Download presentation

Presentation is loading. Please wait.

Published bySarah Rodriguez Modified over 3 years ago

1
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

2
(Undirected) Connectivity

3
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).

4
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 …

5
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

6
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

7
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) 3 2 1 1 2 4 3 4 u v

8
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

9
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.)

10
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]

11
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]

12
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

13
Replacement Product Somewhat easier to describe. Somewhat weaker expansion properties [RVW00,MR00] u 8 7 2 3 6 5 4 1 u (u, 8 ) (u, 7 ) (u, 2 ) (u, 3 ) (u, 6 ) (u, 5 ) (u, 4 ) (u,1) H

14
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

15
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!

16
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]

17
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

18
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

19
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

20
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).

21
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

22
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|

Similar presentations

OK

Randomness Conductors (II) Expander Graphs Randomness Extractors Condensers Universal Hash Functions............

Randomness Conductors (II) Expander Graphs Randomness Extractors Condensers Universal Hash Functions............

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on machine translation online Ppt on coalition government canada Ppt on human nutrition and digestion of sponges Ppt on eddy current resistance Ppt on census 2001 people Animated ppt on magnetism quiz Ppt on health and medicine class 10 Ppt on health tourism in india Ppt on nature and human relationship Ppt on body language management