Very fast construction of bounded-degree spanning graphs via the semi-random graph process

Semi-random processes involve an adaptive decision-maker, whose goal is to achieve some predetermined objective in an online randomized environment. They have algorithmic implications in various areas of computer science, as well as connections to biological processes involving decision making. In this paper, we consider a recently proposed semi-random graph process, described as follows: we start with an empty graph on $n$ vertices, and in each round, the decision-maker, called Builder, receives a uniformly random vertex $v$, and must immediately (in an online manner) choose another vertex $u$, adding the edge $\{u,v\}$ to the graph. Builder's end goal is to make the constructed graph satisfy some predetermined monotone graph property. We consider the property of containing a spanning graph $H$ as a subgraph. It was asked by N. Alon whether for any bounded-degree $H$, Builder can construct a copy of $H$ w.h.p. in $O(n)$ rounds. We answer this question positively in a strong sense, showing that any graph with maximum degree $\Delta$ can be constructed w.h.p. in $(3\Delta/2 + o(\Delta)) n$ rounds. This is tight (even for the offline case) up to a multiplicative factor of $3 + o_{\Delta}(1)$. Furthermore, for the special case where $H$ is a spanning forest of maximum degree $\Delta$, we show that $H$ can be constructed w.h.p. in $O(n \log \Delta)$ rounds. This is tight up to a multiplicative constant, even for the offline setting. Finally, we show a separation between adaptive and non-adaptive strategies, proving a lower bound of $\Omega(n\sqrt{\log n})$ on the number of rounds necessary to eliminate all isolated vertices w.h.p. using a non-adaptive strategy. This bound is tight, and in fact there are non-adaptive strategies for constructing a Hamilton cycle or a $K_r$-factor, which are successful w.h.p. within $O(n\sqrt{\log n})$ rounds.