TY - JOUR

T1 - Almost-Smooth Histograms and Sliding-Window Graph Algorithms

AU - Krauthgamer, Robert

AU - Reitblat, David

PY - 2022/6/24

Y1 - 2022/6/24

N2 - We study algorithms for the sliding-window model, an important variant of the data-stream model, in which the goal is to compute some function of a fixed-length suffix of the stream. We extend the smooth-histogram framework of Braverman and Ostrovsky (FOCS 2007) to almost-smooth functions, which includes all subadditive functions. Specifically, we show that if a subadditive function can be (1+ε)-approximated in the insertion-only streaming model, then it can be (2+ε)-approximated also in the sliding-window model with space complexity larger by factor O(ε−1logw), where w is the window size. We demonstrate how our framework yields new approximation algorithms with relatively little effort for a variety of problems that do not admit the smooth-histogram technique. For example, in the frequency-vector model, a symmetric norm is subadditive and thus we obtain a sliding-window (2+ε)-approximation algorithm for it. Another example is for streaming matrices, where we derive a new sliding-window (2–√+ε)-approximation algorithm for Schatten 4-norm. We then consider graph streams and show that many graph problems are subadditive, including maximum submodular matching, minimum vertex-cover, and maximum k-cover, thereby deriving sliding-window O(1)-approximation algorithms for them almost for free (using known insertion-only algorithms). Finally, we design for every d∈(1,2] an artificial function, based on the maximum-matching size, whose almost-smoothness parameter is exactly d.

AB - We study algorithms for the sliding-window model, an important variant of the data-stream model, in which the goal is to compute some function of a fixed-length suffix of the stream. We extend the smooth-histogram framework of Braverman and Ostrovsky (FOCS 2007) to almost-smooth functions, which includes all subadditive functions. Specifically, we show that if a subadditive function can be (1+ε)-approximated in the insertion-only streaming model, then it can be (2+ε)-approximated also in the sliding-window model with space complexity larger by factor O(ε−1logw), where w is the window size. We demonstrate how our framework yields new approximation algorithms with relatively little effort for a variety of problems that do not admit the smooth-histogram technique. For example, in the frequency-vector model, a symmetric norm is subadditive and thus we obtain a sliding-window (2+ε)-approximation algorithm for it. Another example is for streaming matrices, where we derive a new sliding-window (2–√+ε)-approximation algorithm for Schatten 4-norm. We then consider graph streams and show that many graph problems are subadditive, including maximum submodular matching, minimum vertex-cover, and maximum k-cover, thereby deriving sliding-window O(1)-approximation algorithms for them almost for free (using known insertion-only algorithms). Finally, we design for every d∈(1,2] an artificial function, based on the maximum-matching size, whose almost-smoothness parameter is exactly d.

U2 - https://doi.org/10.1007/s00453-022-00988-y

DO - https://doi.org/10.1007/s00453-022-00988-y

M3 - مقالة

SN - 0178-4617

VL - 84

JO - Algorithmica

JF - Algorithmica

IS - 10

ER -