1. Analysis & Design of Algorithms (CSCE 321)
Prof. Amr Goneid
Department of Computer Science, AUC
Part 8. Greedy Algorithms
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2. Greedy Algorithms
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3. Greedy Algorithms Microsoft Interview
From:
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4. Greedy Algorithms Greedy Algorithms The General Method
Continuous Knapsack Problem
Optimal Merge Patterns
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5. 1. Greedy Algorithms Methodology:
Start with a solution to a small sub-problem
Build up to the whole problem
Make choices that look good in the short term but not necessarily in the long term
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6. Greedy Algorithms Disadvantages: They do not always work.
Short term choices may be disastrous on the long term.
Correctness is hard to prove
Advantages:
When they work, they work fast
Simple and easy to implement
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7. 2. The General method
Let a[ ] be an array of elements that may contribute to a solution. Let S be a solution,
Greedy (a[ ],n) {
S = empty;
for each element (i) from a[ ], i = 1:n
x = Select (a,i);
if (Feasible(S,x)) S = Union(S,x); }
return S;
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8. The General method (continued)
Select:
Selects an element from a[ ] and removes it.Selection is optimized to satisfy an objective function.
Feasible:
True if selected value can be included in the solution vector, False otherwise.
Union:
Combines value with solution and updates objective function.
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9. 3. Continuous Knapsack Problem
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10. Continuous Knapsack Problem
Environment
Object (i):
Total Weight wi
Total Profit pi
Fraction of object (i) is continuous (0 =< xi <= 1)
A Number of Objects
1 =< i <= n
A knapsack
Capacity m 1 2 n m
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11. The problem Problem Statement:
For n objects with weights wi and profits pi, obtain the set of fractions of objects xi which will maximize the total profit without exceeding a total weight m.
Formally:
Obtain the set X = (x1 , x2 , … , xn) that will maximize 1 i  n pi xi subject to the constraints:
1 i  n wi xi  m , 0 xi  1 , 1 i  n
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12. Optimal Solution Feasible Solution: by satisfying constraints.
Feasible solution and maximizing profit.
Lemma 1:
If 1 i  n wi = m then xi = 1 is optimal.
Lemma 2:
An optimal solution will give 1 i  n wi xi = m
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13. Greedy Algorithm To maximize profit, choose highest p first.
Also choose highest x , i.e., smallest w first.
In other words, let us define the “value” of an object (i) to be the ratio vi = pi/wi and so we choose first the object with the highest vi value.
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14. Algorithm GreedyKnapsack ( p[ ] , w[ ] , m , n ,x[ ] ) {
insert indices (i) of items in a maximum heap on value vi = pi / wi ;
Zero the vector x; Rem = m ;
For k = 1..n
{ remove top of heap to get index (i);
if (w[i] > Rem) then break;
x[i] = 1.0 ; Rem = Rem – w[i] ; }
if (k < = n ) x[i] = Rem / w[i] ;
// T(n) = O(n log n)
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15. Example n = 3 objects, m = 20 P = (25 , 24 , 15) , W = (18 , 15 , 10),
V = (1.39 , 1.6 ,1.5)
Objects in decreasing order of V are {2 , 3 , 1}
Set X = {0 ,0 ,0} and Rem = m = 20
K = 1, Choose object i = 2:
w2 < Rem, Set x2 = 1, w2 x2 = 15 , Rem = 5
K = 2, Choose object i = 3:
w3 > Rem, break;
K < n , x3 = Rem / w3 = 0.5
Optimal solution is X = (0 , 1.0 , 0.5) ,
Total profit is 1 i  n pi xi = 31.5
Total weight is 1 i  n wi xi = m = 20
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16. 4. Optimal Merge Patterns (a) Definitions
Binary Merge Tree:
A binary tree with external nodes representing entities and internal nodes representing merges of these entities.
Optimal Binary Merge Tree:
The sum of paths from root to external nodes is optimal (e.g. minimum). Assuming that the node (i) contributes to the cost by pi and the path from root to such node has length Li, then optimality requires a pattern that minimizes
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17. Optimal Binary Merge Tree
If the items {A,B,C} contribute to the merge cost by PA , PB , PC, respectively, then the following 3 different patterns will cost:
P1= 2(PA+PB)+PC P2 = PA+2(PB+PC) P3 = 2PA+PB+2PC
Which of these merge patterns is optimal?
