Algorithm for shortest path search in Geographic Information Systems by using reduced graphs

Definition 1

Definition 2

Definition 3

A graph rewrite rule also can be defined over an undirected graph, in this case, the sets ψ _in and ψ _out must be represented as an only set called ψ.

The set of edges that join vertex v _i with the vertices of the graph G − G _i are called pre-embedding edges. After applying a rewrite rule, the edges that join a vertex of the graph G _j with a vertex of the graph G − G _j are called post-embedding edges. The function ψ _in transforms the set of pre-embedding edges that are incident in a vertex v _i in post-embedding edges that are incident in one or more vertices v _j  ∈ V _j . Similarly, the function ψ _out transforms pre-embedding outgoing edges from a vertex v _i in one or more post-embedding outgoing edges from several vertices v _j  ∈ V _j .

Definition 4

This definition is particularly important when it is associated with another graph, i.e., when a graph is reduced from another graph. We can state that a graph G _r  = (V _r , E _r , f, R) is reduced from a graph G = (V, E, f _c ), when applying the set of rewrite rules R to the graph G _r , the graph G is obtained.

In the case of function f, for all 3-tuple of vertices v _i , v _j , v _k  ∈ V _r it holds that f(v _i , v _j , v _k ) = f _c (v _i , v _j ) + f _c (v _j , v _k ). Notice that f(v _i , v _i , v _j ) = f _c (v _i , v _j ). If v _i and v _j are not adjacent, the image of both functions would be infinite. This is the way in which we specify that two vertices are not adjacent.

Definition 5

Let a graph G = (V, E) and a partition P on V, a vertex v _i  ∈ V is internal if ∀v _j  ∈ V, such that v _i and v _j are adjacent, it holds that v _i and v _j are in the same class of P; i.e. [v _i ] = [v _j ] otherwise v _i is external.

In Figure 1, we show an example of how to refine a partition using definitions of internal and external vertex.

Next, we create a new vertex w _i for each A _i ∈P,|A _i |>1, i=1..s. V ^′=w _i is a set of reduced vertices and V − V ^′ is the set of unreduced vertices in the reduced graph.

We add a vertex in the reduced graph for each class of the partition calculated in the previous step. If the cardinality of the class is 1, the vertex is considered as an unreduced vertex; in any other case, it is considered as a reduced one (GetReducedVertices method). Next, a set of edges is calculated. One edge can be added to the reduced graph if the two vertices of the edge belong to different equivalence classes (GetEdges method). With the addition of edges to the reduced graph, the cost function f _r of the reduced graph must be updated.

The creation of the set of rewrite rules is an essential step in the reduction algorithm. With the rewrite rules, the original graph can be obtained from the reduced graph. Therefore, rewrite rules guarantee no loss of information, and so the reduction process is reversible.

The previous explanation corresponds to the implementation of GetRewriteRules method.

Another step that contributes to obtain the optimal path is the calculation of function f _r . Function f _r stores the cost of the shortest path from one vertex to another, traversing a reduced one.

Additionally, for all 3-tuples of vertices v _i , v _j , v _k  ∈ V, where v _j is a non-reduced vertex, f _r (v _i , v _j , v _k ) = f(v _i , v _j , v _k ).

Path from v _i to v _k is also stored, with the goal of avoiding additional run time, when the shortest path search in a reduced graph is retrieved.

Algorithm 1 provides the detailed pseudo-code of the graph reduction algorithm.

Algorithm 1 GraphReduction

The complexity of the reduction algorithm would be determined by steps 6-8. According to the above description of Updatefr, this method calculates shortest path from all external vertices (taking into account the original graph) of v _j to all vertices of the auxiliary graph.

In a graph obtained from a network in a map, a vertex represents the intersection of two or more lines and an edge represents the connection between two intersections. That is why, in this kind of graph, there are no edges that intersect among them. Thus, we can assume that graphs representing the modeled network through a map are planar.

Moreover, in a graph with these characteristics, the degree of a vertex is generally equal to 4, except in a few cases. Thus it is assumed, without loss of generalization, that the degree of a graph that represents a network of this type is less than or equal to 10. Let Δ(G ⁺) the degree of G, the auxiliary graph has, at most, a · Δ(G ⁺) vertices. In Updatefr method, MDijkstra algorithm is called for each adjacent vertex to any vertex of v _j (see Shortest path search algorithm section for temporal complexity of this algorithm), so the temporal complexity, in the worst case, is: O(a · Δ(G ⁺) · a · Δ(G ⁺) log(a · Δ(G ⁺))) = O(Δ(G ⁺)² · a ²· log(a) + log(Δ(G ⁺)))

The terms involving Δ(G ⁺) are constant, so the temporal complexity is O(a ²· log(a)).

