The large-error lower-bound conjecture for uniform attachment seed detection

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Let K(k,ℓ,ε)K(k,\ell,\varepsilon) denote the minimum number of vertices needed to identify a set containing the whole seed of a uniform attachment tree with probability at least 1−ε1-\varepsilon, when the seed has kk vertices and ℓ\ell leaves. There are universal constants c1,c2,ε0>0c_1,c_2,\varepsilon_0>0 such that, whenever k,ℓ>c1k,\ell>c_1,

K(k,ℓ,ε0)≥c2kℓ.K(k,\ell,\varepsilon_0)\ge c_2 k\ell.

Large-error lower-bound conjecture. The required confidence-set size is at least a constant multiple of kℓk\ell for sufficiently large seeds and a fixed positive error level. This conjectures a matching-order lower bound to the known upper bound in the regime where the error is bounded away from zero; the source presents it as an expression of the belief that the strategy based on likely vertices is close to optimal.

References

Primary source

Luc Devroye and Tommy Reddad, “On the discovery of the seed in uniform attachment trees”, arXiv:1810.00969 (2019).

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