Warshall's algorithm -- survey and applications

TL;DR

This paper surveys Warshall's algorithm, discussing its generalizations and various applications, providing a comprehensive overview of its utility in graph theory and related fields.

Contribution

It offers a comprehensive survey of Warshall's algorithm, including its generalizations and diverse applications, which was not extensively covered before.

Findings

Detailed overview of Warshall's algorithm and its variants

Examples of applications in graph connectivity and transitive closure

Identification of open problems and future research directions

Abstract

The survey presents the well-known Warshall's algorithm, a generalization and some interesting applications of this.

Figures

Equations

$$A=(a_{ij}){\footnotesize{}_{\begin{array}[]{l}i=\overline{1,n}\\ j=\overline{1,n}\end{array}}},{\rm\ where}\ \ a_{ij}=\left\{\begin{array}[]{ll}1&{\rm if\ }s_{i}Rs_{j}\\ 0&{\rm otherwise.}\\ \end{array}\right.$$

$$A^{\ast }=A+A^2+\cdots +A^n.$$

$$w_{ij}\leftarrow w_{ij}+w_{ik}{w}_{kj}$$

$$d^{0}_{ij}=\left\{\begin{array}[]{ll}0&\textrm{if }i=j\\ \infty&\textrm{if there is no edge from vertex }v_{i}\textrm{ to vertex }v_{j}\\ w_{ij}&\textrm{the weight of the edge from }v_{i}\textrm{ to }v_{j}\end{array}\right.$$

$$A_{ij}=\left\{\begin{array}[]{cl}\{v_{i}v_{j}\},&\textrm{if there exists an arc from }v_{i}\textrm{ to }v_{j},\\ \emptyset,&\textrm{otherwise},\end{array}\quad\textrm{ for }\,i,j=1,2,\ldots,n,\right.$$

$${\cal AB}=\big{\{}ab\,\big{|}\,a\in{\cal A},b\in{\cal B},\textrm{ if the strings }a\textrm{ and }b\textrm{ have no common elements}\big{\}}.$$

$$a_{ij}=\left\{\begin{array}[]{ll}1,&\textrm{if }j-i\in M,\\ 0,&\textrm{otherwise},\end{array}\quad\textrm{ for }i=1,2,\ldots,n,j=1,2,\ldots,n.\right.$$

$$R=I+A+A^2+\cdots +A^k,\text{ where }{A}^{k+1}=O(\text{the null matrix}).$$

$$K(n,M)=\sum _{i=1}^n\sum _{j=1}^n{r}_{ij}.$$

$$A=\left(\begin{array}[]{cccccc}0&0&1&1&1&1\\ 0&0&0&1&1&1\\ 0&0&0&0&1&1\\ 0&0&0&0&0&1\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ \end{array}\right)$$

$$W=\left(\begin{array}[]{cccccc}0&0&1&1&2&3\\ 0&0&0&1&1&2\\ 0&0&0&0&1&1\\ 0&0&0&0&0&1\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ \end{array}\right),\qquad R=\left(\begin{array}[]{cccccc}1&0&1&1&2&3\\ 0&1&0&1&1&2\\ 0&0&1&0&1&1\\ 0&0&0&1&0&1\\ 0&0&0&0&1&0\\ 0&0&0&0&0&1\\ \end{array}\right)$$

$$A_{ij}=\left\{\begin{array}[]{ll}\{a_{i}a_{j}\},&\textrm{if }j-i\in M,\\ \emptyset,&\textrm{otherwise},\end{array}\quad\textrm{ for }\,i=1,2,\ldots,n,\,j=1,2,\ldots,n.\right.$$

$$w_{pq}=\{a\in \Sigma \mid \exists k\ge 1,\delta (p,a^k)=q\}.$$

Full text

Zoltán KÁSA

Sapientia Hungarian University of Transylvania, Cluj-Napoca

Department of Mathematics and Informatics,

Tg. Mureş, Romania

kasa@ms.sapientia.ro

1 Introduction

Let $R$ be a binary relation on the set $S=\{s_{1},s_{2},\ldots,s_{n}\}$, we write $s_{i}Rs_{j}$ if $s_{i}$ is in relation to $s_{j}$. The relation $R$ can be represented by the so called relation matrix, which is

[TABLE]

The transitive closure of the relation $R$ is the binary relation $R^{*}$ defined as:

$s_{i}R^{*}s_{j}$ if and only if there exists $s_{p_{1}}$, $s_{p_{2}}$, $\ldots$, $s_{p_{r}},r\geq 2$ such that $s_{i}=s_{p_{1}}$, $s_{p_{1}}Rs_{p_{2}}$, $s_{p_{2}}Rs_{p_{3}},\ldots,$ $s_{p_{r-1}}Rs_{p_{r}},$ $s_{p_{r}}=s_{j}$. The relation matrix of $R^{*}$ is $A^{*}=(a^{*}_{ij})$.

