meljun cortes algorithm_coping with the limitations of algorithm power_ii

8/21/2019 MELJUN CORTES ALGORITHM_Coping With the Limitations of Algorithm Power_II

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Coping with the Limitations of algorithm Power

Design and Analysis of Algorithm

* Property of STI

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The following are the topics to be discussed under

coping with the limitations of algorithm power:

Understand the idea of coping up with the

limitation of algorithm power

Understand the concept of Backtracking

algorithm design technique.

Understand the application of Backtrackingusing algorithm design technique N-queen

problem.

Know the definition of branch and bound

algorithm design technique.


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* Property of STI

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There are problems that are difficult to solve

where algorithms takes too much time for arbitrary

instances of problem.

To cope up with this, it is important to understand

what makes a problem difficult and makes a

problem easy.

Theoretically, a problem will turn out to be difficult

if there is only one instance of the problem that

demands an exponential time to solve by anyalgorithm.


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* Property of STI

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Problems can be solved using

back-tracking

branch-and-bound

Both strategies are improvements over exhaustive


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* Property of STI

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Backtracking: is a refinement of the exhaustive

search strategy where solutions to a problem are

developed and evaluated one step

at a time.

Each partial solution is evaluated for feasibility

and is said to be feasible if it can be developed by

further choices without violating any of the

problem’s constraints.


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The following are the steps in evaluating a

problem thru backtracking:


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Example: N-Queen problem

A standard chess board is an eight-by-eight array

consisting eight rows of eight columns each.

To further understand the constraints on thisproblem, we shall examine the simpler 4-queens

problem that is set on a four-by-four board.


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The possible moves for this queen are the

following:


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Solution to the problem can be attained thru the

following:


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It can be argued that other solutions can be

generated by symmetry arguments, once we have

one applicable solution we shall use symmetry to

get the others as illustrated below:


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The empty board is considered as the root node of

the tree which is the starting position of the

problem. The figure below are the first two levels

of the tree:


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Each of the two leaf nodes in the partial tree is the

root node of a tree with the following leaf count:


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We now consider the subtree on the left and see

backtracking in action. Think an incorrect solution

that places another queen in position (2, 2) as

illustrated below:

You will observe that we have a diagonal conflict

since the first queen at position (1, 1) can attack

the second queen at position (2, 2) by using a

diagonal move.


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This method of stating every node in the subtree

to violate a constraint and therefore dropping the

set of solutions is called pruning the state tree.

It is simple to see that there are 6! = 720 trialsolutions with one queen at (1, 1) and the other at

(2, 2).


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Again, we will look at the other subtree of the

solution space, beginning with the node illustrated

below:


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Only possible placement is at (2, 4), Let’s follow

the search tree for the (2, 4) placement, then

again, we draw it horizontally as illustrated below:


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Remember that there is only one alternative for

positioning the queen in the third column because

of the following reasons:


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Alternative for placing the queen in the fourth

column since:


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Branch and Bound is most commonly applied to

optimization problems.

Bounding solution does not satisfy the constraintsbut can be shown to be a bound for any optimal

solution.


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Assessing the Problem State Space Tree

Branch-and-bound algorithm design technique

is useful to a problem in which a solution can

be viewed as generating nodes in a state-space tree.

The leaf nodes can represent a solution to the

problem, which may not be a reasonable

solution.

Usually, the root node denotes the problem

prior to any decisions made, like the chessboard before any queens has been placed or

the job status before anybody has been

assigned a job.


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* Property of STI

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Usually, in the state-space tree the previously

generated node must be able to assess the node

and specifically we ask the following questions.

1. How does the partial solution compare to the

best existing solution?

2. Is the solution represented by this node a

feasible solution?


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To make the branch-and-bound work, it is

necessary to assess each node just generated as

the root node of a tree that represents all

solutions that can be generated based on the

partial solution we have just generated.

Therefore, we “prune” the search tree if any of the

following holds:

1. It can prove that no solution in the subtree will

gratify the problem constraints.

2. It can prove that no solution in the subtree will

be better than the best solution produced

thus far.

3. The node created symbolizes an absolute

feasible solution that can be compared to the

best solution so far generated and possibly

replace it.


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Live node: is a node that can possibly represent a

partial solution leading to an optimal solution of

the whole problem.

