A　number　of　algorithms　based　on　the　use　of　either　buckets　or　inverse　rules　have　been　proposed　to　address　the　problem　of　query　rewriting　using　views.　Some　inverse　rule-based　algorithms　have　considered　this　problem　in　the　presence　of　inclusion　dependencies.　However,　no　bucket-based　algorithms　have　considered　the　influence　from　inclusion　dependencies,　resulting　in　missing　some　query　rewritings　under　this　condition.　In　a　bucket-based　algorithm,　if　a　view　does　not　contain　any　subgoals　of　a　query,　then　the　algorithm　cannot　form　any　bucket　over　the　view.　In　this　paper,　we　utilize　inclusion　dependencies　to　overcome　this　deficiency.　We　present　two　novel　algorithms　in　the　buckets　framework.　In　the　first　algorithm,　we　apply　a　chase　procedure/rule　to　a　query　to　get　a　set　of　equivalent　or　contained　queries　relative　to　inclusion　dependencies,　and　then　generate　rewritings　for　each　of　the　revised　queries.　In　the　second　algorithm,　we　apply　a　chase　procedure/rule　to　such　a　view　that　does　not　contain　subgoals　of　a　query　but　still　contains　subgoals　to　which　the　chase　procedure/rule　can　be　applied.　We　prove　that　both　algorithms　can　find　a　maximally-contained　rewriting　relative　to　inclusion　dependencies.　Hence,　the　problem　of　missing　rewritings　in　the　previous　bucket-based　algorithms　is　avoided.　ACM　Classification:　H.2.4　(Information　Systems):　Systems-query　processing:　H.2.5　(Information　Systems):　Heterogeneous　Databases.　Copyright　©　2006,　Australian　Computer　Society　Inc.