By K. Ribnikov

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**Example text**

As a conclusion of Phase 1, the algorithm rejects the placements of Cu that imply a large variation distance between the above distributions, and passes all other placements of Cu to Phase 2. Naturally, if Phase 1 rejects all placements of Ck due to distribution test failures or due to the existence of labels in Gu that do not exist in Gk , then Gu is rejected without moving to Phase 2 at all. First we observe the following. 9. With probability 1 − o(1), all of the placements that passed Phase 1 imply ε/10-close distributions, and all placements that imply identical distributions passed Phase 1.

Clearly, if E occurs then {p1 , . . , p n4 } will be a set of chosen random bits. 19. The event E as defined above occurs with probability at least 3/4. D. Thesis PHD-2008-09 - 2008 Proof. For a single pair ei and a random permutation σ, the probability that ei = σ(fj ) for some 1 ≤ j ≤ l is bounded by n . 2(n 2) Hence by the union bound, Pr[E] ≥ 1− 2knn > 3/4. 2. 5 Future work While our two-sided error algorithms run in time quasi-polynomial in n (like the general approximation algorithm of [AFK96]), the one-sided algorithms presented here require an exponential running time.

3. For every large enough n, there are two graphs G and H on n vertices, such that: 1. G is 1/100-far from being isomorphic to H 2. Every knowledge chart from IG,n3/2 /200 can be knowledge-packed with any knowledge chart from IH,n3/2 /200 Proof. We set both G and H to be the union of a complete bipartite graph with a set of isolated vertices. Formally, G has three vertex sets L, Rf , Re , where |L| = n/2, |Rf | = 26n/100 and |Re | = 24n/100, and it has the following edges: {{u, v} : u ∈ L ∧ v ∈ Rf }.