{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,23]],"date-time":"2026-07-23T23:50:52Z","timestamp":1784850652861,"version":"3.55.0"},"reference-count":42,"publisher":"Association for Computing Machinery (ACM)","issue":"4","license":[{"start":{"date-parts":[[2017,8,16]],"date-time":"2017-08-16T00:00:00Z","timestamp":1502841600000},"content-version":"vor","delay-in-days":365,"URL":"http:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"funder":[{"DOI":"10.13039\/100000183","name":"Army Research Office","doi-asserted-by":"publisher","award":["DAAD19-02-1-0389, DAAD19 02 1 0389"],"award-info":[{"award-number":["DAAD19-02-1-0389, DAAD19 02 1 0389"]}],"id":[{"id":"10.13039\/100000183","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["CNF 0435382, CNF 0433540, ANI 0331653, CCR 0205523"],"award-info":[{"award-number":["CNF 0435382, CNF 0433540, ANI 0331653, CCR 0205523"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Algorithms"],"published-print":{"date-parts":[[2016,9,2]]},"abstract":"<jats:p>\n                    We study the problem of routing in doubling metrics and show how to perform hierarchical routing in such metrics with small stretch and compact routing tables (i.e., with a small amount of routing information stored at each vertex). We say that a metric (\n                    <jats:italic toggle=\"yes\">X<\/jats:italic>\n                    ,\n                    <jats:italic toggle=\"yes\">d<\/jats:italic>\n                    ) has\n                    <jats:italic toggle=\"yes\">doubling dimension<\/jats:italic>\n                    dim(&lt;i&lt;X&lt;\/i&lt;) at most \u03b1 if every ball can be covered by 2\n                    <jats:sup>\u03b1<\/jats:sup>\n                    balls of half its radius. (A\n                    <jats:italic toggle=\"yes\">doubling metric<\/jats:italic>\n                    is one whose doubling dimension dim(&lt;i&lt;X&lt;\/i&lt;) is a constant.) We consider the metric space induced by the shortest-path distance in an underlying undirected graph\n                    <jats:italic toggle=\"yes\">G<\/jats:italic>\n                    . We show how to perform (1 + \u03c4)-stretch routing on such a metric for any 0 &lt; \u03c4 \u2264 1 with routing tables of size at most (\u03b1\/\u03c4)\n                    <jats:sup>\n                      <jats:italic toggle=\"yes\">O<\/jats:italic>\n                      (\u03b1)\n                    <\/jats:sup>\n                    log\u2009\u0394log\u2009\u03b4 bits with only\n                    <jats:italic toggle=\"yes\">\n                      (\u03b1\/\u03c4)\n                      <jats:sup>\n                        <jats:italic toggle=\"yes\">O<\/jats:italic>\n                        (\u03b1)\n                      <\/jats:sup>\n                      log\u2009\u0394 entries\n                    <\/jats:italic>\n                    , where \u0394 is the diameter of the graph, and \u03b4 is the maximum degree of the graph\n                    <jats:italic toggle=\"yes\">G<\/jats:italic>\n                    ; hence, the number of routing table entries is just \u03c4\n                    <jats:sup>\n                      \u2212\n                      <jats:italic toggle=\"yes\">O<\/jats:italic>\n                      (1)\n                    <\/jats:sup>\n                    log\u2009\u0394 for doubling metrics. These results extend and improve on those of Talwar (2004).\n                  <\/jats:p>\n                  <jats:p>\n                    We also give better constructions of sparse\n                    <jats:italic toggle=\"yes\">spanners<\/jats:italic>\n                    for doubling metrics than those obtained from the routing tables earlier; for \u03c4 &gt; 0, we give algorithms to construct (1 + \u03c4)-stretch spanners for a metric (\n                    <jats:italic toggle=\"yes\">X<\/jats:italic>\n                    ,\n                    <jats:italic toggle=\"yes\">d<\/jats:italic>\n                    ) with maximum degree at most (2 + 1\/\u03c4)\n                    <jats:sup>\n                      <jats:italic toggle=\"yes\">O(dim(X))<\/jats:italic>\n                    <\/jats:sup>\n                    , matching the results of Das et al. for Euclidean metrics.\n                  <\/jats:p>","DOI":"10.1145\/2915183","type":"journal-article","created":{"date-parts":[[2016,8,16]],"date-time":"2016-08-16T08:14:20Z","timestamp":1471335260000},"page":"1-22","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":16,"title":["On Hierarchical Routing in Doubling Metrics"],"prefix":"10.1145","volume":"12","author":[{"given":"T.-H. Hubert","family":"Chan","sequence":"first","affiliation":[{"name":"The University of Hong Kong, Pokfulam Road, Hong Kong"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Anupam","family":"Gupta","sequence":"additional","affiliation":[{"name":"Carnegie Mellon University, Pittsburgh PA, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Bruce M.","family":"Maggs","sequence":"additional","affiliation":[{"name":"Duke University and Akamai Technologies, NC, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shuheng","family":"Zhou","sequence":"additional","affiliation":[{"name":"University of Michigan, MI, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"320","published-online":{"date-parts":[[2016,8,16]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1145\/1250790.1250883"},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.1109\/ICDCS.2006.72"},{"key":"e_1_2_1_3_1","doi-asserted-by":"publisher","DOI":"10.1002\/0471722154"},{"key":"e_1_2_1_4_1","doi-asserted-by":"publisher","DOI":"10.1145\/225058.225191"},{"key":"e_1_2_1_5_1","doi-asserted-by":"publisher","DOI":"10.1016\/0196-6774(90)90017-9"},{"key":"e_1_2_1_6_1","doi-asserted-by":"publisher","unstructured":"Baruch Awerbuch and David Peleg. 