lowest path

src/SUMMARY.md

@@ -84,7 +84,7 @@
         - [Prim’s algorithm](prim_s_algorithm.md)
     - [Directed graphs](directed_graphs.md)
         - [Topological sorting](topological_sorting.md)
-<!--         - [Dynamic programming](README.md) -->
+        - [Dynamic programming](dynamic_programming_directed_graphs.md)
 <!--         - [Successor paths](README.md) -->
 <!--         - [Cycle detection](README.md) -->
 <!--     - [Strong connectivity](README.md) -->

src/dynamic_programming.md

@@ -29,3 +29,5 @@ the challenge is how to apply
 dynamic programming to different problems.
 This chapter introduces a set of classic problems
 that are a good starting point.
+
+

src/dynamic_programming_directed_graphs.md

@@ -0,0 +1,237 @@
+# Dynamic programming
+
+If a directed graph is acyclic,
+dynamic programming can be applied to it.
+For example, we can efficiently solve the following
+problems concerning paths from a starting node
+to an ending node:
+
+- how many different paths are there?
+- what is the shortest/longest path?
+- what is the minimum/maximum number of edges in a path?
+- which nodes certainly appear in any path?
+
+## Counting the number of paths
+
+As an example, let us calculate the number of paths
+from node 1 to node 6 in the following graph:
+
+<script type="text/tikz">
+\begin{tikzpicture}[scale=0.9]
+\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,->,>=latex] (1) -- (2);
+\path[draw,thick,->,>=latex] (2) -- (3);
+\path[draw,thick,->,>=latex] (1) -- (4);
+\path[draw,thick,->,>=latex] (4) -- (5);
+\path[draw,thick,->,>=latex] (5) -- (2);
+\path[draw,thick,->,>=latex] (5) -- (3);
+\path[draw,thick,->,>=latex] (3) -- (6);
+\end{tikzpicture}
+</script>
+
+There are a total of three such paths:
+
+- $1 \rightarrow 2 \rightarrow 3 \rightarrow 6$
+- $1 \rightarrow 4 \rightarrow 5 \rightarrow 2 \rightarrow 3 \rightarrow 6$
+- $1 \rightarrow 4 \rightarrow 5 \rightarrow 3 \rightarrow 6$
+
+Let `paths(x)` denote the number of paths from
+node 1 to node $x$.
+As a base case, `paths(1)=1`.
+Then, to calculate other values of `paths(x)`,
+we may use the recursion
+
+\\[
+\\texttt{paths}(x) = \\texttt{paths}(a_1)+\\texttt{paths}(a_2)+\\cdots+\\texttt{paths}(a_k)
+\\]
+
+where $a_1,a_2,\ldots,a_k$ are the nodes from which there
+is an edge to $x$.
+Since the graph is acyclic, the values of $\texttt{paths}(x)$
+can be calculated in the order of a topological sort.
+A topological sort for the above graph is as follows:
+
+<script type="text/tikz">
+\begin{tikzpicture}[scale=0.9]
+\node[draw, circle] (1) at (0,0) {1};
+\node[draw, circle] (2) at (4.5,0) {2};
+\node[draw, circle] (3) at (6,0) {3};
+\node[draw, circle] (4) at (1.5,0) {4};
+\node[draw, circle] (5) at (3,0) {5};
+\node[draw, circle] (6) at (7.5,0) {6};
+
+\path[draw,thick,->,>=latex] (1) edge [bend left=30] (2);
+\path[draw,thick,->,>=latex] (2) -- (3);
+\path[draw,thick,->,>=latex] (1) -- (4);
+\path[draw,thick,->,>=latex] (4) -- (5);
+\path[draw,thick,->,>=latex] (5) -- (2);
+\path[draw,thick,->,>=latex] (5) edge [bend right=30] (3);
+\path[draw,thick,->,>=latex] (3) -- (6);
+\end{tikzpicture}
+</script>
+
+Hence, the numbers of paths are as follows:
+
+<script type="text/tikz">
+\begin{tikzpicture}[scale=0.9]
+\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,->,>=latex] (1) -- (2);
+\path[draw,thick,->,>=latex] (2) -- (3);
+\path[draw,thick,->,>=latex] (1) -- (4);
+\path[draw,thick,->,>=latex] (4) -- (5);
+\path[draw,thick,->,>=latex] (5) -- (2);
+\path[draw,thick,->,>=latex] (5) -- (3);
+\path[draw,thick,->,>=latex] (3) -- (6);
+
+\node[color=red] at (1,2.3) {1};
+\node[color=red] at (3,2.3) {1};
+\node[color=red] at (5,2.3) {3};
+\node[color=red] at (1,5.7) {1};
+\node[color=red] at (3,5.7) {2};
+\node[color=red] at (5,5.7) {3};
+\end{tikzpicture}
+</script>
+
+For example, to calculate the value of `paths(3)`,
+we can use the formula `paths(2)+paths(5)`,
+because there are edges from nodes 2 and 5
+to node 3.
+Since `paths}(2)=2` and `paths}(5)=1`, we conclude that `paths}(3)=3`.
+
+## Extending Dijkstra's algorithm
+
+A by-product of Dijkstra's algorithm is a directed, acyclic
+graph that indicates for each node of the original graph
+the possible ways to reach the node using a shortest path
+from the starting node.
+Dynamic programming can be applied to that graph.
