<filename>tex/shortestpath.tex<gh_stars>1-10
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% shortestpath.tex last revised 29 Jan 2014 by <NAME>.
% Explanation of correctness of bidirectional breadth first search algorithm for
% finding a shortest path between two nodes in an unweighted undirected graph.
% Copyright (c) 2014 <NAME>.
% Permission is granted to copy, distribute and/or modify this document
% under the terms of the GNU Free Documentation License, Version 1.3
% or any later version published by the Free Software Foundation;
% with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts.
% A copy of the license is included in the section entitled "GNU
% Free Documentation License".
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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\title{Why the Bidirectional Breadth First Search Algorithm Finds a Shortest Path}
\author{<NAME>}
\maketitle

Suppose you have an unweighted undirected graph, given for every node $x$ of the graph the set $E_x$ of nodes on edges meeting $x$ (i.e. $E_x$ is the edgelist of $x$), and want a shortest path between two nodes $a$ and $b$.

A \textbf{unidirectional breadth first search} starting from $a$ will find one.  Let $A_0 = \{ a \}$, and for $k > 0$, $A_k = (\cup_{x \in A_{k-1}} E_x) \setminus \cup_{0 \le j < k} A_j$ (all the nodes in all the edgelists of all the nodes in $A_{k-1}$, but removing nodes in all previous $A_j$).  Define $d(x,a)$ as the distance from $x$ to $a$, the length of the shortest path from $x$ to $a$.  This is a metric, in particular for any $x, y, z$, $d(x,y) = 0 \Leftrightarrow x = y$ and $d(x,z) \le d(x,y) + d(y,z)$.  The latter is true because you can make a path from $x$ to $z$ by concatenating shortest paths from $x$ to $y$ and $y$ to $z$, though this concatenation may not be the shortest path from $x$ to $z$.

Claim: $A_k = \{ x | d(x, a) = k \}$.  If $d(a,x) = k$, then $x \in A_k$ because there is a path of length $k$ from $a$, and no shorter path (which would cause $x \in A_j$ for some $j < k$).

On the other hand, if $x \in A_k$, there is a path of length $k$ from $a$ to $x$, so $d(a,x) \le k$.  It's not possible that $d(a,x) = j < k$, because if it were, then $x \in A_j$ and thus not in $A_k$.  Thus $d(a,x) = k$.

A breadth first search for $b$ starting from $a$, implemented as in bfs1.py, in effect creates the sets $A_j$ for $j < k = d(a,b)$, and some part of $A_k$ until it encounters $b$ in that set.  This is because nodes are added to the queue from edgelists of nodes popped off the queue (which is initialized with $a$ and is FIFO), but only if they have not been previously encountered.  The code implicitly shifts from $A_j$ to $A_{j+1}$ when all the edgelists of nodes belonging to $A_{j-1}$ have been processed.

For every node in $A_j$ added to the queue, the code keeps track of which node in $A_{j-1}$ owned the edgelist on which it was first encountered.  A shortest path from $a$ to $b$ is then built by reversing what is obtained by following these parent relationships backwards.

A \textbf{bidirectional breadth first search} can be faster, due to the potentially exponential fan-out of breadth first search.  Define the $A_k$ as before, and the $B_k$ similarly starting from $B_0 = \{ b \}$.  Supposing that we start from the one end with $A_0$, from the other with $B_0$, that some choice is made at each iteration whether to move from $A_r$ to $A_{r+1}$ or $B_s$ to $B_{s+1}$, and that we stop when $A_r \cap B_s \ne \emptyset$ for the first time.  From the two paths from the ends we get a path between the endpoints $a$ and $b$ of length $r+s$.  In fact, there is no shorter path.  To see this, use the fact that if the equations below for $A_0$ through $A_k$ hold, then the logical equivalence for $A_{k+1}$ holds.
\begin{align}
&\text{If}  \notag \\
A_0 \cap B_0 &= \emptyset \notag \\
A_1 \cap B_0 &= \emptyset \notag \\
&\cdots \notag \\
A_k \cap B_0 &= \emptyset \notag \\
&\text{then}  \notag \\
A_{k+1} \cap B_0 &= \emptyset \Leftrightarrow A_r \cap B_s = \emptyset, \notag 
\end{align}
for any nonnegative $r, s$ with $r + s = k + 1$.

