{"id":1063305,"date":"2024-07-30T12:53:34","date_gmt":"2024-07-30T19:53:34","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/?post_type=msr-research-item&p=1063305"},"modified":"2024-07-30T12:53:34","modified_gmt":"2024-07-30T19:53:34","slug":"streaming-algorithms-for-connectivity-augmentation","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/streaming-algorithms-for-connectivity-augmentation\/","title":{"rendered":"Streaming Algorithms for Connectivity Augmentation"},"content":{"rendered":"

We study the k<\/span><\/span><\/span><\/span>-connectivity augmentation problem (k<\/span><\/span><\/span><\/span>-CAP) in the single-pass streaming model. Given a (<\/span>k<\/span>\u2212<\/span>1<\/span>)<\/span><\/span><\/span><\/span>-edge connected graph G<\/span>=<\/span>(<\/span>V<\/span>,<\/span>E<\/span>)<\/span><\/span><\/span><\/span> that is stored in memory, and a stream of weighted edges L<\/span><\/span><\/span><\/span> with weights in {<\/span>0<\/span>,<\/span>1<\/span>,<\/span>\u2026<\/span>,<\/span>W<\/span>}<\/span><\/span><\/span><\/span>, the goal is to choose a minimum weight subset L<\/span>\u2032<\/span><\/span>\u2286<\/span>L<\/span><\/span><\/span><\/span> such that G<\/span>\u2032<\/span><\/span>=<\/span>(<\/span>V<\/span>,<\/span>E<\/span>\u222a<\/span>L<\/span>\u2032<\/span><\/span>)<\/span><\/span><\/span><\/span> is k<\/span><\/span><\/span><\/span>-edge connected. We give a (<\/span>2<\/span>+<\/span>\u03f5<\/span>)<\/span><\/span><\/span><\/span>-approximation algorithm for this problem which requires to store O<\/span>(<\/span>\u03f5^(<\/span>\u2212<\/span>1) <\/span><\/span><\/span><\/span>n <\/span>log <\/span><\/span>n<\/span>)<\/span><\/span><\/span><\/span> words. Moreover, we show our result is tight: Any algorithm with better than 2<\/span><\/span><\/span><\/span>-approximation for the problem requires \u03a9<\/span>(<\/span>n^<\/span>2<\/span><\/span>)<\/span><\/span><\/span><\/span> bits of space even when k<\/span>=<\/span>2<\/span><\/span><\/span><\/span>. This establishes a gap between the optimal approximation factor one can obtain in the streaming vs the offline setting for k<\/span><\/span><\/span><\/span>-CAP.
\nWe further consider a natural generalization to the fully streaming model where both E<\/span><\/span><\/span><\/span> and L<\/span><\/span><\/span><\/span> arrive in the stream in an arbitrary order. We show that this problem has a space lower bound that matches the best possible size of a spanner of the same approximation ratio. Following this, we give improved results for spanners on weighted graphs: We show a streaming algorithm that finds a (<\/span>2<\/span>t<\/span>\u2212<\/span>1<\/span>+<\/span>\u03f5<\/span>)<\/span><\/span><\/span><\/span>-approximate weighted spanner of size at most O<\/span>(<\/span>\u03f5^(<\/span>\u2212<\/span>1) <\/span><\/span><\/span><\/span>n^(<\/span>1<\/span>+<\/span>1<\/span>\/<\/span><\/span><\/span>t) <\/span><\/span><\/span><\/span>log <\/span><\/span>n<\/span>)<\/span><\/span><\/span><\/span> for integer t<\/span><\/span><\/span><\/span>, whereas the best prior streaming algorithm for spanner on weighted graphs had size depending on log <\/span><\/span>W<\/span><\/span><\/span><\/span>. Using our spanner result, we provide an optimal O<\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span>-approximation for k<\/span><\/span><\/span><\/span>-CAP in the fully streaming model with O<\/span>(<\/span>n<\/span>k<\/span>+<\/span>n^(<\/span>1<\/span>+<\/span>1<\/span>\/<\/span><\/span><\/span>t) <\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span> words of space.
\nFinally we apply our results to network design problems such as Steiner tree augmentation problem (STAP), k<\/span><\/span><\/span><\/span>-edge connected spanning subgraph (k<\/span><\/span><\/span><\/span>-ECSS), and the general Survivable Network Design problem (SNDP). In particular, we show a single-pass O<\/span>(<\/span>t <\/span>log<\/span><\/span>k<\/span>)<\/span><\/span><\/span><\/span>-approximation for SNDP using O<\/span>(<\/span>k <\/span>n^(<\/span>1<\/span>+<\/span>1<\/span>\/<\/span><\/span><\/span>t)<\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span> words of space, where k<\/span><\/span><\/span><\/span> is the maximum connectivity requirement<\/p>\n","protected":false},"excerpt":{"rendered":"

We study the k-connectivity augmentation problem (k-CAP) in the single-pass streaming model. Given a (k\u22121)-edge connected graph G=(V,E) that is stored in memory, and a stream of weighted edges L with weights in {0,1,\u2026,W}, the goal is to choose a minimum weight subset L\u2032\u2286L such that G\u2032=(V,E\u222aL\u2032) is k-edge connected. We give a (2+\u03f5)-approximation algorithm […]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"","msr_number":"","msr_organization":"","msr_pages_string":"","msr_page_range_start":"","msr_page_range_end":"","msr_series":"","msr_volume":"","msr_copyright":"","msr_conference_name":"International Colloquium on Automata, Languages, and Programming 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