Many　practical　problems　emphasize　the　importance　of　not　only　knowing　whether　an　element　is　selected　but　also　deciding　to　what　extent　it　is　selected,　which　imposes　a　challenge　on　submodule　optimization.　In　this　study,　we　consider　the　monotone,　nondecreasing,　and　non-submodular　maximization　on　the　integer　lattice　with　a　cardinality　constraint.　We　first　design　a　two-pass　streaming　algorithm　by　refining　the　estimation　interval　of　the　optimal　value.　For　each　element,　the　algorithm　not　only　decides　whether　to　save　the　element　but　also　gives　the　number　of　reservations.　Then,　we　introduce　the　binary　search　as　a　subroutine　to　reduce　the　time　complexity.　Next,　we　obtain　a　one-pass　streaming　algorithm　by　dynamically　updating　the　estimation　interval　of　optimal　value.　Finally,　we　improve　the　memory　complexity　of　this　algorithm.　　©　1996-2012　Tsinghua　University　Press.