A　new　variant　of　HEAPSORT　is　presented　in　this　paper.　The　algorithm　is　not　an　internal　sorting　algorithm　in　the　strong　sense,　since　extra　storage　for　n　integers　is　necessary.　The　basic　idea　of　the　new　algorithm　is　similar　to　the　classical　sorting　algorithm　HEAPSORT,　but　the　algorithm　rebuilds　the　heap　in　another　way.　The　basic　idea　of　the　new　algorithm　is　it　uses　only　one　comparison　at　each　node.　The　new　algorithm　shift　walks　down　a　path　in　the　heap　until　a　leaf　is　reached.　The　request　of　placing　the　element　in　the　root　immediately　to　its　destination　is　relaxed.　The　new　algorithm　requires　about　n　log　n∈-∈0.788928n　comparisons　in　the　worst　case　and　n　log　n∈-∈n　comparisons　on　the　average　which　is　only　about　0.4n　more　than　necessary.　It　beats　on　average　even　the　clever　variants　of　QUICKSORT,　if　n　is　not　very　small.　The　difference　between　the　worst　case　and　the　best　case　indicates　that　there　is　still　room　for　improvement　of　the　new　algorithm　by　constructing　heap　more　carefully.　©　2007　Science　Press,　Beijing,　China　and　Springer　Science　+　Business　Media,　LLC,　USA.