Dynamic Programming [Dr. Ashraf Iqbal] ()

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Dynamic Programming

M. Ashraf Iqbal

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The Knapsack Problem


The 0/1 Knapsack Problem

• The 0-1 knapsack problem: Either put an item once in the knapsack or ignore it so as to maximize total profit while keeping the weight inside the knapsack within its capacity. Alternately we may like to ignore profits and try to maximize the weight in the knapsack such that it does not exceed its capacity. This is quite similar to the subset sum problem in mathematics.


The Fractional knapsack problem: Either put an item or its fraction in the knapsack or ignore it so as to maximize total profit while keeping the weight inside the knapsack exactly equal to its capacity.

You can Cut an Item to Fit

http://www.google.com.pk/imgres?imgurl=http://images.clipartof.com/small/5669-Male-Carpenter-Cutting-Wood-On-A-Sawhorse-Clipart-Illustration.jpg&imgrefurl=http://www.clipartof.com/details/clipart/5669.html&h=368&w=450&sz=55&tbnid=QzoxPWsL_w7nAM:&tbnh=104&tbnw=127&prev=/images%3Fq%3Dimage%2Bof%2Bcutting%2Bwood&zoom=1&q=image+of+cutting+wood&hl=en&usg=__13lywSPGfp84b9pE_jq9zFTGznw=&sa=X&ei=1KuRTLvNGoGyvgO81ICJBA&ved=0CBgQ9QEwAA


Un-Bounded 0/1 Knapsack

The unbounded knapsack problem assumes that we have an unlimited supply of each weight. We intend to fill the knapsack as far as possible while keeping the total number of weights minimum (also known as the Change Problem). We can also maximize profit provided profit for each weight is also given.


Un-bounded 0/1 Knapsack

http://www.fotosearch.com/BLD007/jp2005_0002521/


Another Knapsack Problem

The bounded 0/1 knapsack problem restricts the number of each item to a fixed maximum value.


Objectives1. We may like to maximize the number of weights to be placed in the knapsack

such that the total weight remains within the knapsack capacity in a 0/1 scenario.

2. We may like to maximize the profit provided the profit of each weight is the same.

3. We may like to maximize profit provided the weight of each item is the same.


How Many Knapsack Problems?

• What Strategies should be used?• Greedy?• Dynamic Programming?• Brute Force – look at all options?• What is Greedy• What is Dynamic Programming• What is the relationship between Greedy, Dynamic

Programming, & Brute force Strategies?

How Many Knapsack?


Which Strategy should be used?• Brute Force is expensive but?• Dynamic Programming is intelligent but?• Greedy is FAST but?


Which Strategy should be used?• Brute Force can solve every thing?• Dynamic Programming can solve some thing?• How about Greedy?


Start with the Change Problem?


Start with the Change Problem?



The Recursion Tree


The Change Problem?Recursive Formulation? Recursion Tree?


The Change Problem transforms into a Graph Problem?

Find a Change = Find a Path?



Find a Change = Find a Shortest Path?


Where is Brute Force, DP, & Greedy?


Do it by Brute Force?


Do it by Dynamic Programming?


Do it by Greedy?


The Change Problem?

• Brute Force provides an Optimal answer?

• Dynamic Programming provides an Optimal answer?

• Greedy strategy provides an Optimal answer?


Profits & Weights in a Knapsack?

How to start now?


Profits & Weights in a Knapsack?



Where is Brute Force, DP, & Greedy?


Do it by Brute Force?


Greedy: The Graph collapses?


Dynamic Programming: It Collapses?


Do it by Dynamic Programming?


The Change ProblemDo it by Dynamic

Programming?



Programming?



Programming?


Maximizing ProfitDo it by Dynamic

Programming?

Weights = 4 3 2 Profit = 10 8 9


Maximizing ProfitDo it by Dynamic

Programming?

Weights = 4 3 2 Profit = 10 8 9


Time complexity calculations?


The Knapsack Problem?






The Knapsack Problem?

• The above diagram shows a recursive implementation of which recursive equation? Discuss briefly. We have not yet gone into dynamic programming.

• How can you find (from the above diagram) the combination of weights that exactly fits the knap sack of given capacity equal to 11. Discuss briefly.

• The answer to the above part requires a cute reduction of this problem into a graph problem? What is that graph problem? Please discuss.

• What will be the time complexity of this recursive implementation?


A New Recursive Formulation?

P(5,11) = P(4,5) + 6 = P(4,6) + 5 = P(4,7) + 4 = P(4,8) + 3 = P(4,9) + 2

max


A “Last Year” Mistake??


