Handling updates and failures: Dynamic graph algorithms and distributed computing on dynamic networks

In this thesis, we took a look at networks, and more specifically, at networks that change over time, whether those are networks in the distributed algorithms sense of the word, or the graph algorithm sense. In distributed algorithms, we looked at two main problems. First, the broadcast problem: given n agents, each agent is tasked to forward a (unique) message to every other agent. Agents collaborate and can copy and forward all messages they have received up until that point. Broadcast is achieved when one agent has successfully broadcast its message to everyone else. We studied the case where the communication network is controlled by an adversary, under the condition that the graph is rooted in every round of communication. We show that the adversary can delay broadcast for at most l (1 + √ 2)n m rounds, improving on the $O(n\log\log n)$ previous upper bound~\cite{fugger2020radius}, and asymptotically matching the $\sim 1.5n$ lower bound~\cite{schwarz2017linear}. We then looked at the stochastic version of the problem: here, the adversary -- parametrized by $k$ where $k=0$ signifies that the adversary has no control, and $k=n$ that the adversary has full control -- can choose parts of the graph, and the graph is then completed stochastically. Here, we are able to look at a stronger version of broadcast: instead of having $n$ messages trying to be broadcast in parallel, we can assume that only one message needs to be broadcasted. We show the bound $\Theta(k+\log n)$. Then, we looked at undecided states dynamics in population protocols: given a population of $n$ agents, where each initially holds an opinion among $k$ different ones. In each round, two agents are chosen uniformly at random, and can interact. If they have different opinions, they forget their opinions and become undecided. If one of them is undecided while the other has an opinion, they undecided agent copies they opinion of the decided one. The question is then, how many interactions does it take for the whole population to share the same opinion? We show a $\Omega(kn\log \frac {\sqrt n} {k \log n})$ lower bound for any $k = o\left(\frac {\sqrt n}{\log n}\right)$. This is tight for any $ k \le n^{\frac 1 2 - \epsilon}$, where $\epsilon >0$ can be any small constant, matching the known $O(kn\log n)$ upper bound for $k = O\left(\frac {\sqrt n} {\log ^2 n}\right)$~\cite{DBLP:conf/podc/AmirABBHKL23}. Finally, in dynamic algorithms, we study the minimum cut problem: we are given a graph, whose vertex set we want to partition into two subsets such that the number of edges crossing from one subset to the other is minimized. Then, the graph can be updated via edge insertions or deletions, and we must update the solution without recomputing everything from scratch. We present an exact fully-dynamic minimum cut algorithm that runs in $n^{o(1)}$ deterministic update time when the minimum cut size is at most $2^{\Theta(\log^{3/4-c}n)}$ for any $c>0$, improving on the previous algorithm~\cite{DBLP:conf/soda/JinST24} whose minimum cut size limit is $(\log n)^{o(1)}$. Using sparsification and randomization techniques, we are able to extend this to all values of the minimum cut in weighted graphs, at the cost of a $(1+o(1))$-approximation ratio.