PATH FINDING
ALGORITHMS
12/02/2020
PACMAN
1980
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CITIES SKYLINES
2015
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AGE OF EMPIRES 3
2004
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FAR CRY 4
2014
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AGENDA
What are we going to discuss?
6 ALGORITHMS
WE WILL DISCUSS THE FOLLOWING ALGORITHMS
Breadth First Search
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Depth First Search
Dijkstra's Algorithm
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6 ALGORITHMS
WE WILL DISCUSS THE FOLLOWING ALGORITHMS
Greedy Search
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A*
D*
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PROBLEM STATEMENT
o Shizuka has invited Nobita to have cake
o But because Gian had forced Nobita to play baseball, he
is already late
o Shizuka will get upset if he gets any more late.
o There are chances that she will invite Dekisuki if
Nobita doesn’t show up
o As Nobita’s friend, Doraemon (you) have to find the
optimal path to Shizuka’s home.
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BREADTH FIRST
SEARCH
Treat the neighbourhood as layers
o Explore the neighbourhood layer wise
o Proceed to next layer once all nodes in a layer are
complete
o If destination is found, no need to search further. That is
the nearest distance.
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BIDIRECTIONAL
BREADTH FIRST
SEARCH
LET’S COOPERATE
o Both start state and goal state search towards each other
o Approximately halves the search time
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DFS
“Don’t care about results just choose a path which you see first, if you
fail trace back”
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DFS
• Depth-first Search (DFS) is an algorithm for searching a graph or
tree data structure.
• The algorithm starts at the root (top) node of a tree and goes as
far as it can down a given branch (path), and then backtracks
until it finds an unexplored path, and then explores it.
• The algorithm does this until the entire graph has been explored.
• Depth-first searches are often used as subroutines in other more
complex algorithms.
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Depth first search in Trees
A tree is an undirected graph in which any two vertices are connected by exactly one path. In other words,
any acyclic connected graph is a tree. For a tree, we have below traversal methods –
• Preorder: visit each node before its children.
• Postorder: visit each node after its children.
• Inorder (for binary trees only): visit left subtree, node, right subtree.
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Depth first search in Graph
Recursive Approach:Depth first search is a way of traversing
graphs, which is closely related to preorder
traversal of a tree. Below is recursive
implementation of preorder traversal
To turn this into a graph traversal algorithm, we
basically replace “child” by “neighbor”. But to
prevent infinite loops we keep track of the vertices
are already discovered and not visit them again.
procedure preorder(treeNode v)
{
visit(v);
for each child u of v
preorder(u);
}
procedure dfs(vertex v)
{
visit(v);
for each neighbor u of v
if u is undiscovered
call dfs(u);
}
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Depth first search in Graph
Iterative Approach:The non-recursive implementation of DFS is similar to the non-recursive implementation of BFS,
but differs from it in two ways:
• It uses a stack instead of a queue
• The DFS should mark discovered only after popping the vertex not before pushing it.
• It uses reverse iterator instead of iterator to produce same results as recursive DFS.
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Tic Tac Toe Using DFS
Using Recursive Method:1. Check if the game is over — if either player won or if the board is fully filled
1. If so, return the result of the game
2. Iterate through all board squares
1. If the square is occupied, continue to the next one
2. Set the square to either an ‘X’ or an ‘O’ depending on the current player’s color
3. Get the outcome of the game by recursing, calling the same method with the updated board, and with
the turn boolean swapped
4. Update the best result of this branch, trying to maximize the result for the current player
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Dijkstra’s Algorithm
“Finding shortest path from source by building a set of nodes
that have minimum distance from the source.”
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Dijkstra’s Algorithm
• Create a set of shortest path tree that keeps track of vertices included in
shortest path tree, i.e., whose minimum distance from source is calculated and
finalized. Initially, this set is empty.
• Assign a distance value to all vertices in the input graph. Initialize all distance
values as ∞. Assign distance value as 0 for the source vertex so that it is picked
first.
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Continue…
• While the set doesn’t include all vertices
• Pick a vertex u which is not there in set and has minimum distance value.
• Include u to set.
• Update distance value of all adjacent vertices of u. To update the distance
values, iterate through all adjacent vertices. For every adjacent vertex v, if
sum of distance value of u (from source) and weight of edge u-v, is less
than the distance value of v, then update the distance value of v.
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Application
Google maps!!
It uses more complex and efficient algorithms.. But Dijkstra is the
basis.
Network Routing Protocols!!!
Dijkstra's algorithm is widely used in the routing protocols required by
the routers to update their forwarding table. The algorithm provide
the shortest cost path from source router to other routers in the
network.
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Greedy
“Focus on choosing the next best choice whether it gives the best
solution or not”
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Greedy
• Greedy algorithms are fast
• A greedy algorithm is an algorithmic paradigm that follows the problem solving heuristic of making the
locally optimal choice at each stage with the intent of finding a global optimum.
