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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. In the deterministic semistreaming (i.e., O(n · polylog n) space) regime, we present an algorithm that achieves a combinatorially optimal (Δ+1)coloring using O(logΔ log logΔ) passes. This improves upon the prior O(Δ)coloring algorithm of Assadi, Chen, and Sun (STOC 2022) at the cost of only an O(log logΔ) factor in the number of passes. In the adversarially robust semistreaming regime, we design an O(Δ5/2)coloring algorithm that improves upon the previously best O(Δ3)coloring algorithm of Chakrabarti, Ghosh, and Stoeckl (ITCS 2022). Further, we obtain a smooth colors/space tradeoff that improves upon another algorithm of the said work: whereas their algorithm uses O(Δ2) colors and O(nΔ1/2) space, ours, in particular, achieves (i)~O(Δ2) colors in O(nΔ1/3) space, and (ii)~O(Δ7/4) colors in O(nΔ1/2) space.more » « less

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. In order to better understand the differences between these models, we study a streaming task called “Missing Item Finding”. In this problem, for r < n, one is given a data stream a1 , . . . , ar of elements in [n], (possibly with repetitions), and must output some x ∈ [n] which does not equal any of the ai. We prove that, for r = nΘ(1) and δ = 1/poly(n), the space required for randomized algorithms that solve this problem in the static setting with error δ is Θ(polylog(n)); for algorithms in the adversarial setting with error δ, Θ((1 + r2/n)polylog(n)); and for deterministic algorithms, Θ(r/polylog(n)). Because our adversarially robust algorithm relies on free access to a string of O(r log n) random bits, we investigate a “random start” model of streaming algorithms where all random bits used are included in the space cost. Here we find a conditional lower bound on the space usage, which depends on the space that would be needed for a pseudodeterministic algorithm to solve the problem. We also prove an Ω(r/polylog(n)) lower bound for the space needed by a streaming algorithm with < 1/2polylog(n) error against “whitebox” adversaries that can see the internal state of the algorithm, but not predict its future random decisions.more » « less

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 nvertex 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, nonrobust, streaming setting, (Δ+1)colorings can be obtained while using only Õ(n) space. Here, we prove that an adversarially robust algorithm running under a similar space bound must spend almost Ω(Δ²) 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 complexitytheoretic standpoint, these lower bounds provide (i) the first significant separation between adversarially robust algorithms and ordinary randomized algorithms for a natural problem on insertiononly 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(Δ²)coloring using Õ(n √Δ) space and an O(Δ³)coloring using Õ(n) space.more » « less