# Greedy Algorithms

### Overview

• Greedy Algorithms:
• General design technique
• Used for optimization problems
• Simply choose best option at each step
• Solve remaining subproblems after making greedy step

• We look at:
• Knapsack Problem (again): 0-1 and Fractional
• Huffman Encoding
• Activity Selection

### Kinds of Knapsack Problems

• Two main kinds of Knapsack Problems:
1. 0-1 Knapsack:
• N items (can be the same or different)
• Have only one of each
• Must leave or take (ie 0-1) each item (eg ingots of gold)
• DP works, greedy does not

2. Fractional Knapsack:
• N items (can be the same or different)
• Can take fractional part of each item (eg bags of gold dust)
• Greedy works and DP algorithms work

• Knapsack Problem that we did earlier with DP:
• N kinds of items
• Have unlimited supply of each item
• Equivalent to a 0-1 problem in which there are enough of each item to fill the knapsack

### Fractional Knapsack: Greedy Solution

• Algorithm:
• Assume knapsack holds weight W and items have value vi and weight wi
• Rank items by value/weight ratio: vi / wi
• Thus: vi / wi ≥ vj / wj, for all i ≤ j
• Consider items in order of decreasing ratio
• Take as much of each item as possible

• Code:
•     -- Assumes value and weight arrays are sorted by vi/wi
Fractional-Knapsack(v, w, W)
i := 1
while load < W and i ≤ n loop
if wi ≤ W - load then
take all of item i
else
take (W-load) / wi of item i
end if
add weight of what was taken to load
i := i + 1
end loop

• Example: Knapsack Capacity W = 30 and  Item A B C D Value 50 140 60 60 Size 5 20 10 12 Ratio 10 7 6 5

• Solution:
• All of A, all of B, and ((30-25)/10) of C (and none of D)
• Size: 5 + 20 + 10*(5/10) = 30
• Value: 50 + 140 + 60*(5/10) = 190 + 30 = 220

• Time: Θ(n), if already sorted

### Greedy Algorithms Don't Work for 0-1 Knapsack Problems

• Greedy doesn't work for 0-1 Knapsack Problem:
• Example 1: Knapsack Capacity W = 25 and  Item A B C D Price 12 9 9 5 Size 24 10 10 7

• Optimal: B and C. Size=10+10=20. Price=9+9=18
• Possible greedy approach: take largest Price first (Price=12, not optimal)
• Possible greedy approach: take smallest size item first (Price=9+5=14, not optimal)

• Example 2: Knapsack Capacity = 30
•  Item A B C Price 50 140 60 Size 5 20 10 Ratio 10 7 6

• Possible greedy approach: take largest ratio: (Solution: A and B. Size=5+20=25. Price=50+140=190

• Optimal: B and C. Size=20+10=30. Price=140+60=200

• Greedy fractional: A, B, and half of C. Size=5+20+10*(5/10)=30. Price 50+140+60*(5/10) = 190+30 = 220

• For comparison: DP algorithm gives 18
• Use 2D array: rows 0..25, columns 0..4
• Initialize first row and column to 0
• Solve a row at a time, subtracting off added size as needed
• What is the best way to fill:
• With A only: sizes 0..23, 24, 25
• With A,B only: sizes 0..9, 10, 11..23, 24, 25
• With A,B,C only: sizes 0..9, 10, 11..19, 20, 21..23, 24, 25
• With A,B,C,D: sizes 0..6, 7, 8..9, 10, 11..16, 17, 18..19, 20, 21..23, 24, 25

### Greedy vs DP (Overview)

• With DP: solve subproblems first, then use those solutions to make an optimal choice

• With Greedy: make an optimal choice (without knowing solutions to subproblems) and then solve remaining subproblem(s)

• DP solutions are bottom up; greedy are top down

• Both apply to problems with optimal substructure: solution to larger problem contains solutions to (1 or more) subproblems

### Another Greedy Algorithm: Huffman Coding

• Goal: Compress data

• Assumptions:
• Data are a sequence of characters, encoded with some fixed length scheme
• Frequency of characters is known

• Basic technique: Compress by encoding each character as a specific, variable length bit string

• Key idea: encode common characters with short codewords

• Efficient encoding with a variable length code requires a prefix code

### Prefix Code

• In a prefix code, no codeword is a prefix of any other codeword
• A codeword is a word in the code
• In the codes below, what are the codewords?
• Is the first code below a prefix code?
• Is the second code below a prefix code?
• No codeword has a prefix of 0, 101, 100, 111, etc