• In many problems, a greedy strategy does not usually produce an optimal solution, but nonetheless a
greedy heuristic may yield locally optimal solutions that approximate a globally optimal solution in a
reasonable amount of time.
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Crystal Quest
• Crystal Quest is an action game originally for the Macintosh. It was written by Patrick Buckland for Casady
& Greene in 1987, and was ported to the Apple IIgs in 1989 by Bill Heineman. Ports were also made to
the Amiga, Game Boy, iOS, and Palm. It was notable for being the first game to support the color displays
on the Macintosh II.
• In the Macintosh computer game Crystal Quest the objective is to collect crystals, in a fashion similar to
the travelling salesman problem. The game has a demo mode, where the game uses a greedy algorithm
to go to every crystal. The artificial intelligence does not account for obstacles, so the demo mode often
ends quickly.
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A
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H=2
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H=6
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A* Algorithm
“Cost and Heuristics both matter”
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A* Algorithm
• Informally speaking, A* Search algorithms, unlike other traversal techniques, it has
“brains”.
• What it means is that it is really a smart algorithm which separates it from the
other conventional algorithms.
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A* Algorithm - What it does?
What A* Search Algorithm does is that at each step it picks the node according to a
value-‘f’ which is a parameter equal to the sum of two other parameters – ‘g’ and ‘h’.
At each step it picks the node/cell having the lowest ‘f’, and process that node/cell.
g = the movement cost to move from the starting point to a given square on the
grid, following the path generated to get there.
h = the estimated movement cost to move from that given square on the grid to the
final destination. This is often referred to as the heuristic (smart guess)
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A* Algorithm
1. Initialize the open list
successor.h = distance from goal to successor
2. Initialize the closed list
successor.f = successor.g + successor.h
put the starting node on the open
ii) if a node with the same position as
list (you can leave its f at zero)
successor is in the OPEN list which has a
3. while the open list is not empty
lower f than successor, skip this successor
a) find the node with the least f on
the open list, call it "q"
iii) if a node with the same position as
successor is in the CLOSED list which has
b) pop q off the open list
a lower f than successor, skip this successor
c) generate q's 8 successors and set their
otherwise, add the node to the open list
parents to q
end (for loop)
d) for each successor
i) if successor is the goal, stop search
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successor.g = q.g + distance between
successor and q
e) push q on the closed list
end (while loop)
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A* Algorithm
Games don’t need lot of accuracy, but speed is a crucial thing.
So most of the games use A* algorithm which uses heuristics
to speed up the calculations but with accuracy trade off. In
most cases A* is more than sufficient.
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A
H=2
G=6
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H=6
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H=3
G=9
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H=5
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H=3
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D* Algorithm
behaves like A* except that the arc costs can change as the algorithm runs
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D* Algorithm
• This is the algo that most games use for games when the terrain is not
known/visible to the player.
• There are three types to this algo viz., original D* (informed incremental search
algorithm), Focussed D* (informed incremental heuristic search algorithm) and the
D* Lite (Lifelong Planning A* algorithm).
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D* Algorithm – An Example
Imagine exploring an unknown planet using a robotic vehicle. The robot moves
along the rugged terrain while using a range scanner to make precise measurements
of the ground in its vicinity. As the robot moves, it may discover that some parts were
easier to traverse than it originally thought. In other cases, it might realize that some
direction it was intending to go is impassable due to a large bolder or a ravine. If the
goal is to arrive at some specified coordinates, this problem can be viewed as a
navigation problem in an unknown environment. Sounds like pure Artificial
Intelligence isn’t?
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The Automated Cross-Country Unmanned Vehicle (XUV) is
equipped with laser radar and other sensors, and uses
Stentz's algorithm (D*) to navigate (courtesy of General
Dynamics Robotic Systems).
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D* Algorithm
Games don’t need lot of accuracy, but speed is a crucial thing.
So most of the games use A* algorithm which uses heuristics
to speed up the calculations but with accuracy trade off. In
most cases A* is more than sufficient.
Very well, but what if the heuristic of a path keeps changing?
Or there is a enemy roaming around that needs to be
avoided. Or the position of the goal state itself changes
without you having any knowledge of the situation? This is a
problem of dynamic heuristic
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Bye – Bye Nobita!
I am not going to help you anymore, you have do it yourself now!
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Bye – Bye Nobita!
I am not going to help you anymore, you have do it yourself now!
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D* Algorithm
D* and its variants have been widely used for mobile robot and autonomous
vehicle navigation. Such navigation systems include a prototype system tested on
the Mars rovers Opportunity and Spirit and the navigation system of the winning
entry in the DARPA Urban Challenge, both developed at Carnegie Mellon
University.
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THANK YOU!
Guess which of these algorithms did Dr Strange
use to search for the winning possibility!