The document describes the backtracking method for solving problems that require finding optimal solutions. Backtracking involves building a solution one component at a time and using bounding functions to prune partial solutions that cannot lead to an optimal solution. It then provides examples of applying backtracking to solve the 8 queens problem by placing queens on a chessboard with no attacks. The general backtracking method and a recursive backtracking algorithm are also outlined.
BackTracking Algorithm: Technique and ExamplesFahim Ferdous
This slides gives a strong overview of backtracking algorithm. How it came and general approaches of the techniques. Also some well-known problem and solution of backtracking algorithm.
Backtracking is a general algorithm for finding all (or some) solutions to some computational problems, notably constraint satisfaction problems, that incrementally builds candidates to the solutions, and abandons each partial candidate c ("backtracks") as soon as it determines that c cannot possibly be completed to a valid solution.
BackTracking Algorithm: Technique and ExamplesFahim Ferdous
This slides gives a strong overview of backtracking algorithm. How it came and general approaches of the techniques. Also some well-known problem and solution of backtracking algorithm.
Backtracking is a general algorithm for finding all (or some) solutions to some computational problems, notably constraint satisfaction problems, that incrementally builds candidates to the solutions, and abandons each partial candidate c ("backtracks") as soon as it determines that c cannot possibly be completed to a valid solution.
P, NP, NP-Complete, and NP-Hard
Reductionism in Algorithms
NP-Completeness and Cooks Theorem
NP-Complete and NP-Hard Problems
Travelling Salesman Problem (TSP)
Travelling Salesman Problem (TSP) - Approximation Algorithms
PRIMES is in P - (A hope for NP problems in P)
Millennium Problems
Conclusions
Artificial Intelligence: Introduction, Typical Applications. State Space Search: Depth Bounded
DFS, Depth First Iterative Deepening. Heuristic Search: Heuristic Functions, Best First Search,
Hill Climbing, Variable Neighborhood Descent, Beam Search, Tabu Search. Optimal Search: A
*
algorithm, Iterative Deepening A*
, Recursive Best First Search, Pruning the CLOSED and OPEN
Lists
P, NP, NP-Complete, and NP-Hard
Reductionism in Algorithms
NP-Completeness and Cooks Theorem
NP-Complete and NP-Hard Problems
Travelling Salesman Problem (TSP)
Travelling Salesman Problem (TSP) - Approximation Algorithms
PRIMES is in P - (A hope for NP problems in P)
Millennium Problems
Conclusions
Artificial Intelligence: Introduction, Typical Applications. State Space Search: Depth Bounded
DFS, Depth First Iterative Deepening. Heuristic Search: Heuristic Functions, Best First Search,
Hill Climbing, Variable Neighborhood Descent, Beam Search, Tabu Search. Optimal Search: A
*
algorithm, Iterative Deepening A*
, Recursive Best First Search, Pruning the CLOSED and OPEN
Lists
An introduction to Cassandra, including replication + partitioning options, data center awareness, local storage model, data modeling example. Presented by Andrew Byde on 25th August 2011 at NoSQLNow! in San Jose , California
Online learning, Vowpal Wabbit and HadoopHéloïse Nonne
Online learning, Vowpal Wabbit and Hadoop
Online learning has recently caught a lot of attention, following some competitions, and especially after Criteo released 11GB for the training set of a Kaggle contest.
Online learning allows to process massive data as the learner processes data in a sequential way using up a low amount of memory and limited CPU ressources. It is also particularly suited for handling time-evolving date.
Vowpal Wabbit has become quite popular: it is a handy, light and efficient command line tool allowing to do online learning on GB of data, even on a standard laptop with standard memory. After a reminder of the online learning principles, we present how to run Vowpal Wabbit on Hadoop in a distributed fashion.
TensorFlow Korea 논문읽기모임 PR12 243째 논문 review입니다
이번 논문은 RegNet으로 알려진 Facebook AI Research의 Designing Network Design Spaces 입니다.
