graph exploration

src/SUMMARY.md

@@ -65,9 +65,9 @@
     - [Basics of graphs](basic_of_graph.md)
         - [ Graph terminology](graph_terminology.md)
         - [ Graph representation](graph_representation.md)
-<!--     - [Graph traversal](README.md) -->
-<!--         - [ Depth-first search](README.md) -->
-<!--         - [ Breadth-first search](README.md) -->
+    - [Graph traversal](graph_traversal.md)
+        - [ Depth-first search](depth_first_search.md)
+        - [ Breadth-first search](breadth_first_search.md)
 <!--         - [ Applications](README.md) -->
 <!--     - [Shortest paths](README.md) -->
 <!--         - [Bellman–Ford algorithm](README.md) -->

src/breadth_first_search.md

@@ -0,0 +1,145 @@
+#  Breadth-first search
+
+**Breadth-first search** (BFS) visits the nodes
+in increasing order of their distance
+from the starting node.
+Thus, we can calculate the distance
+from the starting node to all other
+nodes using breadth-first search.
+However, breadth-first search is more difficult
+to implement than depth-first search.
+
+Breadth-first search goes through the nodes
+one level after another.
+First the search explores the nodes whose
+distance from the starting node is 1,
+then the nodes whose distance is 2, and so on.
+This process continues until all nodes
+have been visited.
+
+## Example
+
+Let us consider how breadth-first search processes
+the following graph:
+
+<script type="text/tikz">
+\begin{tikzpicture}
+\node[draw, circle] (1) at (1,5) {1};
+\node[draw, circle] (2) at (3,5) {2};
+\node[draw, circle] (3) at (5,5) {3};
+\node[draw, circle] (4) at (1,3) {4};
+\node[draw, circle] (5) at (3,3) {5};
+\node[draw, circle] (6) at (5,3) {6};
+
+\path[draw,thick,-] (1) -- (2);
+\path[draw,thick,-] (2) -- (3);
+\path[draw,thick,-] (1) -- (4);
+\path[draw,thick,-] (3) -- (6);
+\path[draw,thick,-] (2) -- (5);
+\path[draw,thick,-] (5) -- (6);
+\end{tikzpicture}
+</script>
+
+Suppose that the search begins at node 1.
+First, we process all nodes that can be reached
+from node 1 using a single edge:
+
+<script type="text/tikz">
+\begin{tikzpicture}
+\node[draw, circle,fill=lightgray] (1) at (1,5) {1};
+\node[draw, circle,fill=lightgray] (2) at (3,5) {2};
+\node[draw, circle] (3) at (5,5) {3};
+\node[draw, circle,fill=lightgray] (4) at (1,3) {4};
+\node[draw, circle] (5) at (3,3) {5};
+\node[draw, circle] (6) at (5,3) {6};
+
+\path[draw,thick,-] (1) -- (2);
+\path[draw,thick,-] (2) -- (3);
+\path[draw,thick,-] (1) -- (4);
