A Distributed Algorithm For Large-Scale Graph Partitioning

Detta kandidatarbete har sin placering på Skeppsbron/Skeppsbrokajen i centrala Stockholm. Inriktningen jag valde var att rita ett förslag för en fiskmarknad som skulle placeras på denna plats. Mitt arbete har fått inspireras av Sveriges största och mest kända fiskmarknad, Feskekörkan, i Göteborg. Analyser av Feskekörkan som organisation och dess planlösning har i mitt arbete lett till en tektoniskt uppbyggd struktur där material byggnadskonstruktion var viktiga element. Med bland annat en fiskfjällsfasad i mässing och en bärande skelett av storskaliga limträbalkar. Platsen som byggnaden ligger på är ett välbesökt promenadstråk med en bred och lång kajkant som används flitigt av såväl, turister som besöker gamla stan och det kungliga slottet, och Stockholmare som tar sig mellan Södermalm och Norrmalm. Jag har valt att bebygga platsen på ett sätt som både tar vara på det vackra promenadstråket men också ger möjlighet för besökande att stanna upp och ta del av fiskmarknaden.This candidate's work has its placement on Skeppsbron/Skeppsbrokajen in the central area of Stockholm. The focus I chose was to draw a proposal for a fishmarket that would be placed at this location. My work has been inspired by the largest and most famous fish market in Sweden, Feskekörkan, in Gothenburg. Analyses of Feskekörkan’s organization and its plan has, in my work, led to a tectonically constructed structure where building materials were important elements. Including a fish scale facade made of brass and a bearing skeleton of large glulam beams. The place which the building is situated on a popular promenade with a broad and long quay which is widely used by both, tourists visiting the Old Town and the Royal Palace, and the Stockholm citizens who ride between Södermalm and Norrmalm. I have chosen to build on the site in a way that both takes advantage of the beautiful promenade but also provides the opportunity for visitors to stop and take some of the fish market

Finding all Convex Cuts of a Plane Graph in Polynomial Time

Convexity is a notion that has been defined for subsets of \RR^n and for subsets of general graphs. A convex cut of a graph $G=(V, E)$ is a $2$-partition $V_1 \dot{\cup} V_2=V$ such that both $V_1$ and $V_2$ are convex, \ie shortest paths between vertices in $V_i$ never leave $V_i$, $i \in \{1, 2\}$. Finding convex cuts is $\mathcal{NP}$-hard for general graphs. To characterize convex cuts, we employ the Djokovic relation, a reflexive and symmetric relation on the edges of a graph that is based on shortest paths between the edges' end vertices. It is known for a long time that, if $G$ is bipartite and the Djokovic relation is transitive on $G$, \ie $G$ is a partial cube, then the cut-sets of $G$'s convex cuts are precisely the equivalence classes of the Djokovic relation. In particular, any edge of $G$ is contained in the cut-set of exactly one convex cut. We first characterize a class of plane graphs that we call {\em well-arranged}. These graphs are not necessarily partial cubes, but any edge of a well-arranged graph is contained in the cut-set(s) of at least one convex cut. We also present an algorithm that uses the Djokovic relation for computing all convex cuts of a (not necessarily plane) bipartite graph in \bigO(|E|^3) time. Specifically, a cut-set is the cut-set of a convex cut if and only if the Djokovic relation holds for any pair of edges in the cut-set. We then characterize the cut-sets of the convex cuts of a general graph $H$ using two binary relations on edges: (i) the Djokovic relation on the edges of a subdivision of $H$, where any edge of $H$ is subdivided into exactly two edges and (ii) a relation on the edges of $H$ itself that is not the Djokovic relation. Finally, we use this characterization to present the first algorithm for finding all convex cuts of a plane graph in polynomial time.Comment: 23 pages. Submitted to Journal of Discrete Algorithms (JDA