Adversarially robust streaming algorithms are required to process a stream of elements and produce correct outputs, even when each stream element can be chosen as a function of earlier algorithm outputs. As with classic streaming algorithms, which must only be correct for the worst-case fixed stream, adversarially robust algorithms with access to randomness can use significantly less space than deterministic algorithms. We prove that for the Missing Item Finding problem in streaming, the space complexity also significantly depends on how adversarially robust algorithms are permitted to use randomness. (In contrast, the space complexity of classic streaming algorithms does not depend as strongly on the way randomness is used.) For Missing Item Finding on streams of length $\ell$ with elements in $\{1,\ldots,n\}$, and $\le 1/\text{poly}(\ell)$ error, we show that when $\ell = O(2^{\sqrt{\log n}})$, "random seed" adversarially robust algorithms, which only use randomness at initialization, require $\ell^{Ω(1)}$ bits of space, while "random tape" adversarially robust algorithms, which may make random decisions at any time, may use $O(\text{polylog}(\ell))$ space. When $\ell$ is between $n^{Ω(1)}$ and $O(\sqrt{n})$, "random tape" adversarially robust algorithms need $\ell^{Ω(1)}$ space, while "random oracle" adversarially robust algorithms, which can read from a long random string for free, may use $O(\text{polylog}(\ell))$ space. The space lower bound for the "random seed" case follows, by a reduction given in prior work, from a lower bound for pseudo-deterministic streaming algorithms given in this paper.

For edge coloring, the online and the W-streaming models seem somewhat orthogonal: the former needs edges to be assigned colors immediately after insertion, typically without any space restrictions, while the latter limits memory to sublinear in the input size but allows an edge's color to be announced any time after its insertion. We aim for the best of both worlds by designing small-space online algorithms for edge coloring. We study the problem under both (adversarial) edge arrivals and vertex arrivals. Our results significantly improve upon the memory used by prior online algorithms while achieving an $O(1)$-competitive ratio. In particular, for $n$-node graphs with maximum vertex-degree $Δ$ under edge arrivals, we obtain an online $O(Δ)$-coloring in $\tilde{O}(n\sqrtΔ)$ space. This is also the first W-streaming edge-coloring algorithm using $O(Δ)$ colors (in sublinear memory). All prior works either used linear memory or $ω(Δ)$ colors. We also achieve a smooth color-space tradeoff: for any $t=O(Δ)$, we get an $O(Δt (\log^2 Δ))$-coloring in $\tilde{O}(n\sqrt{Δ/t})$ space, improving upon the state of the art that used $\tilde{O}(nΔ/t)$ space for the same number of colors (the $\tilde{O}(.)$ notation hides polylog$(n)$ factors). The improvements stem from extensive use of random permutations that enable us to avoid previously used colors. Most of our algorithms can be derandomized and extended to multigraphs, where edge coloring is known to be considerably harder than for simple graphs.

Graph coloring is a fundamental problem with wide reaching applications in various areas including ata mining and databases, e.g., in parallel query optimization. In recent years, there has been a growing interest in solving various graph coloring problems in the streaming model. The initial algorithms in this line of work are all crucially randomized, raising natural questions about how important a role randomization plays in streaming graph coloring. A couple of very recent works prove that deterministic or even adversarially robust coloring algorithms (that work on streams whose updates may depend on the algorithm's past outputs) are considerably weaker than standard randomized ones. However, there is still a significant gap between the upper and lower bounds for the number of colors needed (as a function of the maximum degree Δ) for robust coloring and multipass deterministic coloring. We contribute to this line of work by proving the following results.

Many problems on data streams have been studied at two extremes of difficulty: either allowing randomized algorithms, in the static setting (where they should err with bounded probability on the worst case stream); or when only deterministic and infallible algorithms are required. Some recent works have considered the adversarial setting, in which a randomized streaming algorithm must succeed even on data streams provided by an adaptive adversary that can see the intermediate outputs of the algorithm.

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 $Δ$ and aim for colorings using a small number $f(Δ)$ of colors. A recent breakthrough (Assadi, Chen, and Khanna; SODA~2019) shows that in the standard, non-robust, streaming setting, $(Δ+1)$-colorings can be obtained while using only $\widetilde{O}(n)$ space. Here, we prove that an adversarially robust algorithm running under a similar space bound must spend almost $Ω(Δ^2)$ colors and that robust $O(Δ)$-coloring requires a linear amount of space, namely $Ω(nΔ)$. 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 $O(Δ^2)$-coloring using $\widetilde{O}(n \sqrtΔ)$ space and an $O(Δ^3)$-coloring using $\widetilde{O}(n)$ space.