动态规划详解01背包问题knapsack

Lecture13:The Knapsack Problem

Outline of this Lecture

Introduction of the0-1Knapsack Problem.

A dynamic programming solution to this problem.

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0-1Knapsack Problem

Informal Description:We have computed data?les that we want to store,and we have available bytes of storage.

File has size bytes and takes minutes to re-compute.

We want to avoid as much recomputing as possible, so we want to?nd a subset of?les to store such that The?les have combined size at most.

The total computing time of the stored?les is as large as possible.

We can not store parts of?les,it is the whole?le or nothing.

How should we select the?les?

2

0-1Knapsack Problem

Formal description:Given two-tuples of positive numbers

and

and,we wish to determine the subset

(of?les to store)that

maximizes

subject to

Remark:This is an optimization problem.

Brute force:T ry all possible subsets. Question:Any solution better than the brute-force?

3

Recall of the Divide-and-Conquer

1.Partition the problem into subproblems.

2.Solve the subproblems.

https://www.360docs.net/doc/f83578886.html,bine the solutions to solve the original one.

Remark:If the subproblems are not independent,i.e. subproblems share subsubproblems,then a divide-and-conquer algorithm repeatedly solves the common subsubproblems.

Thus,it does more work than necessary!

Question:Any better solution?

Y es–Dynamic programming(DP)!

4

The Idea of Dynamic Programming

Dynamic programming is a method for solving optimization problems.

The idea:Compute the solutions to the subsub-problems once and store the solutions in a table,so that they can be reused(repeatedly)later.

Remark:We trade space for time.

5

The Idea of Developing a DP Algorithm

Step1:Structure:Characterize the structure of an optimal solution.

–Decompose the problem into smaller problems, and?nd a relation between the structure of the optimal solution of the original problem and the solutions of the smaller problems.

Step2:Principle of Optimality:Recursively de?ne the value of an optimal solution.

–Express the solution of the original problem in terms of optimal solutions for smaller problems.

6

The Idea of Developing a DP Algorithm

Step3:Bottom-up computation:Compute the value of an optimal solution in a bottom-up fashion by using a table structure.

Step4:Construction of optimal solution:Construct an optimal solution from computed information.

Steps3and4may often be combined.

7

Remarks on the Dynamic Programming Approach

Steps1-3form the basis of a dynamic-programming solution to a problem.

Step4can be omitted if only the value of an opti-

mal solution is required.

8

Developing a DP Algorithm for Knapsack

Step1:Decompose the problem into smaller problems.

We construct an array.

For,and,the entry will store the maximum(combined) computing time of any subset of?les

of(combined)size at most.

If we can compute all the entries of this array,then the array entry will contain the maximum computing time of?les that can?t into the

storage,that is,the solution to our problem.

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Developing a DP Algorithm for Knapsack

Step2:Recursively de?ne the value of an optimal solution in terms of solutions to smaller problems.

Initial Settings:Set

for,no item

for,illegal Recursive Step:Use

for,.

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Correctness of the Method for Computing Lemma:For,,

Proof:T o compute we note that we have only two choices for?le:

Leave?le:The best we can do with?les

and storage limit is. Take?le(only possible if):Then we gain of computing time,but have spent bytes of our storage.The best we can do with remaining ?les and storage is

.

T otally,we get.

Note that if,then

so the lemma is correct in any case.

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Developing a DP Algorithm for Knapsack Step 3:Bottom-up computing (using iteration,

not recursion).

Bottom:

for all

.

Bottom-up computation:Computing the table using

row by

row.

1n

20

000

.........bottom

up

i= 0V[i,w]w=01

23......W 12

Example of the Bottom-up computation

Let and

24

1030

56

12345678910 00000000000 1

000040404040405050 3

0005050505090909090 Remarks:

The?nal output is.

The method described does not tell which subset gives the optimal solution.(It is in this example).

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The Dynamic Programming Algorithm KnapSack()

for(to);

for(to)

for(to)

if()

;

else

;

return;

Time complexity:Clearly,.

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Constructing the Optimal Solution

The algorithm for computing described in the previous slide does not keep record of which subset of items gives the optimal solution.

T o compute the actual subset,we can add an auxiliary boolean array which is1if we decide to take the-th?le in and0other-wise.

Question:How do we use all the values to determine the subset of?les having the maximum computing time?

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Constructing the Optimal Solution

Question:How do we use the values to determine the subset of items having the maximum computing time?

If keep[]is1,then.We can now repeat this argument for keep[].

If keep[]is0,the and we repeat the argu-ment for keep[].

Therefore,the following partial program will output the elements of:

;

for(downto1)

if(keep)

output i;

;

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The Complete Algorithm for the Knapsack Problem KnapSack()

for(to);

for(to)

for(to)

if(()and())

;

keep;

else

;

keep;

;

for(downto1)

if(keep)

output i;

;

return;

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