CNN을 디자인할 때, bottleneck layer는 정말 좋을까요? layer 수는 많을 수록 높은 성능을 낼까요? activation map의 width, height를 절반으로 줄일 때(stride 2 혹은 pooling), channel을 2배로 늘려주는데 이게 최선일까요? 혹시 bottleneck layer가 없는 게 더 좋지는 않은지, 최고 성능을 내는 layer 수에 magic number가 있는 건 아닐지, activation이 절반으로 줄어들 때 channel을 2배가 아니라 3배로 늘리는 게 더 좋은건 아닌지?
이 논문에서는 하나의 neural network을 잘 design하는 것이 아니라 Auto ML과 같은 기술로 좋은 neural network을 찾을 수 있는 즉 좋은 neural network들이 살고 있는 좋은 design space를 design하는 방법에 대해서 얘기하고 있습니다. constraint이 거의 없는 design space에서 human-in-the-loop을 통해 좋은 design space로 그 공간을 좁혀나가는 방법을 제안하였는데요, EfficientNet보다 더 좋은 성능을 보여주는 RegNet은 어떤 design space에서 탄생하였는지 그리고 그 과정에서 우리가 당연하게 여기고 있었던 design choice들이 잘못된 부분은 없었는지 아래 동영상에서 확인하실 수 있습니다~
영상링크: https://youtu.be/bnbKQRae_u4
논문링크: https://arxiv.org/abs/2003.13678
Realtime Per Face Texture Mapping (PTEX)basisspace
This presentation shows the original method for implementing Per-Face Texture Mapping (PTEX) in real-time on commodity hardware. PTEX is used throughout the film industry to handle texture seams robustly while simultaneously easing artist workflow.
PR-183: MixNet: Mixed Depthwise Convolutional KernelsJinwon Lee
TensorFlow-KR 논문읽기모임 PR12(12PR) 183번째 논문 review입니다.
이번에 살펴볼 논문은 Google Brain에서 발표한 MixNet입니다. Efficiency를 추구하는 CNN에서 depthwise convolution이 많이 사용되는데, 이 때 depthwise convolution filter의 size를 다양하게 해서 성능도 높이고 efficiency도 높이는 방법을 제안한 논문입니다. 자세한 내용은 영상을 참고해주세요
논문링크 : https://arxiv.org/abs/1907.09595
발표영상 : https://youtu.be/252YxqpHzsg
Artificial Intelligence, Machine Learning and Deep LearningSujit Pal
Slides for talk Abhishek Sharma and I gave at the Gennovation tech talks (https://gennovationtalks.com/) at Genesis. The talk was part of outreach for the Deep Learning Enthusiasts meetup group at San Francisco. My part of the talk is covered from slides 19-34.
On how to change the utility curve of deep learning to make deep learning projects deliver an ROI no matter how accurate the machine learning system is - presented at the Nasscom Analytics Summit 2018.
"Practical Machine Learning With Ruby" by Iqbal Farabi (ID Ruby Community)Tech in Asia ID
This slide was shared on Tech in Asia Jakarta 2016 @ 17 November 2016.
Get updates about our dev events delivered straight to your inbox by signing up here: http://bit.ly/tia-dev ! Be the first to know when new information is available!
Interest in Deep Learning has been growing in the past few years. With advances in software and hardware technologies, Neural Networks are making a resurgence. With interest in AI based applications growing, and companies like IBM, Google, Microsoft, NVidia investing heavily in computing and software applications, it is time to understand Deep Learning better!