Last Year Class: Hisaan to the rescue


Alternate Recursive Formulation?


Old Recursive Formulation?


The Knapsack Problem





Another Knapsack Problem

• Problem: We have k lists of integers and a knapsack of capacity T. We need to select one item from each list such as to maximize the total weight in the knapsack without overflowing it. Here is recursive solution unfolding itself with some common sense.



• Problem: We have k lists of integers and a knapsack of capacity T. We need to select one item from each list such as to maximize the total profit in the knapsack such that the weight in the knapsack remains within its capacity. Here is recursive solution unfolding itself with some common sense.



Problem: We have k lists of integers and a knapsack of capacity T. We need to select one item from each list such as to maximize the total weight in the knapsack without overflowing it. Here is an efficient dynamic programming solution.



The Knapsack Problem• We intend to maximize profit keeping the total weight

within the knapsack capacity. Find a recursive equation corresponding to this scenario. How will you implement this recursive equation on the diagram above?

• How can you find (from the above recursive diagram) the combination of weights that maximizes total profit keeping the total weight within knapsack capacity? Discuss briefly.

• How can you reduce this graph and thus reduce the time complexity of this recursive implementation (without going into dynamic programming).


The Problem of Maximizing Profit?


The Problem of Maximizing Number of Elements in a single Knapsack



The Activity Selection Problem?


Activity Selection Problem?



Activity Selection Problem

• Maximize Total Merit with Start time and Finish time fixed for each activity?

• Maximize Total Merit when only Duration is fixed for each activity?

• Maximize the Total Utilization of the Hall?• Maximize the Total Number of activities?



Activity Selection Problem?


Maximize Merit







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The Partitioning Problem• Problem: Given a fixed sequence of n numbers, we wish

to divide the sequence into at most p partitions as fairly as possible (you know the problem already). Assume that we are given the following sequence (5, 1, 2, 3, 4) and assume that p =2.

• Write down a general recursive formulation of the linear partition problem. Find the optimal answer by inspection for the specific problem given above.

• Draw the recursion tree similar to the one drawn in Fig. 7(attached herewith). Remember that in the present problem p=2. Highlight the edges in your recursion tree, which corresponds to the optimal solution.


The Partitioning Problem

(5, 1, 2, 3, 4, 3)



(5, 1, 2, 3, 4, 3)

Partition the sequence into 4 partitions


Partition the sequence into 4 partitions (5, 1, 2, 3, 4, 3)



Time Complexity?Number of VerticesNumber of Edges




Once again the Partitioning Problem?


The Same Problem but?



Distributed by : www.myUET.net.tcAdd one more level – but why?

Distributed by : www.myUET.net.tcAdd one more level – but why?


The Rod Partitioning Problem

The size - price tableCutting costs?


The Rod Partitioning Problems

1. No cutting cost?2. Uniform cutting

cost?3. Non uniform

cutting cost



The size - price tableCutting Cost = 0


The Rod Partitioning

Problem

The size - price tableUniform Cutting Cost = x



Problem

The size - price tableNon Uniform Cutting Cost



Cutting Cost = 0Maximum Price = 13



The size - price tableCutting Cost = 0



Problem


Length & Price is given – put weights?




Distributed by : www.myUET.net.tcHow many vertices, edges?

Distributed by : www.myUET.net.tcDynamic Programming?


Dynamic Programming


Dynamic Programming


Dynamic Programming

Total vertices and edges?


Dynamic Programming


Directed Acyclic Graph


How about a fixed cutting cost?


Maximizing profits after subtracting total cutting costs








Profits after cutting costs


• Total Vertices?• Total Edges?• Time

Complexity?


An alternate way of solving it?


An alternate way of solving it?



How the DAG will change? In shape? In edge weights?




Once again the partitioning problem

• Given a fixed sequence of n positive numbers how can we partition it in k partitions such that the weight of the heaviest partition is minimum.



ghh


Once again the partitioning problem?


• How about if k=n?




• How about if some numbers are negative?



• How about if some numbers are negative? • What should be the value of k so as to

minimize the weight of the heaviest partition?• The value of k could be just 1?





ghh

Why can not be the DAG like this or that?


How the DAG would look like if some edge weights are negative but k is fixed from above?



Matrix Multiplication Problem?













The Longest or shortest path problem


The Longest path problem


Longest Increasing sequence

• What is this problem?


Longest Increasing Sub-Sequence

• What is this problem?



Dynamic Programming [Dr. Ashraf Iqbal] ()

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