chapter 11

src/graph_representation.md

@@ -124,3 +124,217 @@ for u in adj{
 # i +=1;
 }
 ```
+
+## Adjacency matrix representation
+
+An **adjacency matrix** is a two-dimensional array
+that indicates which edges the graph contains.
+We can efficiently check from an adjacency matrix
+if there is an edge between two nodes.
+The matrix can be stored as an array
+
+```rust
+# const N:usize =4;
+let adj: Vec<Vec<i32>> = Vec::new();
+# println!("{adj:?}");
+```
+
+where each value \\(\\texttt{adj}[a][b]\\) indicates
+whether the graph contains an edge from
+node $a$ to node $b$.
+If the edge is included in the graph,
+then \\(\\texttt{adj}[a][b]=1\\),
+and otherwise \\(\\texttt{adj}[a][b]=0\\).
+For example, the graph
+
+<script type="text/tikz">
+\begin{tikzpicture}[scale=0.9]
+\node[draw, circle] (1) at (1,3) {1};
+\node[draw, circle] (2) at (3,3) {2};
+\node[draw, circle] (3) at (5,3) {3};
+\node[draw, circle] (4) at (3,1) {4};
+
+\path[draw,thick,->,>=latex] (1) -- (2);
+\path[draw,thick,->,>=latex] (2) -- (3);
+\path[draw,thick,->,>=latex] (2) -- (4);
+\path[draw,thick,->,>=latex] (3) -- (4);
+\path[draw,thick,->,>=latex] (4) -- (1);
+\end{tikzpicture}
+\end{center}
+can be represented as follows:
+\begin{center}
+\begin{tikzpicture}[scale=0.7]
+\draw (0,0) grid (4,4);
+\node at (0.5,0.5) {1};
+\node at (1.5,0.5) {0};
+\node at (2.5,0.5) {0};
+\node at (3.5,0.5) {0};
+\node at (0.5,1.5) {0};
+\node at (1.5,1.5) {0};
+\node at (2.5,1.5) {0};
+\node at (3.5,1.5) {1};
+\node at (0.5,2.5) {0};
+\node at (1.5,2.5) {0};
+\node at (2.5,2.5) {1};
+\node at (3.5,2.5) {1};
+\node at (0.5,3.5) {0};
+\node at (1.5,3.5) {1};
+\node at (2.5,3.5) {0};
+\node at (3.5,3.5) {0};
+\node at (-0.5,0.5) {4};
+\node at (-0.5,1.5) {3};
+\node at (-0.5,2.5) {2};
+\node at (-0.5,3.5) {1};
+\node at (0.5,4.5) {1};
+\node at (1.5,4.5) {2};
+\node at (2.5,4.5) {3};
+\node at (3.5,4.5) {4};
+\end{tikzpicture}
+</script>
+
+If the graph is weighted, the adjacency matrix
+representation can be extended so that
+the matrix contains the weight of the edge
+if the edge exists.
+Using this representation, the graph
+
+
+<script type="text/tikz">
+\begin{tikzpicture}[scale=0.9]
+\node[draw, circle] (1) at (1,3) {1};
+\node[draw, circle] (2) at (3,3) {2};
+\node[draw, circle] (3) at (5,3) {3};
+\node[draw, circle] (4) at (3,1) {4};
+
+\path[draw,thick,->,>=latex] (1) -- node[font=\small,label=above:5] {} (2);
+\path[draw,thick,->,>=latex] (2) -- node[font=\small,label=above:7] {} (3);
+\path[draw,thick,->,>=latex] (2) -- node[font=\small,label=left:6] {} (4);
+\path[draw,thick,->,>=latex] (3) -- node[font=\small,label=right:5] {} (4);
+\path[draw,thick,->,>=latex] (4) -- node[font=\small,label=left:2] {} (1);
+\end{tikzpicture}
+</script>
+
+corresponds to the following matrix:
+
+<script type="text/tikz">
+\begin{tikzpicture}[scale=0.7]
+\draw (0,0) grid (4,4);
+\node at (0.5,0.5) {2};
+\node at (1.5,0.5) {0};
+\node at (2.5,0.5) {0};
+\node at (3.5,0.5) {0};
+\node at (0.5,1.5) {0};
+\node at (1.5,1.5) {0};
+\node at (2.5,1.5) {0};
+\node at (3.5,1.5) {5};
+\node at (0.5,2.5) {0};
+\node at (1.5,2.5) {0};
+\node at (2.5,2.5) {7};
+\node at (3.5,2.5) {6};
+\node at (0.5,3.5) {0};
+\node at (1.5,3.5) {5};
+\node at (2.5,3.5) {0};
+\node at (3.5,3.5) {0};
+\node at (-0.5,0.5) {4};
+\node at (-0.5,1.5) {3};
+\node at (-0.5,2.5) {2};
+\node at (-0.5,3.5) {1};
+\node at (0.5,4.5) {1};
+\node at (1.5,4.5) {2};
+\node at (2.5,4.5) {3};
+\node at (3.5,4.5) {4};
+\end{tikzpicture}
+</script>
+
+The drawback of the adjacency matrix representation
+is that the matrix contains $n^2$ elements,
+and usually most of them are zero.
