{"id":{"repo_id":"mit","oai_identifier":"oai:dspace.mit.edu:1721.1/156830"},"canonical_url":"https://search.dev.ndltd.org/etd/mit/oai:dspace.mit.edu:1721.1/156830","repository":{"repo_id":"mit","name":"MIT","base_url":"https://dspace.mit.edu/oai/request"},"display":{"title":"Pigeonhole Equal Subset Sum in O ⃰ (2ⁿ/³)","abstract":"Subset Sum is a well known NP-hard problem. In Subset Sum, we are given a set of n integers S = {a1,··· ,an} and a target integer t, and are asked to find a subset A ⊆ [n] such that [formula]. We study a variant of the Subset Sum problem, Pigeonhole Equal Subset Sum. In Pigeonhole Equal Subset Sum, we are given a set of n integers S ={a₁,··· , aₙ} with the additional restriction that [formula], and want to find two different subsets A,B ⊆ [n] such that [formula]. The naive algorithm where we enumerate over all subset sums and look for a match takes O∗(2ⁿ) time. Horowitz and Sahni improve this to O ⃰ (2ⁿ/²) using a classical meet in the middle algorithm [1]. Recently, Jin and Wu improved this further to [formula] [2]. In this paper, we build on Jin and Wu’s techniques to improve the runtime even further to O ⃰ (2ⁿ/³).","abstract_html":"Subset Sum is a well known NP-hard problem. In Subset Sum, we are given a set of n integers S = {a1,··· ,an} and a target integer t, and are asked to find a subset A ⊆ [n] such that [formula]. We study a variant of the Subset Sum problem, Pigeonhole Equal Subset Sum. In Pigeonhole Equal Subset Sum, we are given a set of n integers S ={a₁,··· , aₙ} with the additional restriction that [formula], and want to find two different subsets A,B ⊆ [n] such that [formula]. The naive algorithm where we enumerate over all subset sums and look for a match takes O∗(2ⁿ) time. Horowitz and Sahni improve this to O ⃰ (2ⁿ/²) using a classical meet in the middle algorithm [1]. Recently, Jin and Wu improved this further to [formula] [2]. In this paper, we build on Jin and Wu’s techniques to improve the runtime even further to O ⃰ (2ⁿ/³).","abstract_has_math":false,"creators":["Zhang, Stan"],"institution":"Massachusetts Institute of Technology","degree_name":"Master","degree_level":null,"degree_discipline":null,"degree_department":"Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science","school":null,"contributors":[],"advisors":["Williams, Ryan"],"committee_chairs":[],"committee_members":[],"year":2024,"date_issued":"2024-05","date_published":"2024-05","updated_at":"2026-07-22T22:22:19Z","subjects":[],"languages":[],"rights":["Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0)","Copyright retained by author(s)"],"rights_urls":["https://creativecommons.org/licenses/by-nc-nd/4.0/"],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/1721.1/156830","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Williams, Ryan"]},{"key":"dc:contributor.department","label":"Department","values":["Massachusetts Institute of Technology. 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In Subset Sum, we are given a set of n integers S = {a1,··· ,an} and a target integer t, and are asked to find a subset A ⊆ [n] such that [formula]. We study a variant of the Subset Sum problem, Pigeonhole Equal Subset Sum. In Pigeonhole Equal Subset Sum, we are given a set of n integers S ={a₁,··· , aₙ} with the additional restriction that [formula], and want to find two different subsets A,B ⊆ [n] such that [formula]. The naive algorithm where we enumerate over all subset sums and look for a match takes O∗(2ⁿ) time. Horowitz and Sahni improve this to O ⃰ (2ⁿ/²) using a classical meet in the middle algorithm [1]. Recently, Jin and Wu improved this further to [formula] [2]. In this paper, we build on Jin and Wu’s techniques to improve the runtime even further to O ⃰ (2ⁿ/³)."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["MNG"]},{"key":"dc:title","label":"Title","values":["Pigeonhole Equal Subset Sum in O ⃰ (2ⁿ/³)"]}]}],"canonical_facts":{"dc:contributor.advisor":["Williams, Ryan"],"dc:contributor.department":["Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science"],"dc:creator":["Zhang, Stan"],"dc:date.accessioned":["2024-09-16T13:51:44Z"],"dc:date.available":["2024-09-16T13:51:44Z"],"dc:date.issued":["2024-05"],"dc:description.abstract":["Subset Sum is a well known NP-hard problem. In Subset Sum, we are given a set of n integers S = {a1,··· ,an} and a target integer t, and are asked to find a subset A ⊆ [n] such that [formula]. We study a variant of the Subset Sum problem, Pigeonhole Equal Subset Sum. In Pigeonhole Equal Subset Sum, we are given a set of n integers S ={a₁,··· , aₙ} with the additional restriction that [formula], and want to find two different subsets A,B ⊆ [n] such that [formula]. The naive algorithm where we enumerate over all subset sums and look for a match takes O∗(2ⁿ) time. Horowitz and Sahni improve this to O ⃰ (2ⁿ/²) using a classical meet in the middle algorithm [1]. Recently, Jin and Wu improved this further to [formula] [2]. In this paper, we build on Jin and Wu’s techniques to improve the runtime even further to O ⃰ (2ⁿ/³)."],"dc:description.degree":["MNG"],"dc:identifier.uri":["https://hdl.handle.net/1721.1/156830"],"dc:publisher":["Massachusetts Institute of Technology"],"dc:rights":["Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0)","Copyright retained by author(s)"],"dc:rights.uri":["https://creativecommons.org/licenses/by-nc-nd/4.0/"],"dc:title":["Pigeonhole Equal Subset Sum in O ⃰ (2ⁿ/³)"],"dc:type":["Thesis"],"thesis:degree_name":["Master"]},"updated_at":"2026-07-22T22:22:19Z"}