In this workshop, we will discuss the basics of Neural Networks and discuss how Deep Learning Neural networks are different from conventional Neural Network architectures. We will review a bit of mathematics that goes into building neural networks and understand the role of GPUs in Deep Learning. We will also get an introduction to Autoencoders, Convolutional Neural Networks, Recurrent Neural Networks and understand the state-of-the-art in hardware and software architectures. Functional Demos will be presented in Keras, a popular Python package with a backend in Theano and Tensorflow.
http://imatge-upc.github.io/telecombcn-2016-dlcv/
Deep learning technologies are at the core of the current revolution in artificial intelligence for multimedia data analysis. The convergence of big annotated data and affordable GPU hardware has allowed the training of neural networks for data analysis tasks which had been addressed until now with hand-crafted features. Architectures such as convolutional neural networks, recurrent neural networks and Q-nets for reinforcement learning have shaped a brand new scenario in signal processing. This course will cover the basic principles and applications of deep learning to computer vision problems, such as image classification, object detection or text captioning.
We all have good and bad thoughts from time to time and situation to situation. We are bombarded daily with spiraling thoughts(both negative and positive) creating all-consuming feel , making us difficult to manage with associated suffering. Good thoughts are like our Mob Signal (Positive thought) amidst noise(negative thought) in the atmosphere. Negative thoughts like noise outweigh positive thoughts. These thoughts often create unwanted confusion, trouble, stress and frustration in our mind as well as chaos in our physical world. Negative thoughts are also known as “distorted thinking”.
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
Students, digital devices and success - Andreas Schleicher - 27 May 2024..pptxEduSkills OECD
Andreas Schleicher presents at the OECD webinar ‘Digital devices in schools: detrimental distraction or secret to success?’ on 27 May 2024. The presentation was based on findings from PISA 2022 results and the webinar helped launch the PISA in Focus ‘Managing screen time: How to protect and equip students against distraction’ https://www.oecd-ilibrary.org/education/managing-screen-time_7c225af4-en and the OECD Education Policy Perspective ‘Students, digital devices and success’ can be found here - https://oe.cd/il/5yV
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
The Indian economy is classified into different sectors to simplify the analysis and understanding of economic activities. For Class 10, it's essential to grasp the sectors of the Indian economy, understand their characteristics, and recognize their importance. This guide will provide detailed notes on the Sectors of the Indian Economy Class 10, using specific long-tail keywords to enhance comprehension.
For more information, visit-www.vavaclasses.com
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
1. BACKTRACKING
GENERAL METHOD
• Problems searching for a set of solutions or which
require an optimal solution can be solved using the
backtracking method .
• To apply the backtrack method, the solution must
be expressible as an n-tuple(x1,…,xn), where the
xi are chosen from some finite set si
• The solution vector must satisfy the criterion
function P(x1 , ….. , xn).
1
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2. BACKTRACKING (Contd..)
• Suppose there are m n-tuples which are
possible candidates for satisfying the
function P.
• Then m= m1, m2…..mn where mi is size of
set si 1<=i<=n.
• The brute force approach would be to form
all of these n-tuples and evaluate each one
with P, saving the optimum.
2
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3. BACKTRACKING (Contd..)
• The backtracking algorithm has the ability to yield
the same answer with far fewer than m-trials.
• In backtracking, the solution is built one
component at a time.
• Modified criterion functions Pi (x1...xn) called
bounding functions are used to test whether the
partial vector (x1,x2,......,xi) can lead to an optimal
solution.
• If (x1,...xi) is not leading to a solution, mi+1,....,mn
possible test vectors may be ignored. 3
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4. BACKTRACKING (Contd..)
• The constraints may be of two categories.
• EXPLICIT CONSTRAINTS are rules which restrict the
values of xi. Examples xi 0 or x1= 0 or 1 or li xi ui.
• All tuples that satisfy the explicit constraints define a
possible solution space for I.
• IMPLICIT CONSTRAINTS describe the way in which
the xi must relate to each other .
• Implicit constraints allow to find those tuples in the
solution space that satisfy the criterion function.
4
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5. Example : 8 queens problem
• The problem is to place eight queens on an 8 x 8
chess board so that no two queens attack i.e. no two
of them are on the same row, column or diagonal.