This can be seen as follows. If $A_{k+1} \cap B_0 \ne \emptyset$, then recalling the properties of unidirectional breadth first search from $a$, there is a shortest path of length $k+1$ from $a$ to $b$.  This can be split into two pieces of lengths $r$ and $s$, reversing one of them, that are shortest paths of those lengths from $a$ to some $x$ and from $b$ to $x$, so that $x \in A_r \cap B_s$, which is thus nonempty.  Conversely, if there is an $x \in A_r \cap B_s$, by concatenating paths, one of them reversed, we get a path of length $r+s = k+1$ from $a$ to $b$, so that $d(a,b) \le k+1$.  But $d(a,b) \le k$ is false because of all the intersections empty by hypothesis.  Thus $d(a,b) = k+1$, and thus $A_{k+1} \cap B_0 \ne \emptyset$.

Then, to see that the path we construct that meets at a point in the first nonempty $A_r \cap B_s$ is a shortest path from $a$ to $b$, if $A_0 \cap B_0 \ne \emptyset$, then $a=b$ and the path is of length $0$.  Otherwise, use the above fact repeatedly, noting that we have some sequence of $A_{r_i} \cap B_{s_i} = \emptyset$, $r_i + s_i = k_i = 1,2,\dots,r+s-1$, implying $A_{k_i} \cap B_0 = \emptyset$ until finally we arrive at $A_r \cap B_s \ne \emptyset \Rightarrow A_{r+s} \cap B_0 \ne \emptyset \Rightarrow d(a,b) = r+s$.

The code in bfs2.py works with the $A_{r_i}$ and $B_{s_i}$ fairly explicitly, using the lists r\_level\_nodes and t\_level\_nodes for them (r for root or $a$, t for target or $b$).  It processes a whole level, or one of these sets, at a time, before going on to the next.  It finds a nonempty intersection of the two by finding a newly encountered node in the one that it's building, that is in the other that it's already built, and stops when this first happens.

\bigskip
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    Copyright \copyright{} 2014 <NAME>.
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in part, as part of another Document.

An MMC is ``eligible for relicensing'' if it is licensed under this
License, and if all works that were first published under this License
somewhere other than this MMC, and subsequently incorporated in whole
or in part into the MMC, (1) had no cover texts or invariant sections,
and (2) were thus incorporated prior to November 1, 2008.

The operator of an MMC Site may republish an MMC contained in the site
under CC-BY-SA on the same site at any time before August 1, 2009,
provided the MMC is eligible for relicensing.


\begin{center}
{\Large\bf ADDENDUM: How to use this License for your documents\par}
\phantomsection
\addcontentsline{toc}{section}{ADDENDUM: How to use this License for your documents}
\end{center}

To use this License in a document you have written, include a copy of
the License in the document and put the following copyright and
license notices just after the title page:

\bigskip
\begin{quote}
    Copyright \copyright{}  YEAR  YOUR NAME.
    Permission is granted to copy, distribute and/or modify this document
    under the terms of the GNU Free Documentation License, Version 1.3
    or any later version published by the Free Software Foundation;
    with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts.
    A copy of the license is included in the section entitled ``GNU
    Free Documentation License''.
\end{quote}
\bigskip
    
If you have Invariant Sections, Front-Cover Texts and Back-Cover Texts,
replace the ``with \dots\ Texts.''\ line with this:

\bigskip
\begin{quote}
    with the Invariant Sections being LIST THEIR TITLES, with the
    Front-Cover Texts being LIST, and with the Back-Cover Texts being LIST.
\end{quote}
\bigskip
    
If you have Invariant Sections without Cover Texts, or some other
combination of the three, merge those two alternatives to suit the
situation.

If your document contains nontrivial examples of program code, we
recommend releasing these examples in parallel under your choice of
free software license, such as the GNU General Public License,
to permit their use in free software.

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\end{document}