If we can assess the live nodes as to the bestsolution that can be associated with the node’s

subtree, we can apply best first branch and bound

The key approach to branch-and-bound is to view

each newly generated node as the root node of a

subtree leading to a set of solutions.


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We first choose the assignment problem, which

amounts to assigning each of N workers to N jobs

so that the total costs of the jobs are minimized.

We are given an N-by-N matrix describing the costfor each worker to do each job

It should be noted that this method can be used to

assign N workers to M jobs, with M < N.

This is done by creating (N – M) dummy jobs, each

with a cost representing the expense of having a

worker do nothing.

We shall assume one job per worker.


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Consider the matrix of costs. Each row in the

matrix represents a worker and each column

represents a job. The constraints of a feasible

solution are:

1. We pick one entry from each row,

corresponding to the assignment of exactly

one worker to a job.

2. We pick one entry from each column, so that

each job has someone working on it.


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Set another method, a feasible solution

corresponds to the selection of one entry in each

row of the matrix so that no two selected elements

are in the same column.

An optimal solution is that feasible solution of

lowest cost.


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For the first bounding solution, find the minimum

cost per row and add it up.


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For the second bounding solution consider the

column minimum.


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We need something generated quickly and

guaranteed to be feasible, so we pick as follows,

just moving down the diagonal of the matrix.


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For the first bounding solution, find the minimum

cost per row and add it up.


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For the second bounding solution consider the

column minimum.

The total cost associated with this feasiblesolution is 9 + 4 + 1 + 4 = 18. Thus we know that

the limits on any optimal solution are 12 and 18


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We need something generated quickly and

guaranteed to be feasible, so we pick as follows,

just moving down the diagonal of the matrix.


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We now generate a version of the search tree and

consider how to improve it.

We label each node with the total cost of the

allocation so far, here just the cost of the job.


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Let’s return to the examination of minimum costs

per row. The table is repeated here for easy

reference.

The lower bound for the rest of the workers, after

Worker A has been assigned a job, is 3 + 1 + 4 =

8.


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Thus, we may immediately generate a lower bound

on any partial solution:

The total cost of the partial solution plus the

minimum cost of the remaining assignments. With that in mind, we arrive at a new start to the

search tree.


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Here are the adjusted matrices to be used in

assessing the subtrees:

Here are the adjusted matrices to be used in

assessing the subtrees.

A gets job1 with cost = 9


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A gets job 2 with cost = 2

The subtree assesses again at 8, so the total

lower limit is 10


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Here we have a new assessment for the subtree –

a new lower limit of 13, so that the total lower limit

evaluation of the node is 20


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Here we also have a new assessment for the

subtree – a new lower limit of 10, so that the totallower limit evaluation is 18.


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Given these more computationally intensive

calculations, we have the following for the

problem state-space tree.


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One of the nodes – “A gets job 3” – has a lower

bound exceeding the cost of a known feasible

solution.

For this reason it is pruned and the remaining treeis as follows.


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Here are the three possibilities for this subtree.

A gets job 2 B gets job 1

The remaining jobs assess at 5, so the node

assesses at a value of 13.


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The total cost is 2 + 3 = 5.

The matrix for the rest of the jobs and workers is


assesses at a value of 14.


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The total cost is 2 + 7 = 9.

The matrix for the rest of the jobs and workers is


assesses at a value of 17


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Here is the state-tree after the node “A gets job 2”

has been expanded.

At this point, we have five live nodes to consider.These are labeled as

1) “A gets job 1”,

2) “B gets job 1”,

3) “B gets job 3”,

4) “B gets job 4”, and

5) “A gets job 4”.


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The best looking node is the one labeled “B gets

job 1” with a lower bound of 13.

We consider two possibilities for the subtree

rooted at this partial solution. Recall our matrix of choices for workers and D.


The total cost is 2 + 6 = 8. The matrix for workers

C and D is


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Here is the status: we have two feasible solutions,

one with total cost of 13.

We have four non-terminal live nodes, none of

which have a lower bound less than 14.


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So we have a solution: A gets job 2, B gets job 1,

C gets job 3, and D gets job 4.

The solution generated 9 of the possible (4 + 12 +24) 40 nodes – an 75% saving.

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