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We use soft heaps to obtain simpler optimal algorithms for selecting the$k$-th smallest item, and the set of~$k$ smallest items, from a heap-orderedtree, from a collection of sorted lists, and from $X+Y$, where $X$ and $Y$ aretwo unsorted sets. Our results match, and in some ways extend and improve,classical results of Frederickson (1993) and Frederickson and Johnson (1982).In particular, for selecting the $k$-th smallest item, or the set of~$k$smallest items, from a collection of~$m$ sorted lists we obtain a new optimal"output-sensitive" algorithm that performs only $O(m+\sum_{i=1}^m \log(k_i+1))$comparisons, where $k_i$ is the number of items of the $i$-th list that belongto the overall set of~$k$ smallest items.

Selection from heaps, row-sorted matrices and $X+Y$ using soft heaps