+For this reason, the representation cannot be used
+if the graph is large.
+
+## Edge list representation
+
+An **edge list** contains all edges of a graph
+in some order.
+This is a convenient way to represent a graph
+if the algorithm processes all edges of the graph
+and it is not needed to find edges that start
+at a given node.
+
+The edge list can be stored in a vector
+
+```rust
+let mut edges: Vec<(i32, i32)> = Vec::new();
+# println!("{edges:?}");
+```
+
+where each pair $(a,b)$ denotes that
+there is an edge from node $a$ to node $b$.
+Thus, the graph
+
+<script type="text/tikz">
+\begin{tikzpicture}[scale=0.9]
+\node[draw, circle] (1) at (1,3) {1};
+\node[draw, circle] (2) at (3,3) {2};
+\node[draw, circle] (3) at (5,3) {3};
+\node[draw, circle] (4) at (3,1) {4};
+
+\path[draw,thick,->,>=latex] (1) -- (2);
+\path[draw,thick,->,>=latex] (2) -- (3);
+\path[draw,thick,->,>=latex] (2) -- (4);
+\path[draw,thick,->,>=latex] (3) -- (4);
+\path[draw,thick,->,>=latex] (4) -- (1);
+\end{tikzpicture}
+</script>
+
+can be represented as follows:
+
+```rust
+# let mut edges: Vec<(i32, i32)> = Vec::new();
+edges.push((1,2));
+edges.push((2,3));
+edges.push((2,4));
+edges.push((3,4));
+edges.push((4,1));
+# println!("{edges:?}");
+```
+
+If the graph is weighted, the structure can
+be extended as follows:
+
+```rust
+let mut edges: Vec<(i32, i32, i32)> = Vec::new();
+# println!("{edges:?}");
+```
+
+Each element in this list is of the
+form $(a,b,w)$, which means that there
+is an edge from node $a$ to node $b$ with weight $w$.
+For example, the graph
+
+<script type="text/tikz">
+\begin{tikzpicture}[scale=0.9]
+\node[draw, circle] (1) at (1,3) {1};
+\node[draw, circle] (2) at (3,3) {2};
+\node[draw, circle] (3) at (5,3) {3};
+\node[draw, circle] (4) at (3,1) {4};
+
+\path[draw,thick,->,>=latex] (1) -- node[font=\small,label=above:5] {} (2);
+\path[draw,thick,->,>=latex] (2) -- node[font=\small,label=above:7] {} (3);
+\path[draw,thick,->,>=latex] (2) -- node[font=\small,label=left:6] {} (4);
+\path[draw,thick,->,>=latex] (3) -- node[font=\small,label=right:5] {} (4);
+\path[draw,thick,->,>=latex] (4) -- node[font=\small,label=left:2] {} (1);
+\end{tikzpicture}
+</script>
+
+can be represented as follows:
+
+```rust
+# let mut edges: Vec<(i32, i32, i32)> = Vec::new();
+edges.push((1,2,5));
+edges.push((2,3,7));
+edges.push((2,4,6));
+edges.push((3,4,5));
+edges.push((4,1,2));
+# println!("{edges:?}");
+```