# The problem definition¶

Given the following query:

pgr_dijkstra($$sql, start_{vid}, end_{vid}, directed$$)

where $$sql = \{(id_i, source_i, target_i, cost_i, reverse\_cost_i)\}$$

and

• $$source = \bigcup source_i$$,
• $$target = \bigcup target_i$$,

The graphs are defined as follows:

Directed graph

The weighted directed graph, $$G_d(V,E)$$, is definied by:

• the set of vertices $$V$$
• $$V = source \cup target \cup {start_{vid}} \cup {end_{vid}}$$
• the set of edges $$E$$
• $$E = \begin{cases} &\{(source_i, target_i, cost_i) \text{ when } cost >=0 \} &\quad \text{ if } reverse\_cost = \varnothing \\ \\ &\{(source_i, target_i, cost_i) \text{ when } cost >=0 \} \\ \cup &\{(target_i, source_i, reverse\_cost_i) \text{ when } reverse\_cost_i >=0)\} &\quad \text{ if } reverse\_cost \neq \varnothing \\ \end{cases}$$

Undirected graph

The weighted undirected graph, $$G_u(V,E)$$, is definied by:

• the set of vertices $$V$$
• $$V = source \cup target \cup {start_v{vid}} \cup {end_{vid}}$$
• the set of edges $$E$$
• $$E = \begin{cases} &\{(source_i, target_i, cost_i) \text{ when } cost >=0 \} \\ \cup &\{(target_i, source_i, cost_i) \text{ when } cost >=0 \} &\quad \text{ if } reverse\_cost = \varnothing \\ \\ &\{(source_i, target_i, cost_i) \text{ when } cost >=0 \} \\ \cup &\{(target_i, source_i, cost_i) \text{ when } cost >=0 \} \\ \cup &\{(target_i, source_i, reverse\_cost_i) \text{ when } reverse\_cost_i >=0)\} \\ \cup &\{(source_i, target_i, reverse\_cost_i) \text{ when } reverse\_cost_i >=0)\} &\quad \text{ if } reverse\_cost \neq \varnothing \\ \end{cases}$$

The problem

Given:

• $$start_{vid} \in V$$ a starting vertex
• $$end_{vid} \in V$$ an ending vertex
• $$G(V,E) = \begin{cases} G_d(V,E) &\quad \text{ if } directed = true \\ G_u(V,E) &\quad \text{ if } directed = false \\ \end{cases}$$

Then:

$\begin{split}\text{pgr_dijkstra}(sql, start_{vid}, end_{vid}, directed) = \begin{cases} \text{shortest path } \boldsymbol{\pi} \text{ between } start_{vid} \text{and } end_{vid} &\quad \text{if } \exists \boldsymbol{\pi} \\ \varnothing &\quad \text{otherwise} \\ \end{cases}\end{split}$

$$\boldsymbol{\pi} = \{(path_\seq_i, node_i, edge_i, cost_i, agg\_cost_i)\}$$

where:
• $$path_\seq_i = i$$
• $$path_\seq_{| \pi |} = | \pi |$$
• $$node_i \in V$$
• $$node_1 = start_{vid}$$
• $$node_{| \pi |} = end_{vid}$$
• $$\forall i \neq | \pi |, \quad (node_i, node_{i+1}, cost_i) \in E$$
• $$edge_i = \begin{cases} id_{(node_i, node_{i+1},cost_i)} &\quad \text{when } i \neq | \pi | \\ -1 &\quad \text{when } i = | \pi | \\ \end{cases}$$
• $$cost_i = cost_{(node_i, node_{i+1})}$$
• $$agg\_cost_i = \begin{cases} 0 &\quad \text{when } i = 1 \\ \displaystyle\sum_{k=1}^{i} cost_{(node_{k-1}, node_k)} &\quad \text{when } i \neq 1 \\ \end{cases}$$
In other words: The algorithm returns a the shortest path between $$start_{vid}$$ and $$end_{vid}$$ , if it exists, in terms of a sequence of nodes and of edges,
• $$path_\seq$$ indicates the relative position in the path of the $$node$$ or $$edge$$.
• $$cost$$ is the cost of the edge to be used to go to the next node.
• $$agg\_cost$$ is the cost from the $$start_{vid}$$ up to the node.

If there is no path, the resulting set is empty.