\chapter{Shortest paths}

\index{shortest path}

Finding a shortest path between two nodes
of a graph
is an important problem that has many
practical applications.
For example, a natural problem related to a road network
is to calculate the shortest possible length of a route
between two cities, given the lengths of the roads.

In an unweighted graph, the length of a path equals
the number of its edges, and we can
simply use breadth-first search to find
a shortest path.
However, in this chapter we focus on
weighted graphs
where more sophisticated algorithms
are needed
for finding shortest paths.

\section{Dijkstra's algorithm}

\index{Dijkstra's algorithm}

\key{Dijkstra's algorithm}\footnote{E. W. Dijkstra published the algorithm in 1959 \cite{dij59};
however, his original paper does not mention how to implement the algorithm efficiently.}
finds shortest
paths from the starting node to all nodes of the graph.
Dijkstra's algorithm is very efficient and can be used for
processing large graphs.
However, the algorithm requires that there
are no negative weight edges in the graph.

Dijkstra's algorithm maintains distances
to the nodes and reduces them during the search.
Dijkstra's algorithm is efficient, because
it only processes
each edge in the graph once, using the fact
that there are no negative edges.

\subsubsection{Example}

Let us consider how Dijkstra's algorithm
works in the following graph when the
starting node is node 1:
\begin{center}
\begin{tikzpicture}[scale=0.9]
\node[draw, circle] (1) at (1,3) {3};
\node[draw, circle] (2) at (4,3) {4};
\node[draw, circle] (3) at (1,1) {2};
\node[draw, circle] (4) at (4,1) {1};
\node[draw, circle] (5) at (6,2) {5};

\node[color=red] at (1,3+0.6) {$\infty$};
\node[color=red] at (4,3+0.6) {$\infty$};
\node[color=red] at (1,1-0.6) {$\infty$};
\node[color=red] at (4,1-0.6) {$0$};
\node[color=red] at (6,2-0.6) {$\infty$};

\path[draw,thick,-] (1) -- node[font=\small,label=above:6] {} (2);
\path[draw,thick,-] (1) -- node[font=\small,label=left:2] {} (3);
\path[draw,thick,-] (3) -- node[font=\small,label=below:5] {} (4);
\path[draw,thick,-] (2) -- node[font=\small,label=left:9] {} (4);
\path[draw,thick,-] (2) -- node[font=\small,label=above:2] {} (5);
\path[draw,thick,-] (4) -- node[font=\small,label=below:1] {} (5);
\end{tikzpicture}
\end{center}
Like in the Bellman–Ford algorithm,
initially the distance to the starting node is 0
and the distance to all other nodes is infinite.

At each step, Dijkstra's algorithm selects a node
that has not been processed yet and whose distance
is as small as possible.
The first such node is node 1 with distance 0.

When a node is selected, the algorithm
goes through all edges that start at the node
and reduces the distances using them:
\begin{center}
\begin{tikzpicture}[scale=0.9]
\node[draw, circle] (1) at (1,3) {3};
\node[draw, circle] (2) at (4,3) {4};
\node[draw, circle] (3) at (1,1) {2};
\node[draw, circle, fill=lightgray] (4) at (4,1) {1};
\node[draw, circle] (5) at (6,2) {5};

\node[color=red] at (1,3+0.6) {$\infty$};
\node[color=red] at (4,3+0.6) {$9$};
\node[color=red] at (1,1-0.6) {$5$};
\node[color=red] at (4,1-0.6) {$0$};
\node[color=red] at (6,2-0.6) {$1$};

\path[draw,thick,-] (1) -- node[font=\small,label=above:6] {} (2);
\path[draw,thick,-] (1) -- node[font=\small,label=left:2] {} (3);
\path[draw,thick,-] (3) -- node[font=\small,label=below:5] {} (4);
\path[draw,thick,-] (2) -- node[font=\small,label=left:9] {} (4);
\path[draw,thick,-] (2) -- node[font=\small,label=above:2] {} (5);
\path[draw,thick,-] (4) -- node[font=\small,label=below:1] {} (5);

\path[draw=red,thick,->,line width=2pt] (4) -- (2);
\path[draw=red,thick,->,line width=2pt] (4) -- (3);
\path[draw=red,thick,->,line width=2pt] (4) -- (5);
\end{tikzpicture}
\end{center}
In this case,
the edges from node 1 reduced the distances of
nodes 2, 4 and 5, whose distances are now 5, 9 and 1.

