Submodular Streaming in All Its Glory: Tight Approximation, Minimum Memory and Low Adaptive Complexity Ehsan Kazemi 1 , Marko Mitrovic 1 , Morteza Zadimoghaddam 2 , Silvio Lattanzi 2 , Amin Karbasi 1 1 Yale University and 2 Google
Streaming Algorithms Many practical scenarios we need to use streaming algorithms: the data arrives at a very fast pace there is only time to read the data once random access to the entire data is not possible and only a small fraction of the data can be loaded to the main memory Video from “Britain’s Got Talent” Summary � 2
Streaming Algorithms Many practical scenarios we need to use streaming algorithms: the data arrives at a very fast pace there is only time to read the data once random access to the entire data is not possible and only a small fraction of the data can be loaded to the main memory Is it possible to summarize a massive data set “on the fly”, i.e., when at any point of time we have access only to a small fraction of data? Video from “Britain’s Got Talent” Summary � 2
Submodularity � 3
Submodularity s 3 s 1 s 1 s 2 s 2 s 4 A = { s 1 , s 2 } B = { s 1 , s 2 , s 3 , s 4 } � 3
Submodularity s 3 s 1 s 1 s 2 s 2 s 4 A = { s 1 , s 2 } B = { s 1 , s 2 , s 3 , s 4 } Submodularity � 3
Submodularity s 3 s 1 s 1 s s s 2 s 2 s 4 A = { s 1 , s 2 } B = { s 1 , s 2 , s 3 , s 4 } Submodularity ∀ A ⊆ B ⊆ V and s ∈ V \ B � 3
Submodularity s 3 s 1 s 1 s s s 2 s 2 s 4 A = { s 1 , s 2 } B = { s 1 , s 2 , s 3 , s 4 } Submodularity ∀ A ⊆ B ⊆ V and s ∈ V \ B � 3
Submodularity s 3 s 1 s 1 s s s 2 s 2 s 4 A = { s 1 , s 2 } B = { s 1 , s 2 , s 3 , s 4 } Submodularity ∀ A ⊆ B ⊆ V and s ∈ V \ B f ( A ∪ { s } ) − f ( A ) ≥ f ( B ∪ { s } ) − f ( B ) � 3
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sha1_base64="uRnSOZXkcndaRjaVnTJE+5nxVNw=">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</latexit> Submodularity s 3 s 1 s 1 s s s 2 s 2 s 4 A = { s 1 , s 2 } B = { s 1 , s 2 , s 3 , s 4 } Submodularity Monotonicity ∀ A ⊆ B ⊆ V and s ∈ V \ B ∀ A ⊆ B ⊆ V f ( A ) ≤ f ( B ) f ( A ∪ { s } ) − f ( A ) ≥ f ( B ∪ { s } ) − f ( B ) � 3
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