This document discusses shortest path analysis and Dijkstra's algorithm. It defines shortest path analysis as finding the minimum cumulative path between nodes on a network. Dijkstra's algorithm is described as finding the shortest paths from a starting node to all other reachable nodes. An example application calculates the shortest path from node A to G on a sample graph. The document concludes that shortest path analysis can identify key walking routes and inform improvements to pedestrian infrastructure.

1. SHORTEST PATH ANALYSIS
1. A K M Anwaruzzaman, Associate Professor
2. Wazeda Khatun, 5-Year Integrated M Sc (10th Semester) Student
Roll No. GEO-13533
Department of Geography, Aliah University, Kolkata-14

2. INTRODUCTION
 Shortest path analysis finds the path with the minimum
cumulative impedance between nodes on a network. The
path may connect just two nodes an origin and a
destination or have specific stops between the nodes.

3. ACCESSIBILITY DEFINED
 In order to understand Shortest Path Analysis (SPA) a conception
of accessibility is required.
 Accessibility is defined as the measure of the capacity of a
location to be reached by, or to reach different locations.
Therefore, the capacity and the structure of transport
infrastructure are key elements in the determination of
accessibility .
 There are two type of accessibility:
a) Geographical Accessibility, and
b) Potential Accessibility.

4. THE NOTION OF ACCESSIBILITY CONSEQUENTLY
RELIES ON TWO CORE CONCEPTS
 The first is location where the relativity of places is estimated in
relation to transport infrastructures, since they offer the mean to
support movements.
 The second is distance, which is derived from the connectivity
between locations. Connectivity can only exist when there is a
possibility to link two locations through transportation. It
expresses the friction of space (or deterrence) and the location
which has the least friction relative to others is likely to be the
most accessible. Commonly, distance is expressed in units such
as in kilometres or in time, but variables such as cost or energy
spent can also be used.

5. EMISSIVENESS, ATTRACTIVENESS
 Emissiveness is the capacity to leave a
location, the sum of the values of a row in the
A(P) matrix.
 Attractiveness is the capacity to reach a
location, the sum of the values of a column in
the A(P) matrix.

6. CONNECTIVITY
The most basic measure of accessibility involves
network connectivity where a network is
represented as a connectivity matrix, which express
the connectivity of each node with it adjacent
nodes. The number of columns and rows in this
network and a value of a 1 is given for each all
where this is a connected pair and a value of 0 for
each well there is an unconnected pain.

7. GEOGRAPHIC ACCESSIBILITY
 Geographic accessibility considers that the
accessibility of a location is the summation of all
distances between other locations divided by the
number of locations.
FORMULA:
A (G)=
A(G) = geographical accessibility matrix.
dij = shortest path distance between location
i and j.
n = number of locations.

8. A B C D E
A 0 8 4 9 15
B 8 0 7 12 18
C 4 7 0 5 11
D 9 12 5 0 6
E 15 18 11 6 0
L
A B C D E ∑/n
A 0 8 4 9 15 7.2
B 8 0 7 12 18 9.0
C 4 7 0 5 11 5.4
D 9 12 5 0 6 6.4
F 15 18 11 6 0 10.
0
∑/n 7.2 9.0 5.4 6.4 10.0 38.
0
A(G)
Figure :Geographic accessibility
8
4 7
5
6

9. CHARACTER OF TRANSPOSITION
This is a transposable matrix as the summation of values are the same for columns
and rows of this matrix.

10.  In this measure of accessibility , the most accessible place has
the lowest summation of distances. As shown in the figure the
construction of a geographic accessibility matrix, A(G), is a rather
simple undertaking. First, build a matrix containing the shortest
distance between the nodes (node A to node E), here labelled as
the L matrix. Second, build the geographic accessibility matrix
A(G) with the summation of rows and columns divided by the
number of locations in the network. The summation values are
the same for columns and rows since this is a transposable
matrix. The most accessible place is node C, since it has the
lowest summation of distances.

11. Potential accessibility is a more complex measure than geographic
accessibility, since it includes the concept of distance weighted by the
attributes of a location. All locations are not equal and thus some are
more important than others.
+
/
/
A(P) = potential accessibility matrix.
dij = distance between place i and j (derived from valued graph matrix).
Pj = attributes of place j, such as its population, retailing surface,
parking space, etc.
n = number of locations
FORMULA:
POTENTIAL ACCESSIBILITY

12. A B C D E
A 0 8 4 9 15
B 8 0 7 12 18
C 4 7 0 5 11
D 9 12 5 0 6
E 15 18 11 6 0
A 1200
B 900
C 1500
D 600
E 800
L P
i/j A B C D E ∑i
A 1200.
0
150.0 300.0 133.3 80.0 1863.3
B 112.5 900.0 128.6 75.0 50.0 1266.1
C 375.0 214.3 1500.
0
300.0 136.4 2525.7
D 66.6 50.0 120.0 600.0 100.0 936.6
E 53.3 44.4 72.7 133.3 800.0 1103.7
∑i 1807.
4
1358.
7
2121.
3
1241.
6
1166.
4
7695.4
P(G)
Figure: Potential accessibility