The next node to be processed is node 5 with distance 1.
This reduces the distance to node 4 from 9 to 3:
\begin{center}
\begin{tikzpicture}
\node[draw, circle] (1) at (1,3) {3};
\node[draw, circle] (2) at (4,3) {4};
\node[draw, circle] (3) at (1,1) {2};
\node[draw, circle, fill=lightgray] (4) at (4,1) {1};
\node[draw, circle, fill=lightgray] (5) at (6,2) {5};

\node[color=red] at (1,3+0.6) {$\infty$};
\node[color=red] at (4,3+0.6) {$3$};
\node[color=red] at (1,1-0.6) {$5$};
\node[color=red] at (4,1-0.6) {$0$};
\node[color=red] at (6,2-0.6) {$1$};

\path[draw,thick,-] (1) -- node[font=\small,label=above:6] {} (2);
\path[draw,thick,-] (1) -- node[font=\small,label=left:2] {} (3);
\path[draw,thick,-] (3) -- node[font=\small,label=below:5] {} (4);
\path[draw,thick,-] (2) -- node[font=\small,label=left:9] {} (4);
\path[draw,thick,-] (2) -- node[font=\small,label=above:2] {} (5);
\path[draw,thick,-] (4) -- node[font=\small,label=below:1] {} (5);

\path[draw=red,thick,->,line width=2pt] (5) -- (2);
\end{tikzpicture}
\end{center}
After this, the next node is node 4, which reduces
the distance to node 3 to 9:
\begin{center}
\begin{tikzpicture}[scale=0.9]
\node[draw, circle] (1) at (1,3) {3};
\node[draw, circle, fill=lightgray] (2) at (4,3) {4};
\node[draw, circle] (3) at (1,1) {2};
\node[draw, circle, fill=lightgray] (4) at (4,1) {1};
\node[draw, circle, fill=lightgray] (5) at (6,2) {5};

\node[color=red] at (1,3+0.6) {$9$};
\node[color=red] at (4,3+0.6) {$3$};
\node[color=red] at (1,1-0.6) {$5$};
\node[color=red] at (4,1-0.6) {$0$};
\node[color=red] at (6,2-0.6) {$1$};

\path[draw,thick,-] (1) -- node[font=\small,label=above:6] {} (2);
\path[draw,thick,-] (1) -- node[font=\small,label=left:2] {} (3);
\path[draw,thick,-] (3) -- node[font=\small,label=below:5] {} (4);
\path[draw,thick,-] (2) -- node[font=\small,label=left:9] {} (4);
\path[draw,thick,-] (2) -- node[font=\small,label=above:2] {} (5);
\path[draw,thick,-] (4) -- node[font=\small,label=below:1] {} (5);

\path[draw=red,thick,->,line width=2pt] (2) -- (1);
\end{tikzpicture}
\end{center}

A remarkable property in Dijkstra's algorithm is that
whenever a node is selected, its distance is final.
For example, at this point of the algorithm,
the distances 0, 1 and 3 are the final distances
to nodes 1, 5 and 4.

After this, the algorithm processes the two
remaining nodes, and the final distances are as follows:

\begin{center}
\begin{tikzpicture}[scale=0.9]
\node[draw, circle, fill=lightgray] (1) at (1,3) {3};
\node[draw, circle, fill=lightgray] (2) at (4,3) {4};
\node[draw, circle, fill=lightgray] (3) at (1,1) {2};
\node[draw, circle, fill=lightgray] (4) at (4,1) {1};
\node[draw, circle, fill=lightgray] (5) at (6,2) {5};

\node[color=red] at (1,3+0.6) {$7$};
\node[color=red] at (4,3+0.6) {$3$};
\node[color=red] at (1,1-0.6) {$5$};
\node[color=red] at (4,1-0.6) {$0$};
\node[color=red] at (6,2-0.6) {$1$};