1990. Sparse partitions (extended abstract). In FOCS. 503--513. 10.1109\/FSCS.1990.89571","DOI":"10.1109\/FSCS.1990.89571"},{"key":"e_1_2_1_7_1","doi-asserted-by":"publisher","DOI":"10.1137\/0405013"},{"key":"e_1_2_1_8_1","doi-asserted-by":"publisher","DOI":"10.5555\/874062.875536"},{"key":"e_1_2_1_9_1","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.3240020402"},{"key":"e_1_2_1_10_1","doi-asserted-by":"publisher","DOI":"10.5555\/1109557.1109566"},{"key":"e_1_2_1_11_1","doi-asserted-by":"publisher","DOI":"10.1145\/1734213.1734215"},{"key":"e_1_2_1_12_1","doi-asserted-by":"publisher","DOI":"10.1137\/130930984"},{"key":"e_1_2_1_13_1","doi-asserted-by":"publisher","DOI":"10.1006\/jagm.2000.1134"},{"key":"e_1_2_1_14_1","doi-asserted-by":"publisher","DOI":"10.5555\/313651.313697"},{"key":"e_1_2_1_15_1","volume-title":"Geometry of Cuts and Metrics. Algorithms and Combinatorics","author":"Deza Michel Marie","unstructured":"Michel Marie Deza and Monique Laurent. 1997. Geometry of Cuts and Metrics. Algorithms and Combinatorics, Vol. 15. Springer-Verlag, Berlin. xii+587 pages."},{"key":"e_1_2_1_16_1","doi-asserted-by":"publisher","DOI":"10.1145\/2488608.2488691"},{"key":"e_1_2_1_17_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01762113"},{"key":"e_1_2_1_18_1","doi-asserted-by":"publisher","DOI":"10.1137\/0218058"},{"key":"e_1_2_1_19_1","doi-asserted-by":"publisher","DOI":"10.1145\/997817.997848"},{"key":"e_1_2_1_20_1","doi-asserted-by":"publisher","DOI":"10.1145\/568438.568451"},{"key":"e_1_2_1_21_1","doi-asserted-by":"publisher","DOI":"10.1006\/jpdc.2000.1705"},{"key":"e_1_2_1_22_1","doi-asserted-by":"publisher","DOI":"10.5555\/1347082.1347148"},{"key":"e_1_2_1_23_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-87744-8_40"},{"key":"e_1_2_1_24_1","doi-asserted-by":"publisher","DOI":"10.5555\/946243.946308"},{"key":"e_1_2_1_25_1","doi-asserted-by":"publisher","DOI":"10.1145\/1064092.1064117"},{"key":"e_1_2_1_26_1","volume-title":"Lectures on Analysis on Metric Spaces","author":"Heinonen Juha","unstructured":"Juha Heinonen. 2001. Lectures on Analysis on Metric Spaces. Springer-Verlag, New York."},{"key":"e_1_2_1_27_1","doi-asserted-by":"publisher","DOI":"10.1145\/564870.564877"},{"key":"e_1_2_1_28_1","doi-asserted-by":"publisher","DOI":"10.1145\/509907.510013"},{"key":"e_1_2_1_29_1","first-page":"155","article-title":"Hierarchical routing for large networks. Performance evaluation and optimization","volume":"1","author":"Kleinrock Leonard","year":"1977","unstructured":"Leonard Kleinrock and Farouk Kamoun. 1977. Hierarchical routing for large networks. Performance evaluation and optimization. Comput. Netw. 1, 3 (1977), 155--174.","journal-title":"Comput. Netw."},{"key":"e_1_2_1_30_1","doi-asserted-by":"publisher","DOI":"10.5555\/1283383.1283484"},{"key":"e_1_2_1_31_1","doi-asserted-by":"publisher","DOI":"10.1145\/780542.780607"},{"key":"e_1_2_1_32_1","doi-asserted-by":"publisher","DOI":"10.5555\/982792.982913"},{"key":"e_1_2_1_33_1","doi-asserted-by":"publisher","DOI":"10.1145\/1667053.1667060"},{"key":"e_1_2_1_34_1","doi-asserted-by":"publisher","DOI":"10.5555\/355459"},{"key":"e_1_2_1_35_1","doi-asserted-by":"publisher","DOI":"10.1145\/65950.65953"},{"key":"e_1_2_1_36_1","doi-asserted-by":"publisher","DOI":"10.1145\/258492.258523"},{"key":"e_1_2_1_37_1","doi-asserted-by":"publisher","DOI":"10.1145\/378580.378670"},{"key":"e_1_2_1_38_1","doi-asserted-by":"publisher","DOI":"10.1145\/142675.142716"},{"key":"e_1_2_1_39_1","doi-asserted-by":"publisher","DOI":"10.1145\/1073814.1073823"},{"key":"e_1_2_1_40_1","doi-asserted-by":"publisher","DOI":"10.1145\/2591796.2591864"},{"key":"e_1_2_1_41_1","doi-asserted-by":"publisher","DOI":"10.1145\/1007352.1007399"},{"key":"e_1_2_1_42_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02574695"}],"container-title":["ACM Transactions on Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/2915183","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/2915183","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/2915183","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,11,18]],"date-time":"2025-11-18T09:27:10Z","timestamp":1763458030000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/2915183"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,8,16]]},"references-count":42,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2016,9,2]]}},"alternative-id":["10.1145\/2915183"],"URL":"https:\/\/doi.org\/10.1145\/2915183","relation":{},"ISSN":["1549-6325","1549-6333"],"issn-type":[{"value":"1549-6325","type":"print"},{"value":"1549-6333","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,8,16]]},"assertion":[{"value":"2012-11-01","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2016-04-01","order":2,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2016-08-16","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}