good combinatorial expanders [CRVW02]Good combinatorial expanders [CRVW02] Undirected connectivity in L [Reingold05]Undirected connectivity in L [Reingold05] Cayley expanders [ALW01, MW02, RSW04]Cayley expanders [ALW01, MW02, RSW04]

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Expander constructions ConstructionTypeExplicitness 2 (G) 2 (G) [ LPS86,Margulis88, Morgenstern94 ] AlgebraicFullRamanujan, 2D -1/2 2D -1/2 [ReingoldVadhan Wigderson00] CombinatorialFull O(D -1/3 ) Full= The neighbors of a vertex are computable in time polylog(|V|)

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Expander constructions ConstructionTypeExplicitness 2 (G) 2 (G) [ LPS86,Margulis88, Morgenstern94 ] AlgebraicFullRamanujan, 2D -1/2 2D -1/2 [ReingoldVadhan Wigderson00] CombinatorialFull O(D -1/3 ) [BiluLinial04]CombinatorialMild O(D -1/2 log 1.5 D)= D -1/2+o(1) Mild= The graph is computable in time poly(|V|)

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Expander constructions ConstructionTypeExplicitness 2 (G) 2 (G) [ LPS86,Margulis88, Morgenstern94 ] AlgebraicFullRamanujan, 2D -1/2 2D -1/2 [ReingoldVadhan Wigderson00] CombinatorialFull O(D -1/3 ) [BiluLinial04]CombinatorialMild O(D -1/2 log 1.5 D)= D -1/2+o(1) Our result CombinatorialFull D -1/2+o(1)

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Outline The result The result The technique: The technique: a new variant of the zig-zag product a new variant of the zig-zag product

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The construction scheme [ReingoldVadhanWigderson00] Small expanderIncrease size Improve 2 Reduce degree

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The construction scheme (N,D, ) = A D-regular graph with N vertices and 2 (G) = (N,D, ) = A D-regular graph with N vertices and 2 (G) = Building blocks: Tensor: (N,D, ) (N,D, )=(N 2,D 2, ) Tensor: (N,D, ) (N,D, )=(N 2,D 2, ) Squaring: (N,D, ) 2 =(N,D 2, 2 ) Squaring: (N,D, ) 2 =(N,D 2, 2 ) Zig-Zag: (N,D G, G ) (D G,D H, H )=(N D G,D H 2, O( G + H )) Zig-Zag: (N,D G, G ) (D G,D H, H )=(N D G,D H 2, O( G + H )) Bottleneck in [RVW]Bottleneck in [RVW] Our contribution: k-step Zig-Zag: Our contribution: k-step Zig-Zag: (N,D G, G ) k (D G 4k,D H, H )=(N D G 4k,D H k, O( G + H k-1 )) (N,D G, G ) k (D G 4k,D H, H )=(N D G 4k,D H k, O( G + H k-1 )) z z

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Heart of [ReingoldVadhanWigderson00] the zig-zag product G H zig-zag G zigzag H (N,D G, G ) (D G,D H, H ) (N·D G, D H 2, H + G ) vertices degree 2

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The replacement product

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Slightly incomplete….

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The replacement product For simplicity assume G is D G edge-colorable

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The replacement product Cloud

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Vertices: same as in replacement product Vertices: same as in replacement product Edges: (v,u) E there is a path of length 3 on the replacement product such that: Edges: (v,u) E there is a path of length 3 on the replacement product such that: The first step is inside a cloudThe first step is inside a cloud The second step is inter-cloudThe second step is inter-cloud The third step is inside a cloudThe third step is inside a cloud The Zig-Zag product

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vu Example: v and u are connected

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Formally: spectral analysis – not in this talk Formally: spectral analysis – not in this talk Intuitively: analyze entropy flow Intuitively: analyze entropy flow Entropy- a measure of randomness (H 2 ( )=-log Pr x,y~ [x=y]) If G is D-regular then H 2 (G( )) H 2 ( )+log(D) When G has a good spectral gap this almost tight Intuition vs. formality