+\path[draw,thick,-] (3) -- (6);
+\path[draw,thick,-] (2) -- (5);
+\path[draw,thick,-] (5) -- (6);
+
+\path[draw,thick,-] (1) -- (2);
+\path[draw,thick,-] (2) -- (3);
+\path[draw,thick,-] (1) -- (4);
+\path[draw,thick,-] (2) -- (5);
+
+\path[draw=red,thick,->,line width=2pt] (1) -- (2);
+\path[draw=red,thick,->,line width=2pt] (1) -- (4);
+\end{tikzpicture}
+</script>
+
+After this, we proceed to nodes 3 and 5:
+
+<script type="text/tikz">
+\begin{tikzpicture}
+\node[draw, circle,fill=lightgray] (1) at (1,5) {1};
+\node[draw, circle,fill=lightgray] (2) at (3,5) {2};
+\node[draw, circle,fill=lightgray] (3) at (5,5) {3};
+\node[draw, circle,fill=lightgray] (4) at (1,3) {4};
+\node[draw, circle,fill=lightgray] (5) at (3,3) {5};
+\node[draw, circle] (6) at (5,3) {6};
+
+\path[draw,thick,-] (1) -- (2);
+\path[draw,thick,-] (2) -- (3);
+\path[draw,thick,-] (1) -- (4);
+\path[draw,thick,-] (3) -- (6);
+\path[draw,thick,-] (2) -- (5);
+\path[draw,thick,-] (5) -- (6);
+
+\path[draw,thick,-] (1) -- (2);
+\path[draw,thick,-] (2) -- (3);
+\path[draw,thick,-] (1) -- (4);
+\path[draw,thick,-] (2) -- (5);
+
+\path[draw=red,thick,->,line width=2pt] (2) -- (3);
+\path[draw=red,thick,->,line width=2pt] (2) -- (5);
+\end{tikzpicture}
+</script>
+
+Finally, we visit node 6:
+
+<script type="text/tikz">
+\begin{tikzpicture}
+\node[draw, circle,fill=lightgray] (1) at (1,5) {1};
+\node[draw, circle,fill=lightgray] (2) at (3,5) {2};
+\node[draw, circle,fill=lightgray] (3) at (5,5) {3};
+\node[draw, circle,fill=lightgray] (4) at (1,3) {4};
+\node[draw, circle,fill=lightgray] (5) at (3,3) {5};
+\node[draw, circle,fill=lightgray] (6) at (5,3) {6};
+
+\path[draw,thick,-] (1) -- (2);
+\path[draw,thick,-] (2) -- (3);
+\path[draw,thick,-] (1) -- (4);
+\path[draw,thick,-] (3) -- (6);
+\path[draw,thick,-] (2) -- (5);
+\path[draw,thick,-] (5) -- (6);
+
+\path[draw,thick,-] (1) -- (2);
+\path[draw,thick,-] (2) -- (3);
+\path[draw,thick,-] (1) -- (4);
+\path[draw,thick,-] (2) -- (5);
+
+\path[draw=red,thick,->,line width=2pt] (3) -- (6);
+\path[draw=red,thick,->,line width=2pt] (5) -- (6);
+\end{tikzpicture}
+</script>
+
+Now we have calculated the distances
+from the starting node to all nodes of the graph.
+The distances are as follows:
+
+| node | distance |
+| --- | --- |
+| 1 | 0 |
+| 2 | 1 |
+| 3 | 2 |
+| 4 | 1 |
+| 5 | 2 |
+| 6 | 3 |
+
+Like in depth-first search,
+the time complexity of breadth-first search
+is $O(n+m)$, where $n$ is the number of nodes
+and $m$ is the number of edges.