Let us define the following operations. If $a,b\in\{0,1\}$ then $a+b=0$ for $a=0,b=0$, and $a+b=1$ otherwise. $a\cdot b=1$ for $a=1,b=1$, and $a\cdot b=0$ otherwise. In this case

[TABLE]

The transitive closure of a relation can be computed easily by the Warshall’s algorithm [6], [1]:

Warshall($A,n$)

[TABLE]

A binary relation can be represented by a directed graph (i.e. digraph) too. he relation matrix is equal to the adjacency matrix of the corresponding graph. See Fig.​ 1 for an example. Fig.​ 2 represents the graph of the corresponding transitive closure relation.

2 Generalization of Warshall’s algorithm

Lines 5 and 6 in the Warshall’s algorithm described above can be changed in

[TABLE]

using the operations defined above. If instead of the operations + and $\cdot$ we use two operations $\oplus$ and $\odot$ from a semiring, a generalized Warshall’s algorithm results [4]:

Generalized-Warshall($A,n$)

[TABLE]

This generalization leads us to a number of interesting applications.

3 Applications

3.1 Distances between vertices. The Floyd-Warshall’s algorithm

Given a weighted (di)graph with the modified adjacency matrix $D_{0}=(d^{0}_{ij})$, we can obtain the distance matrix $D=(d_{ij})$ in which $d_{i}j$ represents the distance between vertices $v_{i}$ and $v_{j}$. The distance between vertices $v_{i}$ and $v_{j}$ is the length of the shortest path between them. The modified adjacency matrix $D_{0}=(d^{0}_{ij})$ is the following:

[TABLE]

Choosing for $\oplus$ the $\min$ operation (minimum between two reals), and for $\odot$ the real addition ($+$), we obtain the well-known Floyd-Warshall’s algorithm as a special case of the generalized Warshall’s algorithm [4, 5] :

Floyd-Warshall($D_{0},n$)

[TABLE]

Figures 3 and 4 contain az example. The shortest paths can be easily obtained if in the description of the algorithm in line 5 we store also the previous vertex $v_{k}$ on the path. In the case of acyclic digraphs, the algorithm can be easily modified to obtain the longest distances between vertices, and consequently the longest paths.

3.2 Number of paths in acyclic digraphs

Here by path we understand directed path. In an acyclic digraph the following algorithm count the number of paths between vertices [3, 2]. The operation $\oplus,\odot$ are the classical add and multiply operations for real numbers.

Warshall-Path($A,n$)

[TABLE]

An example can be seen in Figures 5 and 6. For example between vertices 1 and 3 there are 3 paths: (1,2,3); (1,2,5,3) and (1,6,5,3).

3.3 All paths in digraphs

The Warshall’s algorithm combined with the Latin square method can be used to obtain all paths in a (not necessarily acyclic) digraph [3]. A path will be denoted by a string formed by its vertices in there natural order in the path.

Let us consider a matrix ${\cal A}$ with the elements $A_{ij}$ which are set of strings. Initially elements of this matrix are defined as:

[TABLE]

If ${\cal A}$ and ${\cal B}$ are sets of strings, ${\cal AB}$ will be formed by the set of concatenation of each string from ${\cal A}$ with each string from ${\cal B}$, if they have no common elements:

[TABLE]

If $s=s_{1}s_{2}\cdots s_{p}$ is a string, let us denote by ${}^{\prime}s$ the string obtained from $s$ by eliminating the first character: ${}^{\prime}s=s_{2}s_{3}\cdots s_{p}$. Let us denote by ${}^{\prime}{A_{ij}}$ the set ${A_{ij}}$ in which we eliminate from each element the first character. In this case ${}^{\prime}{\cal A}$ is a matrix with elements ${}^{\prime}A_{ij}.$

Operations are: the set union and set product defined as before.

Starting with the matrix ${\cal A}$ defined as before, the algorithm to obtain all paths is the following:

Warshall-Latin(${\cal A},n$)

[TABLE]

In Figures 7 and 8 an example is given. For example between vertices $v_{1}$ and $v_{3}$ there are two paths: $v_{1}v_{3}$ and $v_{1}v_{2}v_{3}$.