• Strategy : The rows and columns are numbered
through 1 to 8.
• The queens are also numbered through 1 to 8.
• Since each queen is to be on a different row
without loss of generality, we assume queen i is to
be placed on row i .
5
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6. 8 queens problem (Contd..)
• The solution is an 8 tuple (x1,x2,.....,x8)
where xi is the column on which queen i is
placed.
• The explicit constraints are :
Si = {1,2,3,4,5,6,7,8} 1 i n or 1 xi 8
i = 1,.........8
• The solution space consists of 88 8- tuples.
6
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7. 8 queens problem (Contd..)
The implicit constraints are :
(i) no two xis can be the same that is, all queens
must be on different columns.
(ii) no two queens can be on the same diagonal.
(i) reduces the size of solution space from 88 to 8!
8 – tuples.
Two solutions are (4,6,8,2,7,1,3,5) and
(3,8,4,7,1,6,2,5)
7
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8. 8 queens problem (Contd..)
1 2 3 4 5 6 7 8
1 Q
2 Q
3 Q
4 Q
5 Q
6 Q
7 Q
8 Q 8
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9. State space tree representation
Solution Space :
• Tuples that satisfy the explicit constraints define a solution
space.
• The solution space can be organized into a tree.
• Each node in the tree defines a problem state.
• All paths from the root to other nodes define the state-
space of the problem.
• Solution states are those states leading to a tuple in the
solution space.
• Answer nodes are those solution states leading to an
answer-tuple( i.e. tuples which satisfy implicit constraints).
9
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10. State space tree representation contd..
• The problem may be solved by systematically
generating the problem states determining which
are solution states, and determining the answer
states.
• Let us see the following terminology
• LIVE NODE A node which has been generated
and all of whose children are not yet been
generated .
• E-NODE (Node being expanded) - The live node
whose children are currently being generated .
10
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11. State space tree representation contd..
• DEAD NODE - A node that is either not to
be expanded further, or for which all of its
children have been generated.
• DEPTH FIRST NODE GENERATION- In
this, as soon as a new child C of the current
E-node R is generated, C will become the
new E-node.
R will become E-node again when C has
been fully explored.
11
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12. State space tree representation contd..
• BOUNDING FUNCTION - will be used to
kill live nodes without generating all their
children.
• BACTRACKING-is depth – first node
generation with bounding functions.
• BRANCH-and-BOUND is a method in
which E-node remains E-node until it is
dead.
12
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13. State space tree representation contd..
• BREADTH-FIRST-SEARCH : Branch-and
Bound with each new node placed in a
queue .
The front of the queen becomes the new E-
node.
• DEPTH-SEARCH (D-Search) : New nodes
are placed in to a stack.
The last node added is the first to be
explored.
13
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14. Example : 4 Queens problem
1 1 1 1
. . 2 2 2
3
. . . .
1 1
2
3
. . 4
14
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15. State space tree: 4 Queens problem
1
x1 = 1 x1=2
2 18
x2=2 3 4 x2=1 x2=3 x2 = 4
B 3 8 13 19 24 29
x3=3 x3=4 2 3 B B
4 6 14 16 x3 = 1
x4=4 3 B 30
5 7 15 x4 = 3
B 31
15
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16. State space tree: 4 Queens problem contd..
• If (x1….xi) is the path to the current E-node
, a bounding function has the criterion that
(x1..xi+1) represents a chessboard
configuration, in which no queens are
attacking.
• A node that gets killed as a result of the
bounding function has a B under it.
16
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17. State space tree: 4 Queens problem contd..
• We start with root node as the only live node. The
path is ( ); we generate a child node 2.
• The path is (1).This corresponds to placing queen
1 on column 1 .
• Node 2 becomes the E node. Node 3 is generated
and immediately killed. (because x1=1,x2=2).
• As node 3 is killed, nodes 4,5,6,7 need not be
generated.
17
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18. State space tree: 4 Queens problem contd..
