We address the vexing issue of deletions in balanced trees. Rebalancing after a deletion is generally more complicated than
\nrebalancing after an insertion. Textbooks neglect deletion rebalancing, and many B-tree-based database systems do not do
\nit. We describe a relaxation of AVL trees in which rebalancing is done after insertions but not after deletions, yet worst-case
\naccess time remains logarithmic in the number of insertions. For any application of balanced trees in which the number of
\nupdates is polynomial in the tree size, our structure offers performance competitive with that of classical balanced trees.With
\nthe addition of periodic rebuilding, the performance of our structure is theoretically superior to that of many if not all classic
\nbalanced tree structures. Our structure needs lg lgm + 1 bits of balance information per node, where m is the number of
\ninsertions and lg is the base-two logarithm, or lg lg n + O(1) with periodic rebuilding, where n is the number of nodes. An
\ninsertion takes up to two rotations and O(1) amortized time, not counting the time to find the insertion position. This is the
\nsame as in standard AVL trees. Using an analysis that relies on an exponential potential function, we show that rebalancing
\nsteps occur with a frequency that is exponentially small in the height of the affected node. Our techniques apply to other
\ntypes of balanced trees, notably B-trees, as we show in a companion paper, and in particular red-black trees, which can be
\nviewed as a special case of B-trees.<\/p>\n","protected":false},"excerpt":{"rendered":"
We address the vexing issue of deletions in balanced trees. Rebalancing after a deletion is generally more complicated than rebalancing after an insertion. Textbooks neglect deletion rebalancing, and many B-tree-based database systems do not do it. We describe a relaxation of AVL trees in which rebalancing is done after insertions but not after deletions, yet […]<\/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":"","footnotes":""},"msr-content-type":[3],"msr-research-highlight":[],"research-area":[13561],"msr-publication-type":[193715],"msr-product-type":[],"msr-focus-area":[],"msr-platform":[],"msr-download-source":[],"msr-locale":[268875],"msr-post-option":[],"msr-field-of-study":[],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-168830","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-algorithms","msr-locale-en_us"],"msr_publishername":"","msr_edition":"ACM Transactions on Algorithms","msr_affiliation":"","msr_published_date":"2016-09-01","msr_host":"","msr_duration":"","msr_version":"","msr_speaker":"","msr_other_contributors":"","msr_booktitle":"","msr_pages_string":"","msr_chapter":"","msr_isbn":"","msr_journal":"ACM Transactions on Algorithms","msr_volume":"","msr_number":"","msr_editors":"","msr_series":"","msr_issue":"","msr_organization":"","msr_how_published":"","msr_notes":"To appear","msr_highlight_text":"","msr_release_tracker_id":"","msr_original_fields_of_study":"","msr_download_urls":"","msr_external_url":"","msr_secondary_video_url":"","msr_longbiography":"","msr_microsoftintellectualproperty":1,"msr_main_download":"","msr_publicationurl":"http:\/\/sidsen.org\/papers\/ravl-trees-journal.pdf","msr_doi":"","msr_publication_uploader":[{"type":"url","title":"http:\/\/sidsen.org\/papers\/ravl-trees-journal.pdf","viewUrl":false,"id":false,"label_id":0}],"msr_related_uploader":"","msr_attachments":[{"id":0,"url":"http:\/\/sidsen.org\/papers\/ravl-trees-journal.pdf"}],"msr-author-ordering":[{"type":"user_nicename","value":"sidsen","user_id":33656,"rest_url":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/microsoft-research\/v1\/researchers?person=sidsen"},{"type":"text","value":"Robert E. 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