Dijkstra’s algorithm

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Faisal Patel 254
Parth Bharuch 257
Dhavan Shah 160

Introduction
 Our project is about

Introduction Contd.
 Greedy algorithms use problem solving methods based on actions to see if there’s
a better long term strategy.
 Dijkstra’s algorithm uses the greedy approach to solve the single source shortest
problem. It repeatedly selects from the unselected vertices, vertex v nearest to
source s and declares the distance to be the actual shortest distance from s to v.
 The edges of v are then checked to see if their destination can be reached by v
followed by the relevant outgoing edges.
 For a given source node in the graph, the algorithm finds the shortest path
between that node and every other.
 It can also be used for finding the shortest paths from a single node to a single
destination node by stopping the algorithm once the shortest path to the
destination node has been determined.

How It Works??
 Before going into details of the pseudo-code of the algorithm it is important
to know how the algorithm works.
 Dijkstra’s algorithm works by solving the sub-problem k, which computes the
shortest path from the source to vertices among the k closest vertices to the
source.
 For the dijkstra’s algorithm to work it should be directed- weighted graph and
the edges should be non-negative.
 If the edges are negative then the actual shortest path cannot be obtained.

More Detailed Knowledge
 At the kth round, there will be a set called Frontier of k vertices that will
consist of the vertices closest to the source and the vertices that lie outside
frontier are computed and put into New Frontier.
 The shortest distance obtained is maintained in sDist[w].
 It holds the estimate of the distance from s to w.
 Dijkstra’s algorithm finds the next closest vertex by maintaining the New
Frontier vertices in a priority-min queue.

6
Dijkstra's Shortest Path Algorithm
 Find shortest path from s to t.
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distance label
S = { }
PQ = { s, 2, 3, 4, 5, 6, 7, t }

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S = { }
PQ = { s, 2, 3, 4, 5, 6, 7, t }
delmin

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distance label
S = { s }
PQ = { 2, 3, 4, 5, 6, 7, t }
decrease key
X
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distance label
S = { s }
PQ = { 2, 3, 4, 5, 6, 7, t }
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delmin

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S = { s, 2 }
PQ = { 3, 4, 5, 6, 7, t }
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S = { s, 2 }
PQ = { 3, 4, 5, 6, 7, t }
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decrease key
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S = { s, 2 }
PQ = { 3, 4, 5, 6, 7, t }
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delmin

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S = { s, 2, 6 }
PQ = { 3, 4, 5, 7, t }
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S = { s, 2, 6 }
PQ = { 3, 4, 5, 7, t }
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S = { s, 2, 6, 7 }
PQ = { 3, 4, 5, t }
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S = { s, 2, 6, 7 }
PQ = { 3, 4, 5, t }
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S = { s, 2, 3, 6, 7 }
PQ = { 4, 5, t }
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S = { s, 2, 3, 6, 7 }
PQ = { 4, 5, t }
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S = { s, 2, 3, 5, 6, 7 }
PQ = { 4, t }
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S = { s, 2, 3, 5, 6, 7 }
PQ = { 4, t }
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S = { s, 2, 3, 4, 5, 6, 7 }
PQ = { t }
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S = { s, 2, 3, 4, 5, 6, 7 }
PQ = { t }
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S = { s, 2, 3, 4, 5, 6, 7, t }
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S = { s, 2, 3, 4, 5, 6, 7, t }
PQ = { }
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ALgorithm
function Dijkstra(Graph, source):
dist[source] ← 0 // Distance from source to source
prev[source] ← undefined // Previous node in optimal path initialization
for each vertex v in Graph: // Initialization
if v ≠ source // Where v has not yet been removed from Q (unvisited nodes)
dist[v] ← infinity // Unknown distance function from source to v
prev[v] ← undefined // Previous node in optimal path from source
end if
add v to Q // All nodes initially in Q (unvisited nodes)
end for

while Q is not empty:
u ← vertex in Q with min dist[u] // Source node in first case
remove u from Q
for each neighbor v of u: // where v is still in Q.
alt ← dist[u] + length(u, v)
if alt < dist[v]: // A shorter path to v has been found
dist[v] ← alt
prev[v] ← u
end if
end for
end while
return dist[], prev[]
end function

EFFICIENCY
 The complexity efficiency can be expressed in terms of Big-O Notation.
Big-O gives another way of talking about the way input affects the
algorithm’s running time. It gives an upper bound of the running time.
 In Dijkstra’s algorithm, the efficiency varies depending on |V| and |E|
updates for priority queues that were used.
 If a Fibonacci heap was used then the complexity is O( | E | + | V | log |
V | ) , which is the best bound.

DIS-ADVANTAGES
 The major disadvantage of the algorithm is the fact that it does a blind
search there by consuming a lot of time waste of necessary resources.
 Another disadvantage is that it cannot handle negative edges. This leads to
acyclic graphs and most often cannot obtain the right shortest path

APPLICATIONS
 Traffic information systems use Dijkstra’s algorithm in order to track the
source and destinations from a given particular source and destination .
 OSPF- Open Shortest Path First, used in Internet routing.
 It uses a link-state in the individual areas that make up the hierarchy.
 The computation is based on Dijkstra's algorithm which is used to calculate
the shortest path tree inside each area of the network.

Dijkstra’s algorithm

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