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18. (b) Optimal Merging of Lists
Lists {A,B,C} have lengths 30,25,10, respectively. The cost of merging two lists of lengths n,m is n+m. The following 3 different merge patterns will cost:
P1= 2(30+25)+10 = 120 P2 = 30+2(25+10) = 100 P3 = 25+2(30+10) = 105
P2 is optimal so that the merge order is {{B,C},A}.
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19. The Greedy Method C 10 B 25 A 30 BC 35 BCA 65
Insert lists and their lengths in a minimum heap of lengths.
Repeat
Remove the two lowest length lists (pi ,pj) from heap.
Merge lists with lengths (pi,pj) to form a new list with length pij = pi+ pj
Insert pij and its symbols into the heap
until all symbols are merged into one final list C 10 B 25 A 30 BC 35
BCA 65
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20. The Greedy Method
Notice that both Lists (B : 25 elements) and (C : 10 elements) have been merged (moved) twice
List (A : 30 elements) has been merged (moved) only once.
Hence the total number of element moves is 100.
This is optimal among the other merge patterns.
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21. (c) Huffman Coding Terminology
Symbol:
A one-to-one representation of a single entity.
Alphabet:
A finite set of symbols.
Message:
A sequence of symbols.
Encoding:
Translating symbols to a string of bits.
Decoding:
The reverse.
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22. Example: Coding Tree for 4-Symbol Alphabet (a,b,c,d)
Encoding:
a 00
b 01
c 10
d 11
Decoding:
b c a d a
This is fixed length coding
abcd 1 ab cd 1 1 a b c d
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23. Coding Efficiency & Redundancy
Li =Length of path from root to symbol (i) = no. of bits representing that symbol.
Pi = probability of occurrence of symbol (i) in message.
n = size of alphabet.
< L > = Average Symbol Length = 1 i  n Pi Li bits/symbol (bps)
For fixed length coding, Li = L = constant, < L > = L (bps)
Is this optimal (minimum) ? Not necessarily.
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24. Coding Efficiency & Redundancy
The absolute minimum < L > in a message is called the Entropy.
The concept of entropy as a measure of the average content of information in a message has been introduced by Claude Shannon (1948).
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25. Coding Efficiency & Redundancy
Shannon's entropy represents an absolute limit on the best possible lossless compression of any communication. It is computed as:
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26. Coding Efficiency & Redundancy
Coding Efficiency:  = H / < L >    1
Coding Redundancy: R = 1 -   R  1
Actual <L>
Optimal <L> H
Perfect <L>
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27. Example: Fixed Length Coding
4- Symbol Alphabet (a,b,c,d). All symbols have the same length L = 2 bits
Message : abbcaada
< L > = 2 (bps)
Symbol (i) pi
-log pi
-pi log pi
code Li a
0.5 1 00 2 b
0.25 01 c
0.125 3
0.375 10 d 11
H = 1.75
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28. Example Entropy H = 0.5 + 0.5 + 0.375 + 0.375 = 1.75 (bps),
Coding Efficiency
 = H / < L > = 1.75 / 2 = 0.875,
Coding Redundancy
R = 1 – = 0.125
This is not optimal
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29. Result
Fixed length coding is optimal (perfect) only when all symbol probabilities are equal.
To prove this:
With n = 2m symbols, L = m bits and <L> = m (bps).
If all probabilities are equal,
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30. Variable Length Coding (Huffman Coding)
The problem:
Given a set of symbols and their
probabilities
Find a set of binary codewords
that minimize the average
length of the symbols
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31. Variable Length Coding (Huffman Coding)
Formally:
Input: A message M(A,P) with
a symbol alphabet A = {a1,a2,…,an} of size (n)
a set of probabilities for the symbols P = {p1,p2,….pn}
Output: A set of binary codewords C = {c1,c2,….cn} with bit lengths L = {L1,L2,….Ln}
Condition:
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32. Variable Length Coding (Huffman Coding)
To achieve optimality, we use optimal binary merge trees to code symbols of unequal probabilities.
