Presentation on theme: "The Weizmann Institute"— Presentation transcript:
1The Weizmann Institute Expander Graphs: The Unbalanced CaseOmer ReingoldThe Weizmann Institute
2What's in This Talk? Expander Graphs – an array of definitions. Focus on most established notions, and open problems on explicit constructions. Mainly in the unbalanced case since this isWhat applications often requireWhere constructions are very far from optimalWill flash one construction (no details) - Unbalanced expanders based on Parvaresh-Vardy Codes [Guruswami,Umans,Vadhan 06]
3Bipartite GraphsAs a preparation for the unbalanced case we will talk of bipartite expanders.Can also capture undirected expanders:DG - UndirectedNDNSymmetric
4Vertex Expansion Every (not too large) set expands. N S, |S| K D |(S)| A |S|(A > 1)S, |S| KEvery (not too large) set expands.
5Vertex Expansion Goal: minimize D (i.e. constant D) |(S)| A |S|(A > 1)S, |S| KGoal: minimize D (i.e. constant D)Degree 3 random graphs are expanders [Pin73]
6Vertex Expansion Also: maximize A. Trivial upper bound: A D |(S)| A |S|(A > 1)S, |S| KAlso: maximize A.Trivial upper bound: A Deven A ≲ D-1Random graphs: AD-1
72nd Eigenvalue Expansion 2nd eigenvalue (in absolute value) of (normalized) adjacency matrix is bounded away from 1Can be interpreted in terms of Renyi (l2) entropy
8Expanders Add Entropy Vertex expansion: |Support(X’)| A |Support(X)| Prob. dist. XInduced dist. X’Dxx’Vertex expansion: |Support(X’)| A |Support(X)|Some applications rely on “less naïve” measures of entropy.Col(X) = Pr[X(1)=X(2)] = ||X||2
92nd Eigenvalue Expansion Col(X’) –1/N 2 (Col(X) –1/N)Renyi entropy (log 1/Col(X)) increases as long as: < 1 and Col(X) is not too small
102nd Eigenvalue Expansion Interestingly, vertex expansion and 2nd-eigenvalue expansion are essentially equivalent for constant degree graphs [Tan84, AM84, Alo86]
11Explicit Constructions Applications need explicit constructions:Weakly explicit: easy to build the entire graph (in time poly N).Strongly explicit:Given vertex name x and edge label i easy to find the ith neighbor of x (in time poly log N).
13Explicit constructions – Vertex Expansion Optimal 2nd eigenvalue expansion does not imply optimal vertex expansionExist Ramanujan graphs with vertex expansion D/2 [Kah95].Lossless Expander – Expansion > (1-) DWhy should we care?Limitation of previous techniquesMany applications
14Property 1: A Very Strong Unique Neighbor Property S, |S| K, |(S)| 0.9 D |S|Unique neighbor of SSNon Unique neighborS has 0.8 D |S| unique neighbors !We call graphs where every such S has even a single unique neighbor – unique neighbor expanders
15Property 2: Incredibly Fault Tolerant S, |S| K, |(S)| 0.9 D |S|Remains a lossless expander even if adversary removes (0.7 D) edges from each vertex.
16Explicit constructions – Vertex Expansion Open: lossless expanders for the undirected case.Unique neighbor expanders are known [AC02]For the directed case (expansion only from left side), lossless expanders are known [CRVW02]. Expansion D-O(D).Open: expansion D-O(1) (even with non-constant degree).
26Extractors [NZ 93]NM ≪ NX’XD(k,)-extractor if Min-entropy(X) k X’ -close to uniformMin-entropy(X) k if x, Pr[x] 2-kX and Y are -close if maxT | Pr[XT] - Pr[YT] | = ½ ||X-Y||1
27Equivalently Extractors = Mixing | e(S,T)/DK - |T|/N | < S, |S|= KDVertex Expansion – Sets on the left have many neighbors.Mixing Lemma – the neighborhood of S hits any T with roughly the right proportion.
282-Source Extractors EXT Recently – lots of attention and results source of biased correlated bitsEXTalmost uniform outputrandom bitsanother independent weak sourceRecently – lots of attention and resultsRandomness Extractors are a special case, where the 2nd source is truly random.
29Explicit Constructs. of Extractors Extractors are highly motivated in applications. As a general rule of thumb: “Anything expanders can do, extractors can do better” …Lots of progress. Still very far from optimal. Best in one direction [LRVW03, GUV06]: D=Poly(LogN / ), M=2k(1-)Selected open problem: M=2k with D=Poly(LogN / )Interpretation: extracting an arbitrary constant fraction of entropyInterpretation: extracting all the entropy
30A Word About Techniques Research on randomness extractors was invigorated with the discovery of a beautiful and surprising connection to pseudorandom generators [Tre99].This further led to discoveries of connections between extractors and error correcting codes [Tre99, RRV99, TZ01, TZS01, SU01].In particular, [GUV06] relies on Parvaresh-Vardy list-decodable codes
31[GUV06] - Basic Construction Left vertex f Fqn (poly. of degree· n-1 over Fq)Edge Label y FRight vertices = Fqm+1y’th neighbor of f =(y, f(y), (f h mod E)(y), (f h2 mod E)(y), …, (f hm-1 mod E)(y))where E(Y) = irreducible poly of degree n h = a parameterThm: This is a (K,A) expander with K=hm, A = q-hnm.
32Conclusions Many interesting variants of expander graphs Constructions in general – very far from optimalAny clean and useful algebraic characterization?