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Intuitive analysis: start with some - a entropy deficient distribution on its vertices Intuitive analysis: start with some - a entropy deficient distribution on its vertices Show that H 2 (G( ))>>H 2 ( ) Show that H 2 (G( ))>>H 2 ( ) Well analyze only one illuminating distribution Well analyze only one illuminating distribution Formal analysis follows a similar argumentFormal analysis follows a similar argument Intuition vs. formality

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Why does the zig-zag work? cloud The graph after replacement

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Why does the zig-zag work? cloud A uniform distribution over a subset of clouds H step G step H step Wasted

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Why does the zig-zag work? cloud H step G step H step

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Why does the zig-zag work? cloud H step G step H step

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Why does the zig-zag work? Buffer point-of-view Cloud-labelEdge-label [N] [D G ] A vertex in the new graph: A H-step: H

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Why does the zig-zag work? Buffer point-of-view Cloud-labelEdge-label [N] [D G ] A vertex in the new graph: A G-step: G 93 3 For simplicity assume G is D G edge-colorable

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Why does the zig-zag work? Buffer point-of-view Cloud-labelEdge-label The support of the distribution [N] [D G ] A uniform distribution over a subset of clouds: The marginal on is uniform Conditioned on every cloud in the support, is uniform H step G step H step

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Why does the zig-zag work? Buffer point-of-view H step G step H step Wasted Cloud-labelEdge-label H does nothing Cloud=kUniform

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Edge-label Why does the zig-zag work? Buffer point-of-view H step G step H step Cloud-labelEdge-label - Applying G is just taking a step from according to - Since G is an expander, after this step, should have more entropy - G is a permutation, hence the entropy in is reduced

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Why does the zig-zag work? Buffer point-of-view H step G step H step Cloud-labelEdge-label H adds entropy! Cloud=k

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Spectral gap of the zig-zag We take two H-steps, hence the degree is D H 2. We take two H-steps, hence the degree is D H 2. On some inputs, only one of the two steps is useful This incurs a quadratic loss in 2. On some inputs, only one of the two steps is useful This incurs a quadratic loss in 2.

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First Attempt: zig-zag with 3 cloud steps v u

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The problem cloud The graph after replacement

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The problem cloud A uniform distribution over a subset of clouds H step G step H step G step H step Wasted

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The problem cloud H step G step H step G step H step

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The problem cloud H step G step H step G step H step cloud

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Ideally: H step G step H step G step H step cloud

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The problem cloud H step G step H step G step H step cloud

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The problem cloud H step G step H step G step H step Potentially: 1 out of 2 cloud steps is wasted Wasted

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The problem: Buffer point-of-view H step G step H step G step H step A uniform distribution over a subset of clouds Cloud-labelEdge-label Cloud=kUniform H does nothing

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Edge-label The problem: Buffer point-of-view Cloud-labelEdge-label H step G step H step G step H step - Applying G is just taking a step from according to - Since G is an expander, after this step, has more entropy - G is a permutation, hence the entropy in is reduced

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H step G step H step G step H step The problem: Buffer point-of-view Cloud-labelEdge-label H adds entropy! Cloud=k

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H step G step H step G step H step Edge-label The problem: Buffer point-of-view Cloud-label Since the second component is not uniform, we dont know how the entropy flows in the G step (this is like taking a random step over the graph according to an unknown distribution). Edge-label ? ?

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The problem: Buffer point-of-view Cloud-labelEdge-label Cloud=kUniform We might be in this case, in which H does nothing Potentially: 1 out of 2 cloud steps is wasted H step G step H step G step H step

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Our solution Once an H-step is wasted, all the following H-steps are not Once an H-step is wasted, all the following H-steps are not We shall make sure that all the following G-steps are cloud-dispersing. We shall make sure that all the following G-steps are cloud-dispersing. This is done by taking thicker clouds and choosing H in a special wayThis is done by taking thicker clouds and choosing H in a special way

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Replacement with thicker clouds An expander over the cloud vertices

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Replacement with thicker clouds Arbirary matching

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Replacement with thicker clouds Multiple parallel edges

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Why does this work? cloud The graph after modified replacement D G 10 vertices

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Why does this work? cloud A uniform distribution over a subset of clouds H step G step H step G step H step Wasted

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Why does this work? cloud H step G step H step G step H step D G 9 vertices D G 10 vertices

