The paper by H. T. Kung, F. Luccio, and F. P. Preparata addresses the problem of finding all maximal elements in a set of $n$ $d$-dimensional vectors, where the vectors are ordered in a natural partial order. The computational complexity, denoted by $C_d(n)$, is defined as the maximum number of comparisons required to solve the problem. The authors provide upper and lower bounds on $C_d(n)$ for different values of $d$: 1. **Lower Bound**: For $d \geq 2$, $C_d(n) \geq \lceil \log_2 n \rceil!$. 2. **Upper Bound for $d=2,3$**: $C_2(n) = O(n \log n)$ and $C_3(n) = O(n \log n)$. 3. **Upper Bound for $d \geq 4$**: $C_d(n) \leq O(n (\log n)^{d-2})$. The paper also presents algorithms for finding the maxima in these cases, including a general recursive procedure for $d \geq 4$. The algorithms are designed to exploit the partial ordering structure to reduce the number of comparisons required. The results are significant for applications in pattern classification and operations research, where the dimensionality of the vectors can be high.The paper by H. T. Kung, F. Luccio, and F. P. Preparata addresses the problem of finding all maximal elements in a set of $n$ $d$-dimensional vectors, where the vectors are ordered in a natural partial order. The computational complexity, denoted by $C_d(n)$, is defined as the maximum number of comparisons required to solve the problem. The authors provide upper and lower bounds on $C_d(n)$ for different values of $d$: 1. **Lower Bound**: For $d \geq 2$, $C_d(n) \geq \lceil \log_2 n \rceil!$. 2. **Upper Bound for $d=2,3$**: $C_2(n) = O(n \log n)$ and $C_3(n) = O(n \log n)$. 3. **Upper Bound for $d \geq 4$**: $C_d(n) \leq O(n (\log n)^{d-2})$. The paper also presents algorithms for finding the maxima in these cases, including a general recursive procedure for $d \geq 4$. The algorithms are designed to exploit the partial ordering structure to reduce the number of comparisons required. The results are significant for applications in pattern classification and operations research, where the dimensionality of the vectors can be high.