src/depth_first_search.md

@@ -0,0 +1,158 @@
+#  Depth-first search
+
+**Depth-first search** (DFS)
+is a straightforward graph traversal technique.
+The algorithm begins at a starting node,
+and proceeds to all other nodes that are
+reachable from the starting node using
+the edges of the graph.
+
+Depth-first search always follows a single
+path in the graph as long as it finds
+new nodes.
+After this, it returns to previous
+nodes and begins to explore other parts of the graph.
+The algorithm keeps track of visited nodes,
+so that it processes each node only once.
+
+## Example
+
+Let us consider how depth-first search processes
+the following graph:
+
+<script type="text/tikz">
+\begin{tikzpicture}
+\node[draw, circle] (1) at (1,5) {1};
+\node[draw, circle] (2) at (3,5) {2};
+\node[draw, circle] (3) at (5,4) {3};
+\node[draw, circle] (4) at (1,3) {4};
+\node[draw, circle] (5) at (3,3) {5};
+
+\path[draw,thick,-] (1) -- (2);
+\path[draw,thick,-] (2) -- (3);
+\path[draw,thick,-] (1) -- (4);
+\path[draw,thick,-] (3) -- (5);
+\path[draw,thick,-] (2) -- (5);
+\end{tikzpicture}
+</script>
+
+We may begin the search at any node of the graph;
+now we will begin the search at node 1.
+
+The search first proceeds to node 2:
+
+
+<script type="text/tikz">
+\begin{tikzpicture}
+\node[draw, circle,fill=lightgray] (1) at (1,5) {1};
+\node[draw, circle,fill=lightgray] (2) at (3,5) {2};
+\node[draw, circle] (3) at (5,4) {3};
+\node[draw, circle] (4) at (1,3) {4};
+\node[draw, circle] (5) at (3,3) {5};
+
+\path[draw,thick,-] (1) -- (2);
+\path[draw,thick,-] (2) -- (3);
+\path[draw,thick,-] (1) -- (4);
+\path[draw,thick,-] (3) -- (5);
+\path[draw,thick,-] (2) -- (5);
+
+\path[draw=red,thick,->,line width=2pt] (1) -- (2);
+\end{tikzpicture}
+</script>
+
+After this, nodes 3 and 5 will be visited:
+
+<script type="text/tikz">
+\begin{tikzpicture}
+\node[draw, circle,fill=lightgray] (1) at (1,5) {1};
+\node[draw, circle,fill=lightgray] (2) at (3,5) {2};
+\node[draw, circle,fill=lightgray] (3) at (5,4) {3};
+\node[draw, circle] (4) at (1,3) {4};
+\node[draw, circle,fill=lightgray] (5) at (3,3) {5};
+
+\path[draw,thick,-] (1) -- (2);
+\path[draw,thick,-] (2) -- (3);
+\path[draw,thick,-] (1) -- (4);
+\path[draw,thick,-] (3) -- (5);
+\path[draw,thick,-] (2) -- (5);
+
+\path[draw=red,thick,->,line width=2pt] (1) -- (2);
+\path[draw=red,thick,->,line width=2pt] (2) -- (3);
+\path[draw=red,thick,->,line width=2pt] (3) -- (5);
+\end{tikzpicture}
+</script>
+
+The neighbors of node 5 are 2 and 3,
+but the search has already visited both of them,
+so it is time to return to the previous nodes.
+Also the neighbors of nodes 3 and 2
+have been visited, so we next move
+from node 1 to node 4:
+
+<script type="text/tikz">
+\begin{tikzpicture}
+\node[draw, circle,fill=lightgray] (1) at (1,5) {1};
+\node[draw, circle,fill=lightgray] (2) at (3,5) {2};
+\node[draw, circle,fill=lightgray] (3) at (5,4) {3};
+\node[draw, circle,fill=lightgray] (4) at (1,3) {4};
+\node[draw, circle,fill=lightgray] (5) at (3,3) {5};
+
+\path[draw,thick,-] (1) -- (2);
+\path[draw,thick,-] (2) -- (3);
+\path[draw,thick,-] (1) -- (4);
+\path[draw,thick,-] (3) -- (5);
+\path[draw,thick,-] (2) -- (5);
+
+\path[draw=red,thick,->,line width=2pt] (1) -- (4);
+\end{tikzpicture}
+</script>
+
+After this, the search terminates because it has visited
+all nodes.
+
+The time complexity of depth-first search is $O(n+m)$
+where $n$ is the number of nodes and $m$ is the
+number of edges,
+because the algorithm processes each node and edge once.
+
+## Implementation
+
+Depth-first search can be conveniently
+implemented using recursion.
+The following function _dfs_ begins
+a depth-first search at a given node.
+The function assumes that the graph is
+stored as adjacency lists in an array
+
+```rust
+# const N:usize = 10;
+let mut adj: [Vec<i32>; N]= Default::default();
+# println!("{adj:?}");
+```
+
+and also maintains an array
+
+```rust
+let mut visited: Vec<bool> = Vec::new();
+# println!("{visited:?}");
+```
+
+that keeps track of the visited nodes.
+Initially, each array value is `false`,
+and when the search arrives at node $s$,
+the value of `visited[s]` becomes `true`.
+The function can be implemented as follows:
+
+```rust
+# const N:usize = 10;
+# let mut adj: [Vec<i32>; N]= Default::default();
+# let mut visited: Vec<bool> = Vec::new();
+fn dfs(s: usize, adj: &[Vec<i32>],visited: &mut Vec<bool>){
+    if visited[s] {return}
+    visited[s] = true;
+    // process node s
+    for &u in &adj[s] {
+        dfs(u as usize, adj, visited);
+    }
+}
+```

src/graph_traversal.md

@@ -0,0 +1,11 @@
+# Graph traversal
+
+This chapter discusses two fundamental
+graph algorithms:
+depth-first search and breadth-first search.
+Both algorithms are given a starting
+node in the graph,
+and they visit all nodes that can be reached
+from the starting node.
+The difference in the algorithms is the order
+in which they visit the nodes.