Foundation of Computing Systems
Lecture 20
Classic Optimization Problems
08.10.09
IT 60101: Lecture #20
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Some Well Known Problems
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Some problems are well known to learn designing
algorithms
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08.10.09
Sorting
Searching
Coin Changing
Optimal Retrieval Problem
Traveling Salesman Problem
Knapsack Problem
Bin Packing Problem
Minimum Spanning Tree Problem
Shortest Path Problem
Huffman Tree Problem
Optimal Binary Search Tree Problem
IT 60101: Lecture #20
2
Some Well Known Problems
•
Some problems are well known to learn designing
algorithms
•
•
•
•
•
•
•
•
•
•
•
08.10.09
Sorting
Searching
Coin Changing
Optimal Retrieval Problem
Traveling Salesman Problem
Knapsack Problem
Bin Packing Problem
Minimum Spanning Tree Problem
Shortest Path Problem
Huffman Tree Problem
Optimal Binary Search Tree Problem
IT 60101: Lecture #20
3
Coin Changing Problem
=
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IT 60101: Lecture #20
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Coin Changing Problem
• Problem Definition
Given a set of denomination of the coins, say c1, c2, c3, …, cn, such
that c1 > c2 > c3 > …> cn = 1.
Given an amount A, the problem is to determine the minimum
number of coins needed to make A in change.
Note that
• The problem may assume cn=1, so that it is always possible to
make change for any amount A.
• Any number of coins for each denomination is available.
08.10.09
IT 60101: Lecture #20
5
Some Well Known Problems
•
Some problems are well known to learn designing
algorithms
•
•
•
•
•
•
•
•
•
•
•
08.10.09
Sorting
Searching
Coin Changing
Optimal Retrieval Problem
Traveling Salesman Problem
Knapsack Problem
Bin Packing Problem
Minimum Spanning Tree Problem
Shortest Path Problem
Huffman Tree Problem
Optimal Binary Search Tree Problem
IT 60101: Lecture #20
6
Optimal Retrieval Problem

08.10.09
IT 60101: Lecture #20
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Block
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Block
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Optimal Retrieval Problem
Problem Definition
Suppose, that n programs having length l1, l2, …, ln are stored on a tape. If
the programs are stored in the order li1, li2, …, lin, the time to retrieve any
program ik is
k
Tk   li j
j 1
(We assume that retrieval time starts at the beginning of the tape. The
above retrieval time for a program ik results from the fact that in order to
retrieve program ik, we must pass all of the programs stored before program
ik).
The average retrieval time is defined as
1 n
T   Tk
n k 1
The problem is to find an ordering of the program to minimize the average
retrieval time.
08.10.09
IT 60101: Lecture #20
8
Some Well Known Problems
•
Some problems are well known to learn designing
algorithms
•
•
•
•
•
•
•
•
•
•
•
08.10.09
Sorting
Searching
Coin Changing
Optimal Retrieval Problem
Traveling Salesman Problem
Knapsack Problem
Bin Packing Problem
Minimum Spanning Tree Problem
Shortest Path Problem
Huffman Tree Problem
Optimal Binary Search Tree Problem
IT 60101: Lecture #20
9
Traveling Salesman Problem
08.10.09
IT 60101: Lecture #20
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Traveling Salesman Problem
Problem definition
A salesman has a list of cities, each of which he must visit exactly
once. There are direct roads between each pair of cities on the list.
The problem is to find route the salesman should follow so that he
travels the shortest possible distance on a round trip , starting at any
one of the cities and then returning there. He should not travel a city
twice except the starting city.
08.10.09
IT 60101: Lecture #20
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Different Versions of Traveling Salesman
Problems
Hamiltonian Cycle Problem
This problem is to find Hamiltonian cycle with minimum weight graph in
graph theory and stated as below.
Given an connected weighted graph and it is required to find a tour (a cycle
through all the vertices exactly once except the first ) of minimum weight.
Euclidian Traveling Salesman Problem
This problem is to determine the shortest path closed tour that connects a
given set of n-points on a plane.
Bitonic Tour Problem
A tour starts at the left most point, go strictly left to right, and then go strictly
right to left and finally back to the starting point. The objective function and
constraint is same as in the Traveling Salesman Problem.
08.10.09
IT 60101: Lecture #20
12
Some Well Known Problems
•
Some problems are well known to learn designing
algorithms
•
•
•
•
•
•
•
•
•
•
•
08.10.09
Sorting
Searching
Coin Changing
Optimal Retrieval Problem
Traveling Salesman Problem
Knapsack Problem
Bin Packing Problem
Minimum Spanning Tree Problem
Shortest Path Problem
Huffman Tree Problem
Optimal Binary Search Tree Problem
IT 60101: Lecture #20
13
Knapsack Problem
08.10.09
IT 60101: Lecture #20
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Knapsack Problem
The knapsack problem can be posed as follows.
– A thief robbing a store finds n items: the i-th item is worth vi and
weight wi. The thief would take as valuable as possible but he
can carry at most W weights in his knapsack. Which items
should the thief take?
08.10.09
IT 60101: Lecture #20
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Knapsack Problems
The knapsack problem can be generalized and formally stated
as follows.
Given n items and a knapsack. The knapsack has a capacity W. Any item
say i, has a weight wi and a value vi per unit. If a quantity xi of the item i is
placed into the knapsack, then a profit vi.xi is earned.
The objective is to obtain a filling of the knapsack that maximizes the total
profit earned.
Since the knapsack capacity is W, we require the total weight of all chosen
object to be at most W.
Formally the problem may be stated as follows.
n
Maximize
08.10.09
 vi  xi
i 1
n
Subject to
 xi  wi  W
i 1
IT 60101: Lecture #20
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Different Versions Knapsack Problems
Discrete Knapsack Problem
Here, all vi’s, wi’s and W are integers and each item must be taken
or left behind.
This problem is also called “integer” knapsack, and “0-1” knapsack
problem
Continuous Knapsack Problem
In this version of the knapsack problem, we can take any quantity
(i.e. a fraction) of each item (rather that having to make a binary (0
or 1) choice for each item.)
Note:
You can think 0-1 knapsack problem as being like a gold ingot while
an item in continuous problem is more like gold dust.
08.10.09
IT 60101: Lecture #20
17
Some Well Known Problems
•
Some problems are well known to learn designing
algorithms
•
•
•
•
•
•
•
•
•
•
•
08.10.09
Sorting
Searching
Coin Changing
Optimal Retrieval Problem
Traveling Salesman Problem
Knapsack Problem
Bin Packing Problem
Minimum Spanning Tree Problem
Shortest Path Problem
Huffman Tree Problem
Optimal Binary Search Tree Problem
IT 60101: Lecture #20
18
Some Well Known Problems
•
Some problems are well known to learn designing
algorithms
•
•
•
•
•
•
•
•
•
•
•
08.10.09
Sorting
Searching
Coin Changing
Optimal Retrieval Problem
Traveling Salesman Problem
Knapsack Problem
Bin Packing Problem
Minimum Spanning Tree Problem
Shortest Path Problem
Huffman Tree Problem
Optimal Binary Search Tree Problem
IT 60101: Lecture #20
19
Bin Packing Problem
08.10.09
IT 60101: Lecture #20
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Bin Packing Problem
Problem definition
Given n items of sizes s1, s2, …, sn. All sizes satisfy 0<si ≤1.
The problem is to pack these items in the fewest number of bins,
given that each bin has unit capacity.
08.10.09
IT 60101: Lecture #20
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