As a conclusion, the temporal complexity of Algorithm 1 is of polynomial order. The reduction process is made only once, as data preprocessing. This preprocessing task causes an increased in the spatial complexity but, with this approach, we can obtain lower run time in every shortest path computation over the reduced graph.

Algorithm 2 Graph Rewrite Rule Application

After applying the rewrite rule we have obtained the graph G (Figure 2(a)). Thus, in the reduction process does not exist loss of information, that is, the reduction is reversible.

Theorem 1

∀n ∈ {1, 2, .., N − 1}[C a(0) = 0 → ∀ m < n + 1(C a(m) < N) → ∀m < n + 1[D c(C a(m)) + f _c (C a(m), C a(m + 1)) = D c(C a(m + 1))] → D _N−1(C a(n)) ≤ D c(C a(n))]

Proof

(By induction on n)

Base case n = 0 immediate by Lemma 2,

For n = k + 1:

The distance to a vertex v _i is less than or equal to the distance to a visited vertex v _j plus the distance from v _j to v _i , by Lemma 5

By Lemma 3, D c(C a(k v+ 1)) = D c(P c _N−1(P c _N−1(C a(k + 1)))) + f(P c _N−1(P c _N−1(C a(k + 1))), P c _N−1(C a(k + 1)), C a(k + 1)), replacing in (3), D _N−1(C a(k + 1)) ≤ D c(C a(k + 1)). □

We can prove the correctness of Dijkstra’s algorithm with a similar reasoning because the same invariants are satisfied. Thus, for the next proof we assume that Dijkstra’s algorithm is correct and satisfies invariants analogous to those defined for Algorithm 3.

As demonstrated before, Algorithm 3 returns the shortest path in the reduced graph. However, it remains to prove that the cost of the shortest path obtained by the proposed algorithm and the one obtained by Dijkstra’s algorithm (in the original graph without reducing it) are the same.

Theorem 2

Let C a = (v ₁, ..., v _n ) be a path of cost c obtained by applying Dijkstra’s algorithm on the graph G, where v ₁ and v _n are unreduced vertices on the graph G _r , then ∃C a ^′ = (u ₁, u ₂, …, u _t ) with cost c, u ₁ = v ₁, u _t  = v _n , such that C a ^′ is an optimal path on G _r .

Proof

The cost of the path v _i , v _k , v _i+m is equal to the cost of the path v _i , v _i+1, …, v _i+m , by definition of function f. Thus, the paths C a and C a ^′ have the same cost.

Therefore paths C b and C b ^′ have the same cost (c ₁), this leads a contradiction. Thus, there is no path that has less cost than C a. □

Corollary 1

Let C a = (v ₁, ..., v _n ) a path obtained by applying Dijkstra’s algorithm on graph G, ∀i ∈ {1, 2, ..., n} such that C a[i] is an unreduced vertex in G _r , it holds that the distance to C a[i] is equal to the distance obtained by MDijkstra algorithm on the reduced graph from v ₁ to C a[i].

Theorem 2 establishes that the cost of the shortest path from a vertex v _i to any vertex v _j (v _i and v _j being unreduced vertices in G _r ) obtained by applying Algorithm 3 is the same as the cost of the shortest path calculated by Dijkstra’s algorithm in the original graph (without reduction).

The fact that both source and destination must be unreduced vertices could be a limiting factor (in terms of the number of vertices to which one can calculate the shortest path) if one does not have a mechanism that allows obtaining a reduced graph G _{r i} from G _r where v _i  ∈ V (v _i is a vertex in the original graph G = (V, E)) is an unreduced vertex on G _{r i} . This can be accomplished by one or more expansions applying rewrite rules to the reduced vertex that contains vertex v _i .

Lemma 1

∀n < N, |C _n | = n + 1

Proof

(By induction on n)

From the definition of the algorithm, at each step a vertex w is visited, in step 0 vertex v _o is visited, thus in the base case we have C ₀ = {v _o }, |C ₀| = 1,

For n = k + 1, $C_{k+1}=C_k\bigcup \left\{v\right\}$, being v the visited vertex in step k + 1, therefore |C _k+1| = |C _k | + |{v}| = k + 2. □

Lemma 2

∀n < N, D _n (v _o ) = 0 ∧ P _n (v _o ) = v _o

Proof

First, we visit vertex v _o and update D _n (v _o ) = 0, i.e., the minimum distance from v _o to itself is 0, the function D _n has its domain in ${\mathbb{R}}^{+}\bigcup \{0,\infty \}$, so the smallest possible value that can be achieved is 0;

Let c o s t = D _n (P _n (w _n )) + f(P _n (w _n ), w _n , v), ∀w _n , v ∈ V, it holds that 0≤0+c o s t, because the image of the function f is ${\mathbb{R}}^{+}\cup \{0,\infty \}$ and the vector D(V _r ) is initialized from f.