+For example, in the graph
+
+<script type="text/tikz">
+\begin{tikzpicture}
+\node[draw, circle] (1) at (0,0) {1};
+\node[draw, circle] (2) at (2,0) {2};
+\node[draw, circle] (3) at (0,-2) {3};
+\node[draw, circle] (4) at (2,-2) {4};
+\node[draw, circle] (5) at (4,-1) {5};
+
+\path[draw,thick,-] (1) -- node[font=\small,label=above:3] {} (2);
+\path[draw,thick,-] (1) -- node[font=\small,label=left:5] {} (3);
+\path[draw,thick,-] (2) -- node[font=\small,label=right:4] {} (4);
+\path[draw,thick,-] (2) -- node[font=\small,label=above:8] {} (5);
+\path[draw,thick,-] (3) -- node[font=\small,label=below:2] {} (4);
+\path[draw,thick,-] (4) -- node[font=\small,label=below:1] {} (5);
+\path[draw,thick,-] (2) -- node[font=\small,label=above:2] {} (3);
+\end{tikzpicture}
+</script>
+
+the shortest paths from node 1 may use the following edges:
+
+<script type="text/tikz">
+\begin{tikzpicture}
+\node[draw, circle] (1) at (0,0) {1};
+\node[draw, circle] (2) at (2,0) {2};
+\node[draw, circle] (3) at (0,-2) {3};
+\node[draw, circle] (4) at (2,-2) {4};
+\node[draw, circle] (5) at (4,-1) {5};
+
+\path[draw,thick,->] (1) -- node[font=\small,label=above:3] {} (2);
+\path[draw,thick,->] (1) -- node[font=\small,label=left:5] {} (3);
+\path[draw,thick,->] (2) -- node[font=\small,label=right:4] {} (4);
+\path[draw,thick,->] (3) -- node[font=\small,label=below:2] {} (4);
+\path[draw,thick,->] (4) -- node[font=\small,label=below:1] {} (5);
+\path[draw,thick,->] (2) -- node[font=\small,label=above:2] {} (3);
+\end{tikzpicture}
+</script>
+
+Now we can, for example, calculate the number of
+shortest paths from node 1 to node 5
+using dynamic programming:
+
+<script type="text/tikz">
+\begin{tikzpicture}
+\node[draw, circle] (1) at (0,0) {1};
+\node[draw, circle] (2) at (2,0) {2};
+\node[draw, circle] (3) at (0,-2) {3};
+\node[draw, circle] (4) at (2,-2) {4};
+\node[draw, circle] (5) at (4,-1) {5};
+
+\path[draw,thick,->] (1) -- node[font=\small,label=above:3] {} (2);
+\path[draw,thick,->] (1) -- node[font=\small,label=left:5] {} (3);
+\path[draw,thick,->] (2) -- node[font=\small,label=right:4] {} (4);
+\path[draw,thick,->] (3) -- node[font=\small,label=below:2] {} (4);
+\path[draw,thick,->] (4) -- node[font=\small,label=below:1] {} (5);
+\path[draw,thick,->] (2) -- node[font=\small,label=above:2] {} (3);
+
+\node[color=red] at (0,0.7) {1};
+\node[color=red] at (2,0.7) {1};
+\node[color=red] at (0,-2.7) {2};
+\node[color=red] at (2,-2.7) {3};
+\node[color=red] at (4,-1.7) {3};
+\end{tikzpicture}
+</script>
+
+## Representing problems as a graphs
+
+Actually, any dynamic programming problem
+can be represented as a directed, acyclic graph.
+In such a graph, each node corresponds to a dynamic programming state
+and the edges indicate how the states depend on each other.
+
+As an example, consider the problem
+of forming a sum of money $n$
+using coins
+\\(
+\\{c_1,c_2,\\ldots,c_k\\}
+\\).
+In this problem, we can construct a graph where
+each node corresponds to a sum of money,
+and the edges show how the coins can be chosen.
+For example, for coins \\(\\{1,3,4\\}\\) and $n=6$,
+the graph is as follows:
+
+<script type="text/tikz">
+\begin{tikzpicture}[scale=0.9]
+\node[draw, circle] (0) at (0,0) {0};
+\node[draw, circle] (1) at (2,0) {1};
+\node[draw, circle] (2) at (4,0) {2};
+\node[draw, circle] (3) at (6,0) {3};
+\node[draw, circle] (4) at (8,0) {4};
+\node[draw, circle] (5) at (10,0) {5};
+\node[draw, circle] (6) at (12,0) {6};
+
+\path[draw,thick,->] (0) -- (1);
+\path[draw,thick,->] (1) -- (2);
+\path[draw,thick,->] (2) -- (3);
+\path[draw,thick,->] (3) -- (4);
+\path[draw,thick,->] (4) -- (5);
+\path[draw,thick,->] (5) -- (6);
+
+\path[draw,thick,->] (0) edge [bend right=30] (3);
+\path[draw,thick,->] (1) edge [bend right=30] (4);
+\path[draw,thick,->] (2) edge [bend right=30] (5);
+\path[draw,thick,->] (3) edge [bend right=30] (6);
+
+\path[draw,thick,->] (0) edge [bend left=30] (4);
+\path[draw,thick,->] (1) edge [bend left=30] (5);
+\path[draw,thick,->] (2) edge [bend left=30] (6);
+\end{tikzpicture}
+</script>
+
+Using this representation,
+the shortest path from node 0 to node $n$
+corresponds to a solution with the minimum number of coins,
+and the total number of paths from node 0 to node $n$
+equals the total number of solutions.