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Why does this work? cloud H step G step H step G step H step Almost uniform over outgoing edges

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Why does this work? cloud H step G step H step G step H step

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Why does this work? cloud H step G step H step G step H step Not Wasted

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Why does this work? Buffer point-of-view A vertex in the new graph: A H-step: 65 1,… 65 2,… 65 4,… 0.5 3,1,… 10,2,… 7,4,… H H-vertex-label Cloud-label Edge-label [N] [D G ]

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Why does this work? Buffer point-of-view A vertex in the new graph: A G-step: 65 1,… 9 H-vertex-label Cloud-label Edge-label [N] [D G ] G 3 For simplicity assume G is D G edge-colorable

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H-vertex-label Why does this work? Buffer point-of-view Cloud-label Edge-label The support of the distribution [N] [D G ] 10 A uniform distribution over a subset of clouds: The marginal on is uniform Conditioned on every cloud in the support, is uniform H step G step H step G step H step

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Why does this work? Buffer point-of-view H step G step H step G step H step A uniform distribution over a subset of clouds Cloud=kUniform H does nothing H-vertex-label Cloud-label Edge-label

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H-vertex-label Why does this work? Buffer point-of-view H step G step H step G step H step - Applying G is just taking a step from according to - Since G is an expander, after this step, has more entropy - G is a permutation, hence the entropy in is reduced Cloud-label Edge-label

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Why does this work? Buffer point-of-view H step G step H step G step H step Cloud=k H-vertex-label Cloud-label Edge-label We choose H such that it also mixes the second component H adds entropy! Now the Edge-label part is close to uniform!

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H step G step H step G step H step H-vertex-label Why does this work? Buffer point-of-view Cloud-label Edge-label G moves entropy in the right direction again Not Wasted following H adds entropy

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The parameters We take 3 steps and at least 2 We take 3 steps and at least 2 work degree= D H 3 eigenvalue H 2 work degree= D H 3 eigenvalue H 2 The general case: We take k steps and at least k-1 work degree= D H k eigenvalue H k-1 We take k steps and at least k-1 work degree= D H k eigenvalue H k-1

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How to choose H H is a graph with D G 10 vertices that satisfies two properties: Mixing: for every d [D G ], Mixing: for every d [D G ], H(d,U D G 9 ) (U D G,¿) (i.e. almost uniform over the D outgoing edges) (i.e. almost uniform over the D outgoing edges) Expansion Expansion How to achieve both? Choose a random H and verify it has both properties A simple permutation satisfies this

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A problem The previous is only true when G is D G -edge- colorable The previous is only true when G is D G -edge- colorable In general, G may change the edge-label in an arbitrary way (may depend on the cloud) In general, G may change the edge-label in an arbitrary way (may depend on the cloud) The graphs H is of constant size The graphs H is of constant size The number of clouds =The size of the large graph grows to infinity. The number of clouds =The size of the large graph grows to infinity. It seems unlikely that there is a choice for H that is good for all distributions. It seems unlikely that there is a choice for H that is good for all distributions. 653 G 9¿¿¿

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A solution Take the large graph G to be locally invertible Take the large graph G to be locally invertible G(v,i)=(v[i], (i)) This property of G is preserved throughout the construction This property of G is preserved throughout the construction Now things are similar to the D G -edge- colorable case Now things are similar to the D G -edge- colorable case

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The final product G – a locally invertible (N, D G, G ) G – a locally invertible (N, D G, G ) H 1,…,H k – a sequence of mixing expanders (found by exhaustive search) each is (D G 10k, D H, H ) H 1,…,H k – a sequence of mixing expanders (found by exhaustive search) each is (D G 10k, D H, H ) Do modified replacement and take the graph of paths of length k Do modified replacement and take the graph of paths of length k The resulting graph is The resulting graph is (N D G 10k, D H k, O( H k-1 + G ))

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Our result: a fully-explicit combinatorial construction of D-regular graphs with 2 (G)=D -1/2+o(1). Our result: a fully-explicit combinatorial construction of D-regular graphs with 2 (G)=D -1/2+o(1). Can we push this further to O(D -1/2 ) ? Ramanujan+ ? Can we push this further to O(D -1/2 ) ? Ramanujan+ ? Thank you! Conclusions

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