The condition D _n (v _o ) > D _n (w _n ) + f(P _n (w _n ), w _n , v _o ) is never satisfied, thus D _n [v _o ] and P _n [v _o ] never change. □

Lemma 3

∀n < N, D _n (v) = D _n (P _n (P _n (v))) + f(P _n (P _n (v)), P _n (v), v)

Proof

(By induction on n)

The base case n = 0, ∀v ∈ V _r , D ₀(v) = f _c (v _o , v), by preconditions.

f(v _o , v _o , v) = f _c (v _o , v _o ) + f _c (v _o , v) = f _c (v _o , v), by definition of f and f _c , replacing f by f _c :

D ₀(v) = 0 + f(v _o , v _o , v)D ₀(v) = D ₀(v _o ) + f(v _o , v _o , v), by Lemma 2

D ₀(v) = D ₀(P ₀(v _o )) + f(P ₀(v _o ), v _o , v), by Lemma 2

For n = k + 1:

Choose w _k+1 ∈ V ∖ C _k such that D _k+1(w _k+1) is minimal, $C_{k+1}=C_k\bigcup \left\{w_{k+1}\right\}$.

Case 1: If D _k+1(v) > D _k+1(P _k+1(w _k+1)) + f(P _k+1 (w _k+1), w _k+1, v), then D _k+1(v) = D _k+1(P _k+1 (w _k+1)) + f(P _k+1(w _k+1), w _k+1, v) ∧ P _k+1(v) = w _k+1

Case 2: If case 1 is not satisfied, D _k+1(v) = D _k (v), P _k+1(v) = P _k (v), D _k+1(v) = D _k (P _k (P _k (v))) + f(P _k (P _k (v)), P _k (v), v), by induction hypothesis, replacing P _k (v) by P _k+1(v) D _k+1(v) = D _k (P _k+1(P _k+1(v))) + f(P _k+1(P _k+1(v)), P _k+1(v), v). □

Lemma 4

∀v ∈ C _n D _n+1(v) = D _n (v)

Proof

Let v ∈ C _n , w _n  ∈ V ∖ C _n

w _n  ∈ C _n+1 by definition.

D _n (v) ≤ D _n (w _n ), otherwise vertex w _n was visited before vertex v,

D _n (v) ≤ D _n (w _n ) + f _c (w _n , v) = D _n (P _n (w _n )) + f(P _n (w _n ), w _n , v),

D _n+1(v) = D _n (v), by definition of D _n+1(v). □

Lemma 5

∀n < N, ∀v _i , v _j  ∈ V[v _j  ∈ C _n+1 → D _n+1(v _i ) ≤ D _n+1(P _n (v _j )) + f(P _n (v _j ), v _j , v _i )]

Proof

(By induction on n)

The base case n = 0, C ₀ = {v _j }, D ₀(v _j ) = 0, by definition, notice that v _j is the only vertex in C ₀ (in the base case, if v _j  ∈ C ₀, v _j is the origin vertex).

P _n (v _j ) = v _j by Lemma 2.

∀v _i  ∈ V, f(v _j , v _j , v _i ) = f(P ₀(v _j ), v _j , v _i ), by definition of f, and D ₀(v _i ) = f(v _j , v _j , v _i ). Thus D ₀(v _i ) ≤ 0 + f(P ₀(v _j ), v _j , v _i ), D ₀(v _j ) = 0 = D ₀(P ₀(V _j )) D ₀(v _i ) ≤ D ₀(P ₀(v _j )) + f(P ₀(v _j ), v _j , v _i )

D ₁(v _i ) ≤ D ₀(v _i ), from the definition (D _n+1(v) = M in(D _n (w _n ) + f _c (w _n , v), D _n (v))) and D ₁(v _j ) = D ₀(v _j ) = 0 (notice that v _j is the origin vertex). Replacing D ₀ by D ₁:

D ₁(v _i ) ≤ D ₁(P ₀(v _j )) + f(P ₀(v _j ), v _j , v _i )

For n = k + 1:

Case 1: v _j  ∈ C _k D _k+1(v _i ) ≤ D _k (v _i ), by definition D _k+1(v _i ) ≤ D _k (P _k (v _j )) + f(P _k (v _j ), v _j , v _i ), by induction hypothesis D _k+1(v _i ) ≤ D _k+1(P _k (v _j )) + f(P _k (v _j ), v _j , v _i ) by Lemma 4

Case 2: v _j  = w _k .

D _k+1(v _i ) ≤ D _k (P _k (w _k )) + f(P _k (w _k ), w _k , v _i ), by definition D _k+1(v _i ) ≤ D _k+1(P _k (w _k )) + f(P _k (w _k ), w _k , v _i ), by Lemma 4, replacing w _n by v _j :

D _k+1(v _i ) ≤ D _k+1(P _k (v _j )) + f(P _k (v _j ), v _j , v _i ). □