Huffman Coding:
More frequent symbols occur nearer to the root ( shorter code lengths), less frequent symbols occur at deeper levels (longer code lengths).
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33. The Greedy Method
Store each symbol in a parentless node of a binary tree.
Insert symbols and their probabilities in a minimum heap
of probabilities.
Repeat
Remove lowest two probabilities (pi ,pj) from heap.
Merge symbols with (pi,pj) to form a new symbol (aiaj) with
probability pij = pi+ pj
Store symbol (aiaj) in a parentless node with two children ai and aj
Insert pij and its symbols into the heap
until all symbols are merged into one final alphabet (root)
Trace path from root to each leaf (symbol) to form the bit string for that symbol. Concatenate “0” for a left branch, and “1” for a right branch.
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34. Example (1): 4- Symbol Alphabet A = {a, b, c, d} of size (4).
Message M(A,P) : abbcaada, P = {0.5, 0.25, 0.125, 0.125}
H = 1.75
Symbol (i) pi
-log pi
-pi log pi a
0.5 1 b
0.25 2 c
0.125 3
0.375 d
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35. Building The Optimal Merge Tablesi pi d
0.125 c cd
0.25 b
bcd
0.5 a
abcd
1.0
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36. Optimal Merge Tree for Example(1)
a (50%), b (25%), c (12.5%), d (12.5%) a b c d
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37. Optimal Merge Tree for Example(1)
a (50%), b (25%), c (12.5%), d (12.5%) cd 1 a b c d
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38. Optimal Merge Tree for Example(1)
a (50%), b (25%), c (12.5%), d (12.5%)
bcd 1 b cd 1 a c d
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39. Optimal Merge Tree for Example(1)
a (50%), b (25%), c (12.5%), d (12.5%) ai ci Li
(bits) a 1 b 10 2 c
110 3 d
111
abcd 1 a
bcd 1 b cd 1 c d
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40. Coding Efficiency for Example(1)
< L > = ( 1* * * * 0.125) = 1.75 (bps)
H = = (bps),
 = H / < L > = 1.75 / 1.75 = , R = 0.0
Notice that:
Symbols exist at leaves, i.e., no symbol code is the prefix of another symbol code.
This is why the method is also called
“prefix coding”
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41. Analysis The cost of insertion in a minimum heap is O(n logn)
The repeat loop is done (n-1) times.
In each iteration, the worst case removal of the least two elements is 2 logn and the insertion of the merged element is logn
Hence, the complexity of the Huffman algorithm is
O(n logn)
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42. Example (2): 4- Symbol Alphabet A = {a, b, c, d} of size (4).
Symbol (i) pi
-log pi
-pi log pi a
0.40
1.322
0.5288 b
0.25 2
0.5 c
0.18
2.474
0.4453 d
0.17
2.556
0.4345
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43. Example(2): Merge Tablesi pi d
0.17 c
0.18 b
0.25 cd
0.35 a
0.40
cdb
0.60
cdba
1.0
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44. Optimal Merge Tree for Example(2)ai ci Li
(bits) a 1 b 01 2 c
001 3 d
000
cdba 1
cdb a 1 cd b 1 c d
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45. Coding Efficiency for Example(2)
a (40%), b (25%), c (18%), d (17%)
<L> = 1.95 bps (Optimal)
H = 1.909
 = %
R = 2.1 %
Coding is optimal (97.9%) but not perfect
Important Result:
Perfect coding ( = 100 %) can be achieved only for
probability values of the form 2- m (1/2, ¼, 1/8,…etc )
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46. File Compression
Variable Length Codes can be used to compress files. Symbols are initially coded using ASCII (8-bit) fixed length codes.
Steps:
1. Determine Probabilities of symbols in file.
2. Build Merge Tree (or Table)
3. Assign variable length codes to symbols.
4. Encode symbols using new codes.
5. Save coded symbols in another file together with the symbol code table.
The Compression Ratio = < L > / 8
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47. Huffman Coding Animations
For examples of animations of Huffman coding, see:
Huffman.html
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