In the modern age of wireless connectivity, \emph{wireless spectrum} has become a resource that must be shared and allocated among users. The historical approach to this problem has been for a central authority to regulate and assign spectrum rights to users. Some channels would be reserved for special usages, while others would be available for free community usage. But in such a system, the freely available channels will not efficiently utilized. It is likely for this part of the spectrum to suffer from congestion during times of peak usage, or at its best inefficient utilization as users search for unused spectrum to make use of. In response to this growing need, \emph{Cognitive Radio Network} (CRN) technology has emerged as a paradigm. In a CRN, users take a dynamic approach to spectrum allocation, by sensing available spectrum and opportunistically hopping to available channels. This approach to spectrum allocation, where channels do not have predetermined uses, and the available spectrum is constantly changing as users enter and exit the system, requires users to address the problem of \emph{peer discovery}. Achieving peer discovery in the unlicensed spectrum in a fast and robust manner is the central focus of work on \emph{rendezvous problems}. In this thesis, I introduce and address the problem of \emph{Movement Restricted Rendezvous}. In movement restricted rendezvous, users cannot freely move between channels while exploring the spectrum, and instead must only make moves as described by a \emph{movement restriction graph} specific to that user. This requirement , which invalidates many of the approaches for rendezvous in prior work, may stem from a limitation of the radio hardware, or the desire to remain efficient by scanning certain channels in succession. This variation not only demands more robust solutions for CRN inspired rendezvous problems, but also has inspirations and applications in robotics motion planning, distributed algorithms for software agents, and theoretical computer science. We consider the three main variations of rendezvous: \emph{deterministic}, \emph{oblivious}, and \emph{anonymous}. In deterministic rendezvous, all user strategies must be deterministic as a function of their restriction graph and sensed available spectrum, and must also rendezvous in deterministic finite time. In oblivious rendezvous, users do not necessarily have a consistent view of how the channels are labeled, and their strategies must work independently of how they and the other users view the labels of the channels. In anonymous rendezvous, users do not have a unique identity, and must all invoke identical strategies. It can be shown that there is no algorithm that works when all three requirements are simultaneously invoked. We explore all combinations of these variants in this thesis, and where possible show matching upper and lower bounds for what rendezvous time is achievable. For deterministic, oblivious, non-anonymous rendezvous, we show rendezvous between two users is possible in $O(|G||H||l_G||l_H|)$ time. For deterministic, non-oblivious, anonymous rendezvous, we show rendezvous is possible in $O(|G||H|\log N)$ time. For randomized rendezvous and arbitrary restriction graphs for users, we show $\Theta(|G||H|)$ is the best achievable rendezvous time. We make progress past the matching upper and lower bounds for randomized rendezvous by introducing \emph{subgraph constrained rendezvous}, a variant where users movement graphs are sampled as subgraphs from a larger, global graph. In this variant, we derive algorithms that rendezvous optimally in $O(\max {|G|, |H|})$ time for trees and similarly sparse graphs. We also show lower bounds of $\Omega(|G||H|)$ for grid graphs. In our derivation of these upper and lower bounds, we also show how the analysis techniques can be generalized to other classes of graphs.
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