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 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 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(Δ²)-coloring using Õ(n √Δ) space and an O(Δ³)-coloring using Õ(n) space.
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This content will become publicly available on June 18, 2024
Coloring in Graph Streams via Deterministic and Adversarially Robust Algorithms
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 semi-streaming (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 semi-streaming 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.
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- Award ID(s):
- 2006589
- NSF-PAR ID:
- 10463503
- Date Published:
- Journal Name:
- PODS 2023
- Volume:
- 2023
- Page Range / eLocation ID:
- 141 to 153
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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