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Number of paths of fixed length / Shortest paths of fixed length

The following article describes solutions to these two problems built on the same idea: reduce the problem to the construction of matrix and compute the solution with the usual matrix multiplication or with a modified multiplication.

Number of paths of a fixed length

We are given a directed, unweighted graph $G$ with $n$ vertices and we are given an integer $k$. The task is the following: for each pair of vertices $(i, j)$ we have to find the number of paths of length $k$ between these vertices. Paths don't have to be simple, i.e. vertices and edges can be visited any number of times in a single path.

We assume that the graph is specified with an adjacency matrix, i.e. the matrix $G[][]$ of size $n \times n$, where each element $G[i][j]$ equal to $1$ if the vertex $i$ is connected with $j$ by an edge, and $0$ is they are not connected by an edge. The following algorithm works also in the case of multiple edges: if some pair of vertices $(i, j)$ is connected with $m$ edges, then we can record this in the adjacency matrix by setting $G[i][j] = m$. Also the algorithm works if the graph contains loops (a loop is an edge that connect a vertex with itself).

It is obvious that the constructed adjacency matrix if the answer to the problem for the case $k = 1$. It contains the number of paths of length $1$ between each pair of vertices.

We will build the solution iteratively: Let's assume we know the answer for some $k$. Here we describe a method how we can construct the answer for $k + 1$. Denote by $C_k$ the matrix for the case $k$, and by $C_{k+1}$ the matrix we want to construct. With the following formula we can compute every entry of $C_{k+1}$:

$$C_{k+1}[i][j] = \sum_{p = 1}^{n} C_k[i][p] \cdot G[p][j]$$

It is easy to see that the formula computes nothing other than the product of the matrices $C_k$ and $G$:

$$C_{k+1} = C_k \cdot G$$

Thus the solution of the problem can be represented as follows:

$$C_k = \underbrace{G \cdot G \cdots G}_{k \text{ times}} = G^k$$

It remains to note that the matrix products can be raised to a high power efficiently using Binary exponentiation. This gives a solution with $O(n^3 \log k)$ complexity.

Shortest paths of a fixed length

We are given a directed weighted graph $G$ with $n$ vertices and an integer $k$. For each pair of vertices $(i, j)$ we have to find the length of the shortest path between $i$ and $j$ that consists of exactly $k$ edges.

We assume that the graph is specified by an adjacency matrix, i.e. via the matrix $G[][]$ of size $n \times n$ where each element $G[i][j]$ contains the length of the edges from the vertex $i$ to the vertex $j$. If there is no edge between two vertices, then the corresponding element of the matrix will be assigned to infinity $\infty$.

It is obvious that in this form the adjacency matrix is the answer to the problem for $k = 1$. It contains the lengths of shortest paths between each pair of vertices, or $\infty$ if a path consisting of one edge doesn't exist.

Again we can build the solution to the problem iteratively: Let's assume we know the answer for some $k$. We show how we can compute the answer for $k+1$. Let us denote $L_k$ the matrix for $k$ and $L_{k+1}$ the matrix we want to build. Then the following formula computes each entry of $L_{k+1}$:

$$L_{k+1}[i][j] = \min_{p = 1 \ldots n} \left(L_k[i][p] + G[p][j]\right)$$

When looking closer at this formula, we can draw an analogy with the matrix multiplication: in fact the matrix $L_k$ is multiplied by the matrix $G$, the only difference is that instead in the multiplication operation we take the minimum instead of the sum.

$$L_{k+1} = L_k \odot G,$$

where the operation $\odot$ is defined as follows:

$$A \odot B = C~~\Longleftrightarrow~~C_{i j} = \min_{p = 1 \ldots n}\left(A_{i p} + B_{p j}\right)$$

Thus the solution of the task can be represented using the modified multiplication:

$$L_k = \underbrace{G \odot \ldots \odot G}_{k~\text{times}} = G^{\odot k}$$

It remains to note that we also can compute this exponentiation efficiently with Binary exponentiation, because the modified multiplication is obviously associative. So also this solution has $O(n^3 \log k)$ complexity.

Generalization of the problems for paths with length up to $k$

The above solutions solve the problems for a fixed $k$. However the solutions can be adapted for solving problems for which the paths are allowed to contain no more than $k$ edges.

This can be done by slightly modifying the input graph.

We duplicate each vertex: for each vertex $v$ we create one more vertex $v'$ and add the edge $(v, v')$ and the loop $(v', v')$. The number of paths between $i$ and $j$ with at most $k$ edges is the same number as the number of paths between $i$ and $j'$ with exactly $k + 1$ edges, since there is a bijection that maps every path $[p_0 = i,~p_1,~\ldots,~p_{m-1},~p_m = j]$ of length $m \le k$ to the path $[p_0 = i,~p_1,~\ldots,~p_{m-1},~p_m = j, j', \ldots, j']$ of length $k + 1$.

The same trick can be applied to compute the shortest paths with at most $k$ edges. We again duplicate each vertex and add the two mentioned edges with weight $0$.