A streaming algorithm is considered to be adversarially robust if it provides correct outputs with high probability even when the stream updates are chosen by an adversary who may observe and react to the past outputs of the algorithm. We grow the burgeoning body of work on such algorithms in a new direction by studying robust algorithms for the problem of maintaining a valid vertex coloring of an n-vertex graph given as a stream of edges. Following standard practice, we focus on graphs with maximum degree at most ∆ and aim for colorings using a small number f (∆) of colors.A recent breakthrough (Assadi, Chen, and Khanna; SODA 2019) shows that in the standard, nonrobust, streaming setting, (∆ + 1)-colorings can be obtained while using only O(n) space. Here, we prove that an adversarially robust algorithm running under a similar space bound must spend almost Ω(∆ 2 ) colors and that robust O(∆)-coloring requires a linear amount of space, namely Ω(n∆). We in fact obtain a more general lower bound, trading off the space usage against the number of colors used. From a complexity-theoretic standpoint, these lower bounds provide (i) the first significant separation between adversarially robust algorithms and ordinary randomized algorithms for a natural problem on insertion-only streams and (ii) the first significant separation between randomized and deterministic coloring algorithms for graph streams, since deterministic streaming algorithms are automatically robust.We complement our lower bounds with a suite of positive results, giving adversarially robust coloring algorithms using sublinear space. In particular, we can maintain an O(∆ 2 )-coloring using O(n √ ∆) space and an O(∆ 3 )-coloring using O(n) space.

Two widely-used computational paradigms for sublinear algorithms are using linear measurements to perform computations on a high dimensional input and using structured queries to access a massive input. Typically, algorithms in the former paradigm are non-adaptive whereas those in the latter are highly adaptive. This work studies the fundamental search problem of ELEMENT-EXTRACTION in a query model that combines both: linear measurements with bounded adaptivity.In the ELEMENT-EXTRACTION problem, one is given a nonzero vector z = (z 1 , . . . , z n ) ∈ {0, 1} n and must report an index i where z i = 1. The input can be accessed using arbitrary linear functions of it with coefficients in some ring. This problem admits an efficient nonadaptive randomized solution (through the well known technique of ℓ 0 -sampling) and an efficient fully adaptive deterministic solution (through binary search). We prove that when confined to only k rounds of adaptivity, a deterministic ELEMENT-EXTRACTION algorithm must spend Ω(k(n 1/k − 1)) queries, when working in the ring of integers modulo some fixed q. This matches the corresponding upper bound. For queries using integer arithmetic, we prove a 2-round Ω( √ n) lower bound, also tight up to polylogarithmic factors. Our proofs reduce to classic problems in combinatorics, and take advantage of established results on the zero-sum problem as well as recent improvements to the sunflower lemma.

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