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 amore »
Graphs with Tunable Chromatic Numbers for Parallel Coloring
We consider how to generate graphs of arbitrary size whose chromatic numbers can be chosen (or are wellbounded) for testing graph coloring algorithms on parallel computers. For the distance1 graph coloring problem, we identify three classes of graphs with this property. The first is the ErdősRényi random graph with prescribed expected degree, where the chromatic number is known with high probability. It is also known that the Greedy algorithm colors this graph using at most twice the number of colors as the chromatic number. The second is a random geometric graph embedded in hyperbolic space where the size of the maximum clique provides a tight lower bound on the chromatic number. The third is a deterministic graph described by Mycielski, where the graph is recursively constructed such that its chromatic number is known and increases with graph size, although the size of the maximum clique remains two. For Jacobian estimation, we bound the distance2 chromatic number of random bipartite graphs by considering its equivalence to distance1 coloring of an intersection graph. We use a “balls and bins” probabilistic analysis to establish a lower bound and an upper bound on the distance2 chromatic number. The regimes of graph sizes and probabilities more »
 Award ID(s):
 1637534
 Publication Date:
 NSFPAR ID:
 10181552
 Journal Name:
 SIAM 2020 Workshop on Combinatorial Scientific Computing
 Sponsoring Org:
 National Science Foundation
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