3.4 Scattered complexity for rainbow words

The application mentioned here can be found in [3]. Let $\Sigma$ be an alphabet, $\Sigma^{n}$ the set of all length-$n$ words over $\Sigma$, $\Sigma^{*}$ the set of all finite word over $\Sigma$.

Definition 1

Let $n$ and $s$ be positive integers, $M\subseteq\{1,2,\ldots,n-1\}$ and $u=x_{1}x_{2}\ldots x_{n}\in\Sigma^{n}$. An $M$-subword of length $s$ of $u$ is defined as $v=x_{i_{1}}x_{i_{2}}\ldots x_{i_{s}}$ where

$i_{1}\geq 1$,

$i_{j+1}-i_{j}\in M$* for $j=1,2,\ldots,s-1$,*

$i_{s}\leq n.$**

Definition 2

The number of $M$-subwords of a word $u$ for a given set $M$ is the scattered subword complexity, simply $M$-complexity.

Examples. The word $abcd$ has 11 $\{1,3\}$-subwords: $a$, $ab$, $abc$, $abcd$, $ad$, $b$, $bc$, $bcd$, $c$, $cd$, $d$. The $\{2,3\,4,5\}$-subwords of the word $abcdef$ are the following: $a$, $ac$, $ad$, $ae$, $af$, $ace$, $acf$, $adf$, $b$, $bd$, $be$, $bf$, $bdf$, $c$, $ce$, $cf$, $d$, $df$, $e$, $f$.

Words with different letters are called rainbow words. The $M$-complexity of a length-$n$ rainbow word does not depend on what letters it contains, and is denoted by $K(n,M)$.

To compute the $M$-complexity of a rainbow word of length $n$ we will use graph theoretical results. Let us consider the rainbow word $a_{1}a_{2}\ldots a_{n}$ and the corresponding digraph $G=(V,E)$, with

V=\big{\{}a_{1},a_{2},\ldots,a_{n}\big{\}},

E=\big{\{}(a_{i},a_{j})\mid j-i\in M,\,i=1,2,\ldots,n,j=1,2,\ldots,n\big{\}}.

For $n=6,M=\{2,3,4,5\}$ see Fig. 9.

The adjacency matrix A=\big{(}a_{ij}\big{)}_{\!\!\!\tiny\begin{array}[]{c}i\!\!=\!\!\overline{1,\!n}\\ j\!\!=\!\!\overline{1,\!n}\end{array}} of the graph is defined by:

[TABLE]

Because the graph has no directed cycles, the element in row $i$ and column $j$ in $A^{k}$ (where $A^{k}=A^{k-1}A$, with $A^{1}=A$) will represent the number of length-$k$ directed paths from $a_{i}$ to $a_{j}$. If $I$ is the identity matrix (with elements equal to 1 only on the first diagonal, and 0 otherwise), let us define the matrix $R=(r_{ij})$:

[TABLE]

The $M$-complexity of a rainbow word is then

[TABLE]

Matrix $R$ can be better computed using the Warshall-Path algorithm.

From $W$ we obtain easily $R=I+W$.

For example let us consider the graph in Fig. 9. The corresponding adjacency matrix is:

[TABLE]

After applying the Warshall-Path algorithm:

[TABLE]

and then K\big{(}6,\{2,3,4,5\}\big{)}=20, the sum of elements in $R$.

Using the Warshall-Latin algorithm we can obtain all nontrivial (with length at least 2) $M$-subwords of a given length-$n$ rainbow word $a_{1}a_{2}\cdots a_{n}$. Let us consider a matrix ${\cal A}$ with the elements $A_{ij}$ which are set of strings. Initially this matrix is defined as:

[TABLE]

The set of nontrivial $M$-subwords is ${\displaystyle\bigcup_{i,j\in\{1,2,\ldots,n\}}W_{ij}}$.

For $n=8$, $M=\{3,4,5,6,7\}$ the initial matrix is:

\left(\begin{array}[]{cccccccc}\emptyset&\emptyset&\emptyset&\{ad\}&\{ae\}&\{af\}&\{ag\}&\{ah\}\\ \emptyset&\emptyset&\emptyset&\emptyset&\{be\}&\{bf\}&\{bg\}&\{bh\}\\ \emptyset&\emptyset&\emptyset&\emptyset&\emptyset&\{cf\}&\{cg\}&\{ch\}\\ \emptyset&\emptyset&\emptyset&\emptyset&\emptyset&\emptyset&\{dg\}&\{dh\}\\ \emptyset&\emptyset&\emptyset&\emptyset&\emptyset&\emptyset&\emptyset&\{eh\}\\ \emptyset&\emptyset&\emptyset&\emptyset&\emptyset&\emptyset&\emptyset&\emptyset\\ \emptyset&\emptyset&\emptyset&\emptyset&\emptyset&\emptyset&\emptyset&\emptyset\\ \emptyset&\emptyset&\emptyset&\emptyset&\emptyset&\emptyset&\emptyset&\emptyset\\ \end{array}\right).

