Past exam of the year 1 of the computer science course of the University of Oxford / 2022 / Digital systems, linear algebra, introduction to formal proof / 1 / c / Solution

Apply comparator composition recursively in a balanced binary tree, with one-bit digital comparators at its leaves. For a power of two $n$, the circuit size and circuit depth satisfyThere are $n$ leaves and $n-1$ internal nodes, so the circuit size is $O(n)$ and the combinational delay is $O(\log n)$. Here each fixed-fan-in logic gate has bounded delay. For arbitrary $n$, pad both inputs with leading zero bits to the next power of two, which is less than $2n$; this preserves their unsigned interpretation and both bounds.