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Walk the Walk: On Pseudorandomness, Expansion, and Connectivity Omer Reingold Weizmann Institute Based on join works with Michael Capalbo, Kai-Min Chung, Chi-Jen Lu, Luca Trevisan, Salil Vadhan, and Avi Wigderson

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(Undirected) Connectivity

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How to Walk an Undirected Graph? Random walk - when in doubt, flip a coin: At each step, follow a uniformly selected edge. If there is a path between s and t, a random walk will find it (polynomial number of steps). Algorithm uses logarithmic memory (minimal).

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Pseudorandom Walks? Can we invest less randomness in the walk? Can we escape a maze deterministically? (N,D)-Universal Traversal Sequence [Cook]: sequence of edge labels which guides a walk through all of the vertices of any D-regular graph on N vertices. [AKLLR79] poly-long UTS exist (probabilistic). What about explicit (efficient) poly-long UTS? Can connectivity in undirected graphs be solved deterministically using logarithmic memory? Yes! & partial positive answers for the above … Exploits Expander Graphs …

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Log-Space Algorithm [R04] ĜĜ has constant degree. ĜEach connected component of Ĝ an expander. GĜ v in G define the set C v ={ } in Ĝ. u and v are connected C u and C v are in the same connected component. Enough to verify the existence of a path between and (easy in log-space). … G s t Assume wlog G regular and non-bipartite … Ĝ Log-space transformation highly connected; logarithmic diameter; random walk converges to uniform in logarithmic number of steps

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What about PR Walks? Ĝ GAn edge between C u and C v in Ĝ projects to a polynomial path between u and v in G GĜ Ĝ GG is connected Ĝ an expander log path in Ĝ converges to uniform projects to a poly path in G that converges to uniform The projection is logspace GGOblivious of G, if G is consistently labelled … G s t … Ĝ CuCu CvCv vu

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Labellings of Regular Digraphs Denote by i (v) the ith neighbor of v Inconsistently labelled: u,v,i s.t. i (u)= i (v) Consistently labelled: i i is a permutation (Every regular digraph has a consistent labelling) u v

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More Results [R04,RTV05] For consistently-labelled digraphs: Universal-Traversal Sequence (poly long, log- space constructible). Psedorandom Walk Generator: log-long uniform seed poly-long sequence of edge labels s.t. the walk (on any appropriate- size graph) converges to the stationary distribution. In general: Universal Exploration Sequence

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Some Open Problems Pseudorandom-Walk Generator for inconsistently-labelled digraphs Far reaching implication [RTV 05]: Every randomized algorithm can be derandomized with small penalty in space (RL=L). A walk that is pseudorandom all the way (not just in the limit): every node of the walk should be distributed correctly. A very powerful derandomization tool (generalizes eps-bias, expander walks, etc.)

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Connectivity for undirected graphs [R04] Connectivity for regular digraphs [RTV05], Pseudorandom walks for consistently-labelled, regular digraphs [R04, RTV05] Pseudorandom walks for regular digraphs [RTV05] Connectivity for digraphs w/polynomial mixing time [RTV05] RL in L Suffice to prove RL=L Summary on RL vs. L It is not about reversibility but about regularity In fact it is about having estimates on stationary probabilities [CRV07]

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But How to Construct an Expander? Goal in explicit constructions: minimize degree, maximize expansion. Celebrated sequence of algebraic constructions [Mar73,G80,JM85,LPS86, AGM87,Mar88,Mor94,...]. Ramanujan graphs: Optimal 2nd eigenvalue (as a function of degree). More relevant to us: a simple combinatorial construction w/simple analysis of constant degree expanders [RVW00]

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Reducing Degree, Preserving Expansion [RVW 00]: a method to reduce the degree of a graph while not harming its expansion by much. For that, introduced a new graph product - the zig-zag product: H: degree d on D vertices, G: degree D on N vertices G H: degree d 2 on ND vertices If H & G are good expanders so is G H

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Replacement Product Somewhat easier to describe. Somewhat weaker expansion properties [RVW00,MR00] u u (u, 8 ) (u, 7 ) (u, 2 ) (u, 3 ) (u, 6 ) (u, 5 ) (u, 4 ) (u,1) H

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Zig-Zag Construction of Expanders ( ) 1/4.Building Block: H degree d on d 4 vertices, ( H ) 1/4. Construct [RVW00]: family {G i } of d 2 -regular graphs s.t. G i has d 4i vertices and (G i ) ½ G 1 = H 2 G i+1 = (G i ) 2 H Iteratively pulling the blanket from both sizes, stretches the blanket Squaring: : reduces degree: increases #vertices: unchanged Zig-Zag: : increases degree: reduces #vertices: increases

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Usefulness for Connectivity ( ) 1/4.Building Block: H degree d on d 10 vertices, ( H ) 1/4. G 1 = G non-bipartite, d 2 -regular on n vertices G i+1 = (G i ) 5 H Thm [R04]: If G connected then for L=c logn (G L ) ½ Transformation G G L is log-space. Zig-Zag product applied to non-expanders!

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More Consequences of the Zig- Zag Construction Connection with semi-direct product in groups [ALW01] New expanding Cayley graphs for non-simple groups [MW02, RSW04] Vertex Expansion beating eigenvalue bounds [RVW00, CRVW01]

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Vertex Expansion | (S)| A |S| (A > 1) S, |S| K Every (not too large) set expands. Goal: maximize expansion parameter A In random graphs A D-1 D NN

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Explicit constructions – Vertex Expansion Optimal 2 nd eigenvalue expansion does not imply optimal vertex expansionOptimal 2 nd eigenvalue expansion does not imply optimal vertex expansion Exist Ramanujan graphs with vertex expansion D/2 [Kah95].Exist Ramanujan graphs with vertex expansion D/2 [Kah95]. Lossless Expander – Expansion > (1- ) DLossless Expander – Expansion > (1- ) D Why should we care?Why should we care? Limitation of previous techniquesLimitation of previous techniques Many beautiful applicationsMany beautiful applications

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Strong Unique Neighbor Property S, |S| K, | (S)| 0.9 D |S| S Non Unique neighbor S has 0.8 D |S| unique neighbors ! We call graphs where every such S has even a single unique neighbor – unique neighbor expanders Unique neighbor of S

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Explicit Vertex Expansion Current state of knowledge – extremely far from optimal.Current state of knowledge – extremely far from optimal. Open: lossless undirected expanders.Open: lossless undirected expanders. Unique neighbor expanders are known [AC02]Unique neighbor expanders are known [AC02] Based on the zig-zag product: lossless directed expanders [CRVW02]. Expansion D-O(D ).Based on the zig-zag product: lossless directed expanders [CRVW02]. Expansion D-O(D ). Works even if right-hand side is smaller by a constant factor.Works even if right-hand side is smaller by a constant factor. Open: expansion D-O(1) (even with non-constant degree).Open: expansion D-O(1) (even with non-constant degree).

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Open: More Unbalanced D N M Open: D constant, M=N 0.5, and sets of size at most K=N 0.2 expand. More ambitious: Unique neighbor expanders Lossless expanders Minimal Degree

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Super-Constant Degree D N M State of the art [ GUV06 ]: D=Poly(Log N), M=Poly(KD) Open: M=O(KD) (D=Poly(Log N) ) Open: D= O(Log N) (M=Poly(KD) ) S, |S| K | (S)| ¾ D |S|

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