Dijkstra's algorithm
Shortest paths in edge-weighted digraph
Krasin Georgiev
Assistant Professor
Technical University of Sofia
g.krasin at gmail com
Table of Contents
1.
Background
2.
The problem
3.
Properties and assumptions
4.
Applications
5.
Dijkstra's algorithm and Pseudocode
6.
C# Demo
7.
Related problems and algorithms
8.
Resources
2
Background
 Let
G=(V,E) be a (di)graph.
 In directed graphs, edges are
one-way
 An edge-weighted graph is
a graph
where we associate weights or costs with each
edge (or other attributes).
 A shortest path from vertex s
to vertex t is a
directed path from s to t with the property
that no other such path has a lower weight.
3
The Problem
Single-Source Shortest Path Problem
Find the shortest paths
from a source vertex
to all other vertices in the graph
4
Properties and Assumptions
 Paths are directed.
 The weights are not necessarily
distances
(could be time or cost).
 Edge weights are positive (or zero)
 Shortest paths are not necessarily
unique
 Not all vertices need be reachable
 Parallel
edges may be present
5
Algorithm Applications
Application
Vertex
Edge
Solutions
Map
Intersection
Road
Find the shortest route
Find the fastest route
Find the best route
Network
Router
Connection
Internet routing
Flight Agenda Airports
Flights
Find earliest time to reach destination
Epidemiology Individuals
Possible contacts Model the spread of infectious diseases
Arbitrage
Exchange rate
Currency
6
Algorithm Description
Dijkstra's algorithm first initiates all vertex
distances with preliminary values and puts all
vertexes in a priority queue. Then picks the
unvisited vertex with the lowest-distance,
calculates the distance through it to each
unvisited neighbor, and updates the neighbor's
distance if smaller. Mark visited when done with
neighbors.
7
Algorithm Trace Trough
8
Algorithm Trace Trough
9
Algorithm Pseudocode
function Dijkstra(Graph, source):
for each vertex v in Graph:
dist[v] := infinity ;
previous[v] := undefined ;
end for
dist[source] := 0 ;
Q := the set of all nodes in Graph ;
while Q is not empty:
u := vertex in Q with smallest distance in dist[] ;
remove u from Q ;
if dist[u] = infinity:
break ;
end if
for each neighbor v of u in Q:
alt := dist[u] + dist_between(u, v) ;
if alt < dist[v]:
dist[v] := alt ;
previous[v] := u ;
decrease-key v in Q;
end if
end for
end while
return dist;
//
//
//
//
//
Initializations
Unknown distance function from
source to v
Previous node in optimal path
from source
//
//
//
//
//
Distance from source to source
All nodes in the graph are put in a
Priority Queue Q
The main loop
Start node in first case
// all remaining vertices are
// inaccessible from source
// where v has not yet been
// removed from Q.
// Relax (u,v,a)
// Reorder v in the Queue
10
C# Demo
0
6
3
1
5
4
2
Modifications
 We can solve different problems using
modified Dijkstra algorithm elements:
 Graph –vertices, edges and weights meanings
 Distance – definition
 Priority Queue
 Relaxation and Distance Initialization
12
Some Theory
 Dijkstra algorithm
is based on the following
Lemmas:
 Shortest paths are composed of shortest paths.
It is based on the fact that if there was a shorter
path than any sub-path, then the shorter path
should replace that sub-path to make the whole
path shorter.
 The sum of the lengths of any two sides of a
triangle is greater than the length of the third
side.
13
Some Theory
 Analysis
of Dijkstra’s Algorithm:
 The initialization uses only O(n) time.
 Each vertex is processed exactly once.
 The inner loop
is called once for each edge
in the graph. Each call of the inner loop does O(1) work plus, possibly,
one Decrease-Key operation.
 All of the priority queue operations require
time
 Finally
and we get
time
 If unsorted sequence is used instead of priority queue we get
O(n2 + e)
14
Related algorithms
 Breadth-first
search - special-case of Dijkstra's
algorithm on unweighted graphs when all edge costs
are positive and identical. The priority queue
degenerates into a FIFO queue.
 Uniform-cost search - the shortest path to a
particular node
 The A* algorithm
- generalization of Dijkstra's
algorithm that cuts down on the size of the subgraph
that must be explored, if additional information is
available that provides a lower bound on the "distance"
to the target.
15
Resources
 http://algs4.cs.princeton.edu/44sp/
 http://en.wikipedia.org/wiki/Dijkstra%27s_alg
orithm
 http://ocw.mit.edu/OcwWeb/Electrical-
Engineering-and-Computer-Science/6046JFall-2005/CourseHome/
 https://www.udacity.com/course/cs271
 Nakov’s book: Programming = ++Algorithms;
16
Dijkstra's algorithm
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