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The former problem has been introduced by Ge et al. (ACM Trans Knowl Discov Data 2(2):1\u201335, 2008. <jats:ext-link xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" ext-link-type=\"doi\" xlink:href=\"10.1145\/1376815.1376816\">https:\/\/doi.org\/10.1145\/1376815.1376816<\/jats:ext-link>) to model clustering of data sets with both attribute and relationship data. These problems arise from the classical <jats:italic>k<\/jats:italic>-center and <jats:italic>k<\/jats:italic>-diameter problems by adding a side constraint. For the side constraint, we are given an undirected <jats:italic>connectivity graph<\/jats:italic><jats:italic>G<\/jats:italic> on the input points, and a clustering is now only feasible if every cluster induces a connected subgraph in <jats:italic>G<\/jats:italic>. Usually in clustering problems one assumes that the clusters are pairwise disjoint. We study this case but additionally also the case that clusters are allowed to be non-disjoint. This can help to satisfy the connectivity constraints. Our main result is an <jats:inline-formula><jats:alternatives><jats:tex-math>$$O(\\log ^2k)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>O<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:msup>\n                      <mml:mo>log<\/mml:mo>\n                      <mml:mn>2<\/mml:mn>\n                    <\/mml:msup>\n                    <mml:mi>k<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-approximation algorithm for the disjoint connected <jats:italic>k<\/jats:italic>-center and <jats:italic>k<\/jats:italic>-diameter problem. For Euclidean spaces of constant dimension and for metrics with constant doubling dimension, the approximation factor improves to <jats:italic>O<\/jats:italic>(1). Our algorithm works by computing a non-disjoint connected clustering first and transforming it into a disjoint connected clustering. We complement these upper bounds by several upper and lower bounds for variations and special cases of the model.<\/jats:p>","DOI":"10.1007\/s00453-024-01266-9","type":"journal-article","created":{"date-parts":[[2024,9,2]],"date-time":"2024-09-02T06:01:45Z","timestamp":1725256905000},"page":"3425-3464","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Connected k-Center and k-Diameter Clustering"],"prefix":"10.1007","volume":"86","author":[{"given":"Lukas","family":"Drexler","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jan","family":"Eube","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kelin","family":"Luo","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dorian","family":"Reineccius","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Heiko","family":"R\u00f6glin","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Melanie","family":"Schmidt","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Julian","family":"Wargalla","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,9,2]]},"reference":[{"issue":"2","key":"1266_CR1","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1145\/1376815.1376816","volume":"2","author":"R Ge","year":"2008","unstructured":"Ge, R., Ester, M., Gao, B.J., Hu, Z., Bhattacharya, B.K., Ben-Moshe, B.: Joint cluster analysis of attribute data and relationship data: the connected k-center problem, algorithms and applications. 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