• Node 8 is generated, and the path is (1,3).
• Node 8 gets killed as all its children
represent board configurations that cannot
lead to answer. We backtrack to node 2 and
generate another child node 13.
• But the path (1,4) cannot lead to answer
nodes.
• So , we backtrack to 1 and generate the
path (2) with node 18. We observe that the
path to answer node is (2 4 1 3 )
• Other solution is (3, 1, 4, 2)
18
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19. GENERAL BACKTRACKING METHOD
• All answer nodes are to be found
• If (x1…..xi) is a path from root to a node then T
(x1,….,xi) be the set of all possible values for Xi+1,
such that (x1,x2,…….,xi,xi+1) is also a path from
root to a problem state.
• B(x1…xi+1) or Bxi+1is false for the path (x1,..,xi+1)
if the path cannot reach an answer node.
• The solution vectors X (1: n) are those values
which are generated by T and satisfy Bi+1.
19
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20. GENERAL BACKTRACKING METHOD
(Contd..)
Procedure backtrack (n) // solution vectors are X(1:n)//
// and printed as soon as generated //
// T {X(1),….X(k-1)} gives all possible values of X(k)
// given that X(1),…..,X(k-1) have already been chosen //
// The predicates Bk (X(1),….X(k) ) determine those
elements //
// which satisfy the implicit constraints //
// Implicit constraints are “no two X is can be the same” //
Integer k, n; local X (1:n)
20
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21. GENERAL BACKTRACKING METHOD (Contd..)
K 1
while K <> 0 do
if there remained an untried X(k) such that X(k)
T ( X (1) , …..X(k-1) ) and Bk (X (1) ,…,X(k) ) = true then
if (X (1),…, X(k) ) is a path to an answer node then
print ( X (1) ,…X (k) ) // and proceed for another//
//solution with untried value of X(k)//
K K+1 // consider next set//
endif // end of if there remained ….//
else K K-1 // backtrack to previous component as no value of X(k)
//satisfies the constraints//
endif
repeat
end Backtrack
21
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22. EXAMPLE OF ALGORITHM WITH
4 QUEENS PROBLEMS
k 1
loop 1 X0 =1,1 {1,2,3,4} and B1 (X (1)) = true
1 is not a path to answer node
K 1+1=2
repeat K 2
loop 2
X(2){2,3,4}and B2 (1,2) is False but there
remains untried X(k)=3 and X(K) =4
22
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23. EXAMPLE OF ALGORITHM WITH
4 QUEENS PROBLEMS (Contd..)
B2 (1,3)=true , but (1,3) Is not a path to
answer node ; so K K+1=3
There is no X(3) such that
B3(1,3,2) or B3 (1,3,4) is true .
So backtrack K K-1=2
consider X (2) 4
23
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24. EXAMPLE OF ALGORITHM WITH
4 QUEENS PROBLEMS (Contd..)
(1,4) is not a path to answer
(1,4,2) is not a path to answer
B4(1,4,2,3)=false.
Thus X(2)=4 is false
With X(1)=1, we have seen that there is no X(2)
satisfying the constraints so KK-1=2-1=1
There remained untried X(k)=2,3,4.
Repeating with X(1)=2, we observe (2,4,1, 3) is a path to an
answer node .
Similarly the other solution is (3,1,4,2)
24
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25. BACKTRACKING ALGORITHM WITH
RECURSION
Procedure RBACKTRACK (k)
// on entering the first K-1 values X(1),…..,X(k-1)//
// of the solution vector X(1:n) have been assigned //
global n , X(1:n)
for each X(k) such that
X(k) T ( X (1),..X(k-1) ) Do
if Bk (X(1)..,X(k-1), X(k) )= true then {
if ( X (1) ,….,X(k) ) is a path to an answer node
then print ( X(1),……,X(k) ) endif
if (k<n)
CALL RBACKTRACK (k+1)
endif }// end of if Bk (X(1)…//
repeat // end of for each X(k)…//
end RBACKTRACK 25
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26. BACKTRACKING ALGORITHM WITH
RECURSION (Contd..)