\path[draw,thick,-] (1) -- node[font=\small,label=above:6] {} (2);
\path[draw,thick,-] (1) -- node[font=\small,label=left:2] {} (3);
\path[draw,thick,-] (3) -- node[font=\small,label=below:5] {} (4);
\path[draw,thick,-] (2) -- node[font=\small,label=left:9] {} (4);
\path[draw,thick,-] (2) -- node[font=\small,label=above:2] {} (5);
\path[draw,thick,-] (4) -- node[font=\small,label=below:1] {} (5);
\end{tikzpicture}
\end{center}

\subsubsection{Negative edges}

The efficiency of Dijkstra's algorithm is
based on the fact that the graph does not
contain negative edges.
If there is a negative edge,
the algorithm may give incorrect results.
As an example, consider the following graph:

\begin{center}
\begin{tikzpicture}[scale=0.9]
\node[draw, circle] (1) at (0,0) {$1$};
\node[draw, circle] (2) at (2,1) {$2$};
\node[draw, circle] (3) at (2,-1) {$3$};
\node[draw, circle] (4) at (4,0) {$4$};

\path[draw,thick,-] (1) -- node[font=\small,label=above:2] {} (2);
\path[draw,thick,-] (2) -- node[font=\small,label=above:3] {} (4);
\path[draw,thick,-] (1) -- node[font=\small,label=below:6] {} (3);
\path[draw,thick,-] (3) -- node[font=\small,label=below:$-5$] {} (4);
\end{tikzpicture}
\end{center}
\noindent
The shortest path from node 1 to node 4 is
$1 \rightarrow 3 \rightarrow 4$
and its length is 1.
However, Dijkstra's algorithm
finds the path $1 \rightarrow 2 \rightarrow 4$
by following the minimum weight edges.
The algorithm does not take into account that
on the other path, the weight $-5$
compensates the previous large weight $6$.

\subsubsection{Implementation}

The following implementation of Dijkstra's algorithm
calculates the minimum distances from a node $x$
to other nodes of the graph.
The graph is stored as adjacency lists
so that \texttt{g[$v$]} contains a pair $(w,\text{cost})$
always when there is an edge from node $v$ to node $w$
with weight $\text{cost}$.

An efficient implementation of Dijkstra's algorithm
requires that it is possible to efficiently find the
minimum distance node that has not been processed.
An appropriate data structure for this is a priority queue
that contains the nodes ordered by their distances.
Using a priority queue, the next node to be processed
can be retrieved in logarithmic time.

In the following code, the priority queue
\texttt{pq} contains pairs of the form $(d,x)$,
meaning that the current distance to node $x$ is $d$.

The array $\texttt{distance}$ contains the distance to
each node.  Initially, the distance is $0$ to $\text{start}$ and $-1$ to all
other nodes.  We use $-1$ as invalid value to denote that the node
has not been reached yet.

\begin{lstlisting}
vector<int> distance(n, -1);
priority_queue<pair<int, int>,
                 vector<pair<int, int>>,
                 greater<pair<int, int>>> pq;
distance[start] = 0;
pq.emplace(0, start);
while (!pq.empty()) {
    auto [d, v] = q.top();
    q.pop();
    if (distance[v] != -1) continue;
    distance[v] = d;
    for (auto [w, cost] : g[v])
        q.emplace(d + cost, w);
}
\end{lstlisting}

Note that the type of the priority queue is not
\verb|priority_queue<pair<int, int>| but instead
\verb|priority_queue<pair<int,int>,vector<pair<int,int>>,greater<pair<int,int>>>|.
This is because in C++, a priority queue by default puts the
\emph{largest} element on top, so we reverse the ordering by changing
the comparison operator from \verb|less| (the default) to
\verb|greater| (which does the opposite).

In case you forget, you can look up the syntax for the priority queue
in the C++ cheatsheet linked on the camp page.

Also note that there may be several instances of the same
node in the priority queue; however, only the instance with the
minimum distance will be processed.

The time complexity of the above implementation is
$O(n+m \log m)$, because the algorithm goes through
all nodes of the graph and adds for each edge
at most one distance to the priority queue.