13. TRANSPOSITION CHARACTER OF A(P)
The potential accessibility matrix is not transposable
since locations do not have the same attributes,
which brings the notions of emissiveness and
attractiveness:

14. By considering the same shortest distance matrix (L) as in figure and the
population matrix P, the potential accessibility matrix, P(G), can be calculated
Figure. The value of all corresponding cells (A–A, B–B, etc.) equals the value
of their respective attributes (P). The value of all non-corresponding cells
equals their attribute divided by the corresponding cell in the L matrix. The
higher the value, the more a location is accessible, node C being the most
accessible. The matrix being non-transposable, the summation of rows is
different from the summation of columns, bringing forward the issue of
implying different levels of attractiveness and emissiveness. Node C has more
attractiveness than emissiveness (2525.7 versus 2121.3), while Node B has
more emissiveness than attractiveness (1358.7 versus 1266.1). Likewise, a
Geographic Information System can be used to measure potential accessibility,
notably over a surface.

15. FINDING THE SHORTEST PATH, WITH A LITTLE
HELP FROM DIJKSTRA
 Dijkstra’s algorithm is unique for many reasons, which
start to understand how it works. But the one that has
always come as a slight surprise is the fact that is
algorithm isn’t just use to find out the shortest path
between two specific nodes in a graph data structure.
 Dijkstra’s algorithm can be used to determine the
shortest path from one node in a graph to every other
node with in the same group data structure, provided that
the nodes are reachable from the starting node.

16. HOW DOES DIJKSTRA’S ALGORITHM ACTUALLY
WORK?
There are many possible paths between node A and G. We need to find out the
shortest path from node A to node G. We know that we are going to start at node A,
but we don't know if there is a path that will be the shortest one to get to node G, if
such a path even exist.

17. RULES FOR RUNNING DIJKSTRA’S ALGORITHM
1) From the starting node, visit the vertex with the smallest
known distance/cost.
2) Once we have moved to the smallest cost vertex, check each
of its neighbouring nodes.
3) Calculate the distance/cost/time for the neighbouring nodes by
summing the cost of the edges leading from the start vertex.
4) If the distance/cost to a vertex we are checking is less than a
known distance/cost as compared to the available alternatives,
update the shortest distance for that vertex.

18. THE ABSTRACTED RULES ARE AS FOLLOWS:
 Every time that we set out to visit a new node, we will
choose the node with the smallest known distance/cost to
visit first.
 Once we’ve moved to the node we’re going to visit, we
will check each of its neighboring nodes.
 For each neighboring node, we’ll calculate the
distance/cost/time for the neighboring nodes by summing
the distance/cost of the edges that lead to the node we’re
checking from the starting vertex.
 Finally, if the distance/cost to a node is less than a
known distance/cost, we’ll update the shortest distance
that we have on file for that vertex.

19. AB  D  E  G
3+1+2+2=8

20. •From looking at this table, it might not be completely
obvious, but we have actually got every single shortest
path that stems from our starting node available right at
our fingertips.
•Dijkstra’s algorithm can run once and we can rescue all
the values gain and again –provided our graph does not
change. This is exactly how the characteristic become
very powerful. We set out wanting to find the shortest
path from A to G.

21. AB  D  E  G
3+1+2+2=8

22.  Finally another method is guess and check method. We can do it on
simple diagram. We just draw the path and figure and then draw
with different colour for paths and the figure, and now we find the
smallest one and just show the below the path..
Guess and check method

23. AB  D  E  G
3+1+2+2=8
GUESS AND CHECK METHOD OF SHORTEST PATH ANALYSIS

24. APPLICATION OF SHORTEST PATH ANALYSIS
 The most common example of Dijkstra’s algorithm in the
wild is in path-finding problems, like determining
directions or finding a route on Google Maps.
 Dijkstra’s algorithm implemented for path-finding on
a map.
Dijkstra’s algorithm can be used in many fields including
computer networking(Routing systems).It also has
application in Google maps to find the shortest possible
path from one location to other....In Biology it is used to
find the network model in spreading of a infectious
disease.

25. CONCLUSION
 The Shortest Path Analysis identifies direct links
between walking trip origins and destinations, providing
a snapshot of roadway links on which local residents are
likely to walk. This analysis completes the existing
conditions phase of this project, which quantifies several
metrics for each segment of the Memphis roadway
system: the prevalence of pedestrian crashes, the
likelihood of walking trips (demand), the quality of
pedestrian infrastructure (supply), and the number of
routes to key destinations served. These quantitative
metrics provide a data-driven basis for the identification
of locations in need of pedestrian improvements, as well
as for the prioritization of those improvements.

26.  In the next stage of the project, the project team
will generate a project list outlining needed
pedestrian improvements for roadway segments
and intersections across the city, which may
include improvements such as sidewalk repair,
new sidewalks, striped crosswalks, pedestrian
refuges, and enhancements to existing signalized
intersections.