EXAMPLE (RB – RBACKTRACK)
Initially RB(1) is called
for each X(1) X(1) {1,2,3,4} and B1 (1) is
true , but 1 is not an answer node.
RB (2) is called
X(2) {2,3,4} and B2 (1,3) = true and B2 (1,4) = true
(1,3) is not an answer node , RB(3) is called,X(3) = 2,
26
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27. BACKTRACKING ALGORITHM WITH
RECURSION (Contd..)
but (1,3,2) is not Answer node , so , RB(4) is
called B4 (1,3,2,4) = false.
(1,3,4,2) is not bounded so , X(2) = 4 is tried
(1,4,2,3) is not bounded . With X(1)=1 no solution
Now for X(1) = 2,3,4 repeat
(2,4,1,3) is an answer node, other paths with X(1)
= 2 are not leading to answer node.
With X (1) = 3 , (3,1,4,2) is an answer node.
X (1) = 4 - no solution.
27
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28. EFFICIENCY OF BACKTRACKING (BT)
ALGORITHM
• The time required by a backtracking algorithm or the
efficiency depends on four factors
(i) The time to generate the next X(k);
(ii) The number of X(k) satisfying the explicit
constraints
(iii) The time for bounding functions Bi
(iv) The number of X(k) satisfying the Bi for all i.
28
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29. EFFICIENCY OF BACKTRACKING
ALGORITHM (Contd..)
• The first three are relatively independent of
the problem instance being solved.
• The complexity for the first three is of
polynomial complexity .
• If the number of nodes generated is 2n , then
the worst case complexity for a
backtracking algorithm is O(P(n)2n) where
P(n) is a polynomial in n .
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30. Estimation Of Nodes generated in a
BT Algorithm
• Generate a random path in the state space tree.
• Let X be a node at level i on this path.
• Let mi be the children of X ( at level i+1 ) that do
not get bounded. (i.e. mi are the nodes which can
be considered for getting an answer node ).
• Choose randomly one of the mi.
• Continue this until this node is either is a leaf or
all its children are bounded.
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31. Estimation of Nodes generated in a
BT Algorithm (Contd..)
• Let m be the no. of unbounded nodes to be
generated.
• Let us assume that the bounding functions
are static, i.e., the BT algorithm does not
change its bounding functions.
• The number of estimated number of
unbounded nodes
=1+m1+m1m2+….. +m1m2m3..mi where mi is
the estimated no. of nodes at level i+ 1.
31
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32. Estimation of Nodes generated in a
BT Algorithm (Contd..)
• The number of unbounded nodes on level one
is 1.
• The number of unbounded nodes on level 2 is
m1
• The total no. of nodes generated till level 2 is
1 + m1
The total number of nodes generated till level
i+1 is 1+m1+…+m1…mi .
– The above procedure can be written as an algorithm.
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33. Estimation of Nodes generated in a BTAlgorithm (Contd..)
Procedure Estimate
// This procedure follows a random path and estimates//
// number of unbounded nodes //
m1; r1; k1
loop
Tk {X (k): X(k) T ( X(1) ,….X(k-1)) and Bk (X (1), ….,
X(k)) is TRUE}
If size (Tk) = 0 then exit endif // SIZE returns the //
r r * SIZE (Tk) // size of the set Tk //
m m+r
X (k) CHOOSE (Tk) // CHOOSE makes a random
choice of an element in Tk //
k k+1
repeat
return (m)
end estimate
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34. The n-queens problem and solution
• In implementing the n – queens problem we
imagine the chessboard as a two-
dimensional array A (1 : n, 1 : n).
• The condition to test whether two queens, at
positions (i, j) and (k, l) are on the same row
or column is simply to check i = k or j = l
34
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