Graphs
David Johnson
Describing the Environment
• How do you tell a person/robot how to get
somewhere?
– Tell me how to get to the student services building…
World Representation
• Landmarks
• Paths/Roads
• Map
• This is abstracted as a graph
– Not in the sense of a plot
Graphs
• A graph is a set of
landmarks and the paths
between them
V1
E1
• A graph G has
– Set of vertices (or nodes) V
– Set of edges E
• A graph is a classic data
structure in CS
V2
E2
V3
Graphs
• A directed graph means
– E(A,B) does not imply
E(B,A)
– What kind of robot would
need a graph that is not
bidirectional?
• Edges may have weights
– Where would this be
appropriate?
V1
E1
1.4
V2
2.6
E2
V3
Paths
• A path is a sequence of
vertices such that for Vi
and Vi+1 Ei,i+1 exists
• A graph is connected if
there is a path between
all V
– For what kind of robot and
environment would this not
be true?
V1
E1
1.4
V2
2.6
E2
V3
Imagine a simple robot
• What is the simplest
kind of robot you can
think of?
Point Robots
• A point robot can move along the
infinitesimal width edges of a graph.
• Path planning for this simple case is just
searching a graph for a path.
Finding Paths
• Given a start and end,
how do we find a path
between them?
V1
V1
V3
V8
V7
V2
V4
V6
V5
Depth First Search (DFS)
• Always take the first
option
V1
V1
V3
V8
V7
V2
V4
V6
V5
Depth First Search (DFS)
• Always take the first
option
V1
V1
V3
V8
V7
V2
V4
V6
V5
Depth First Search (DFS)
• Always take the first
option
V1
V1
V3
V8
V7
V2
V4
V6
V5
Depth First Search (DFS)
• Always take the first
option
V1
V1
V3
V8
V7
V2
V4
V6
V5
Depth First Search (DFS)
• Always take the first
option
V1
V1
V3
V8
V7
V2
V4
V6
V5
Depth First Search (DFS)
• Always take the first
option
V1
V1
V3
V8
V7
V2
V4
V6
V5
Depth First Search (DFS)
• Always take the first
option
V1
V1
V3
V8
V7
V2
V4
V6
V5
Breadth First Search (BFS)
• Exhaust all
possibilities at same
level first
V1
V1
V3
V8
V7
V2
V4
V6
V5
Breadth First Search (BFS)
• Exhaust all
possibilities at same
level first
V1
V1
V3
V8
V7
V2
V4
V6
V5
Breadth First Search (BFS)
• Exhaust all
possibilities at same
level first
V1
V1
V3
V8
V7
V2
V4
V6
V5
Breadth First Search (BFS)
• Exhaust all
possibilities at same
level first
V1
V1
V3
V8
V7
V2
V4
V6
V5
Representing a graph in Matlab
• Demo
Wavefront planner
• Use BFS on a grid
• Label cells with values
– Start with zero
– Expand from start
– Add +1 to neighbors of current wavefront
– Use gradient descent to search from goal to
start
Representations: A Grid
• Distance is reduced to discrete steps
– For simplicity, we’ll assume distance is uniform
• Direction is now limited from one adjacent cell to another
Representations: Connectivity
• 8-Point Connectivity
– (chessboard metric)
• 4-Point Connectivity
– (Manhattan metric)
The Wavefront Planner: Setup
The Wavefront in Action (Part 1)
• Starting with the goal, set all adjacent cells with “0” to the
current cell + 1
– 4-Point Connectivity or 8-Point Connectivity?
– Your Choice. We’ll use 8-Point Connectivity in our example
The Wavefront in Action (Part 2)
• Now repeat with the modified cells
– This will be repeated until goal is reached
• 0’s will only remain when regions are unreachable
The Wavefront in Action (Part 3)
• Repeat again...
The Wavefront in Action (Part 4)
• And again...
The Wavefront in Action (Part 5)
• And again until...
The Wavefront in Action (Done)
• You’re done
The Wavefront
• To find the shortest path simply always move
toward a cell with a lower number
– The numbers generated by the Wavefront planner are roughly
proportional to their distance from the goal
Two
possible
shortest
paths
shown