The result of the algorithm in this case is:

\left(\begin{array}[]{cccccccc}\emptyset&\emptyset&\emptyset&\{ad\}&\{ae\}&\{af\}&\{ag,adg\}&\{ah,adh,aeh\}\\ \emptyset&\emptyset&\emptyset&\emptyset&\{be\}&\{bf\}&\{bg\}&\{bh,beh\}\\ \emptyset&\emptyset&\emptyset&\emptyset&\emptyset&\{cf\}&\{cg\}&\{ch\}\\ \emptyset&\emptyset&\emptyset&\emptyset&\emptyset&\emptyset&\{dg\}&\{dh\}\\ \emptyset&\emptyset&\emptyset&\emptyset&\emptyset&\emptyset&\emptyset&\{eh\}\\ \emptyset&\emptyset&\emptyset&\emptyset&\emptyset&\emptyset&\emptyset&\emptyset\\ \emptyset&\emptyset&\emptyset&\emptyset&\emptyset&\emptyset&\emptyset&\emptyset\\ \emptyset&\emptyset&\emptyset&\emptyset&\emptyset&\emptyset&\emptyset&\emptyset\\ \end{array}\right).

3.5 Special paths in finite automata

Let us consider a finite automaton $A=(Q,\Sigma,\delta,\{q_{0}\},F)$, where $Q$ is a finite set of states, $\Sigma$ the input alphabet, $\delta:Q\times\Sigma\rightarrow Q$ the transition function, $q_{0}$ the initial state, $F$ the set of finale states. In following we do not need to mark the initial and the finite states.

The transition function can be generalized for words too: $\delta(q,wa)=\delta(\delta(q,w),a)$, where $q\in Q,a\in\Sigma,w\in\Sigma^{*}$.

We are interesting in finding for each pair $p,q$ of states the letters $a$ for which there exists a natural $k\geq 1$ such that we have the transition $\delta(p,a^{k})=q$ [4], i.e.:

[TABLE]

Instead of $\oplus$ we use here set union ($\cup$) and instead of $\odot$ set intersection ($\cap$).

Warshall-Automata($A,n$)

[TABLE]

The transition table of the finite automaton in Fig. 10 is:

\begin{array}[]{c|cccc}\delta&a&b&c&d\\ \hline\cr q_{1}&\{q_{1},q_{2}\}&\{q_{1}\}&\emptyset&\{q_{5}\}\\ q_{2}&\{q_{1}\}&\{q_{3}\}&\{q_{2}\}&\{q_{3}\}\\ q_{3}&\emptyset&\{q_{4}\}&\emptyset&\emptyset\\ q_{4}&\emptyset&\{q_{5}\}&\emptyset&\emptyset\\ q_{5}&\emptyset&\{q_{2}\}&\emptyset&\emptyset\end{array}

Matrices for graph in Fig. 10 are the following:

A=\left(\begin{array}[]{ccccc}\{a,b\}&\{a\}&\emptyset&\emptyset&\{d\}\\ \{a\}&\{c\}&\{b,d\}&\emptyset&\emptyset\\ \emptyset&\emptyset&\emptyset&\{b\}&\emptyset\\ \emptyset&\emptyset&\emptyset&\emptyset&\{b\}\\ \emptyset&\{b\}&\emptyset&\emptyset&\emptyset\end{array}\right)

W=\left(\begin{array}[]{ccccc}\{a,b\}&\{a\}&\emptyset&\emptyset&\{d\}\\ \{a\}&\{a,b,c\}&\{b,d\}&\{b\}&\{b\}\\ \emptyset&\{b\}&\{b\}&\{b\}&\{b\}\\ \emptyset&\{b\}&\{b\}&\{b\}&\{b\}\\ \emptyset&\{b\}&\{b\}&\{b\}&\{b\}\\ \end{array}\right)

For example $\delta(q_{2},bb)=q_{4}$, $\delta(q_{2},bbb)=q_{5}$, $\delta(q_{2},bbbb)=q_{2}$, $\delta(q_{2},c^{k})=q_{2}$ for $k\geq 1$.