50 lines
2.5 KiB
Markdown
50 lines
2.5 KiB
Markdown
# Description
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This provides the `ConnGraph` class, which implements an undirected graph with
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dynamic connectivity. It supports adding and removing edges and determining
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whether two vertices are connected - whether there is a path between them.
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Adding and removing edges take O(log<sup>2</sup>N) amortized time with high
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probability, while checking whether two vertices are connected takes O(log N)
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time with high probability. Check the source code to see the full API and the
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Javadoc documentation.
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# Features
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* Efficiently add and remove edges and determine whether vertices are connected.
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* A vertex can appear in multiple graphs, with a different set of adjacent
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vertices in each graph.
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* `ConnGraph` supports arbitrary vertex augmentation. Given a vertex V,
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`ConnGraph` can quickly report the result of combining the augmentations of
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all of the vertices in the connected component containing V, using a combining
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function provided to the constructor. For example, if a `ConnGraph` represents
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a game map, then given the location of the player, we can quickly determine
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the amount of gold the player can access, or the strongest monster that can
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reach him. Retrieving the combined augmentation for a connected component
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takes O(log N) time with high probability.
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* Compatible with Java 6.0 and above.
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# Limitations
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* `ConnGraph` does not directly support augmenting edges. However, this can be
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accomplished by imputing each edge's augmentation to an adjacent vertex. For
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example, if each edge contains a certain amount of gold, then we can augment
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each vertex with the amount of gold in the adjacent edges. We can then
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calculate the amount of gold in a connected component by retrieving the
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component's augmentation and dividing by two. A more general approach would be
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to store the edges adjacent to each vertex in an augmented self-balancing
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binary search tree (see
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[RedBlackNode](https://github.com/btrekkie/RedBlackNode)), and to use this to
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assign an augmentation to each vertex.
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* Careful attention has been paid to the asymptotic running time of each method.
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However, beyond this, no special effort has been made to optimize performance.
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# Example usage
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```java
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ConnGraph graph = new ConnGraph();
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ConnVertex vertex1 = new ConnVertex();
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ConnVertex vertex2 = new ConnVertex();
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ConnVertex vertex3 = new ConnVertex();
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graph.addEdge(vertex1, vertex2);
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graph.addEdge(vertex2, vertex3);
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graph.connected(vertex1, vertex3); // Returns true
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graph.removeEdge(vertex1, vertex2);
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graph.connected(vertex1, vertex3); // Returns false
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```
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