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Shortest path problems

Single-Pair Shortest path problem

Input:

• A weighted graph/digraph
• A source node(start)
• A destination node(end) Output:
• A shortest path through the graph from the source to the destination

Single-Source shortest path problem

• A weighted graph/digraph
• A source node(start) Output:
• A set of shortest paths through the graph from the source to every node in the graph
• Can be represented as a tree with the source as the node

Dijkstra's algorithm Greedy

Maintain a set of explored nodes $$S$$

let $$d[u]$$ be the length of a shortest $$s \rightarrow u$$ path

initialize $$S \leftarrow {s}$$ , $$d[u] \leftarrow 0$$

repeatedly:

• choose unexplored node $$v\notin S$$ that minimises $$\pi(v)$$ where:

$$ \pi(v) = \min_{e = (u,v):u \in S}

$$

• add $$v$$ to $$S$$ , $$d[v]\leftarrow \pi(v)$$
• prev[v] $$\leftarrow e$$

The shortest path can then be found by traversing from any point to the start by following prev

Proof

Prove:

For every node $$u \in S$$, $$d[u]$$ is the shortest path $$s \rightarrow u$$

strategy: induction

base Case:

when $$\vert S \vert = 1$$ is easy as $$S = {s}$$, as $$s[s]=0$$ there is no shorter solution

Inductive case:

Assume is true for $$\vert S \vert \geq 1$$

Let $$v$$ be the next path added to $$S$$ and let $$(u,v)$$ be the edge

A shortest path $$s \rightarrow u$$ + $$L(u,v)$$ is a $$s \rightarrow v$$ path of length $$\pi(v)$$

Consider anther path $$P$$

• let $$e = (x,y)$$ be the first edge in $$P$$ that leaves $$S$$
  • so $$x \in S \land y \notin nS$$
• let $$P'$$ be the subpath of $$s \rightarrow x$$
• the length of $$P \geq \pi(v)$$ as soon as it reaches y

$$ L(P) \geq L(P')+ L(e) \geq d[x]+L(e) \geq \pi(y) \geq \pi(v) $$

when L determines the length of a path or edge

Dijkstra's algorithm Efficient

Efficient algorithm can be found by implementing some optimizations on the previous strategy.

Optimization 1

For each unexplored node $$v \in S$$

Maintain $$\pi(v)$$ instead of computing form the definition

As elements are added to $$S$$ for some $$v\notin S , \pi(v)$$ can only decrease.

Suppose $$u$$ is added to $$S$$ if there is an edge $$e=(u,v)$$ leaving $$u$$

then:

$$ pi(v) \leftarrow \min {\pi(v), \pi(u)+L(e)} $$

Optimization 2

Use a min-optimized priority queue to chose an unexplored node that minimises $$\pi(v)$$

Implementation

graph <- graph to search
s <- start node

dist <- array of distances
prev <- array of previous nodes

//initialize start values
pq = priorityQueue()
for each vertex in graph{
    dist[vertex] = INFINITY
    prev[vertex] = UNDEFINED
}
dist[s] = 0;
for each vertex in graph{
    pq.insert(s,dist[s]);
}
// calculate minimum path
while (pq.empty =false){
    u = pq.removeMin();
    for edge in graph.edgesFrom(u){
        v = edge.goingTo() 
        if (dist[v] > dist[u]+edge.weight){
            dist[v]=dist[u]+edge.weight
            pq.decreaseKey(v,dist[v])
            prev[v]=u
        }
    }

}

The shortest path for a node can be found by traversing prev from the node to the start

Complexity

The time complexity fo the algorithm depends on the priority queue algorithm chosen and the ratio of edges to nodes.

For a dense graph $$ e $$ is $$ O(n^2)$$ (where $$e$$ is the number of edges and $$n$$ is the number of nodes) using an array bases implementation is optimal as decreaseKey is $$O(1)$$

For a sparse graph $$ e $$ is $$ O(n)$$ then a heap based method is better as removeMin is $$O(n)$$