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79 lines
2.1 KiB
Markdown
79 lines
2.1 KiB
Markdown
---
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title: "graphs"
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aliases:
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tags:
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- cosc201
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- datastructure
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---
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Represents a set of things with relationships between them.
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- a set of *vertices*
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- some *edges* between them
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- edges on some graphs have *weights*
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- edges on some graphs are *directed*
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Some graphs are named e.g, 
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# Implementation
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should:
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- be static
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- say if two vertices are connected in $O(1)$
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- say what the neighbors of a vertice are in $O(1)$
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## Hashset
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array wwith element to each verttex, and each element of the array is a *TreeSet* of its neighbors.
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graph is built up by specifying size, then edges.
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```java
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add edge(int v, int w){
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neighbors[v].add(w);
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neighbors[w].add(v);
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}
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```
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we could also define a vertex class and use a `HashSet\<vertex>`
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There are many options
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# Traversals
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## Breadth first traversal
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Stating from v, initialise an empty queue a, and add a marker to each vertex (or make a set of seen vertices).
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1. add the start vertex to a queue and mark it as seen
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2. while the queue is not empty
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3. visit the first ellemnet of a qeue removing ti
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4. do someting with it (line 3.5)
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5. unseen neighbors neighbors are added to the queue marking them as seen
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6. repeat
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## Depth first
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init empty *stack*, s, and add a marker to each vertex (or make a set of seen vertices)
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1. add the start vertex to a stack and mark it as seen
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2. while the stack is not empty
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3. visit the first ellemnet of a qeue removing ti
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4. do someting with it (line 3.5)
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5. unseen neighbors neighbors are added to the stack marking them as seen
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6. repeat
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## Cost of traversal
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we measure complexity both in terms of veritces V and edges E.
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for each vertex
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- it occurs in some neghbor list ofot he first time at which point it is added to the list, and marked as seen
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- then at soome point it is removed
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- so $O(1)$ per vertice so $O(V)$
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for each edge:
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- edge arise implicityly in the neihbor list
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- each one appreas twice, once per endpoint
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- when one appreas is causes a constant amount of work
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- also $O(1)$ per edge so $O(E)$
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so $O(V + E)$
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