• In a prefix code, you can always (efficiently) determine when one codeword ends and the next begins
• Example: 1011101111011111000101
• Solution: 101 1101 111 0 111 1100 0 101

### Huffman Coding Example

• Example (from CLRS): encode characters a .. f

•  character to encode a b c d e f Frequency 45 13 12 16 9 5 Fixed length codeword 000 001 010 011 100 101 Variable length codeword 0 101 100 111 1101 1100
• Total frequency is 100
• An prefix code that is optimal always exists - but how to find it?

### Codes as Trees

• The trees below represent the codes above
• The path to a leaf represents the code for the character in that leaf
• Non-leaf node values are total frequencies below

### Huffman Code Algorithm

• Algorithm idea:
• Build tree from bottom up
• Repeat until 1 tree results: join 2 smallest nodes and update frequencies
• Keep nodes and subtrees on a priority queue to find smallest 2 nodes
• Sort by frequencies of total in tree

• Algorithm:
•         -- Assume that Q.front removes and returns the front of the queue
Huffman(C)
Q := C
for i in C.size - 1 loop
l = Q.front
r = Q.front
newFreq := l.freq + r.freq
n := new Node'(l, r, newFreq)
Q.insert(n)


• Sequence of building tree:
• f,e,c,b,d,a
• c,b,(f,e), d, a
• (f,e), d, (c, b), a
• (c,b), ((f,e), d), a
• a, ((c,b),((f,e), d))
• (a, ((c,b),((f,e), d)))

### Another Greedy Algorithm: Activity Selection

• Problem:
• Set of n activities that each require exclusive use of a common resource (eg a room)
• S = {a1, a2, ..., an}, S is a set of activties
• Each ai needs the resource during period [si, fi)
• ai needs resource from start time si up to but not including finish time fi
• Objective: Select largest possible set of nonoverlapping (mutually compatible) activities

• Imagine: Spend day at theater. maximize number of films watched.

• Could have other problems: eg longest total time, maximize fees

• Assume S is sorted by finish time (ie f1 ≤ f2 ≤ f3 ≤ ... ≤ fn-1 ≤ fn)

### Example

• Example (from CLRS) - solve for S =
•  i 1 2 3 4 5 6 7 8 9 si 1 2 4 1 5 8 9 11 13 fi 3 5 7 8 9 10 11 14 16

• A diagram:
•             :a5-----.
:a4-----------.
:a2---.       :a7-.   :a9---.
:a1-. :a3---. :a6-. :a8---.
+ + + + + + + + + + + + + + + +
1-2-3-4-5-6-7-8-9-0-1-2-3-4-5-6

• Two maximal solutions: {a1, a3, a6, a8}, and {a2, a5, a7, a9}

### Optimal Substructure

• Define Sij
• Sij = activities that start after ai finishes and finish before aj starts
• Sij = activities can be done between ai and aj
• Sij = {ak ∈ S | fi ≤ sk < fk ≤ sj }
• Diagram:  -ai-: :-ak-: :-aj---
• S18 = {a3, a5, a6, a7}

• Let Aij = an optimal solution to Sij
• eg A18 = {a3, a6}

• Choose any activity ak ∈ Aij. It defines two subproblems:
1. Solve Sik
2. Solve Skj
3. Example: choose a3 ∈ A18, which gives subproblems S13 = {} and S38 = {a6, a7}

• Now let's use ak, to define Aik and Akj and then show that they are optimal:
• Let Aik = Aij ∩ Sik; eg choose a3, A13 = {}
• Let Akj = Aij ∩ Skj eg choose a3, A38 = {a6}
• Now Aij = Aik ∪ {ak} ∪ Akj
• and thus |Aij| = |Aik| + 1 + |Akj|
• But, Aij is optimal, and so Aik and Akj must also be optimal
• Otherwise we could cut and paste and improve Aij

• The provides a basis for a recursive solution

### DP Solutions

• Proved above: Optimal solution Aij contains optimal solutions for Sik and Skj. This gives the following:
• Let c(i, j) = size of optimal solution for Sij
• Then c(i, j) = c(i, k) + 1 + c(k, j), for some ak

• To find the right activity $a_k$ to choose, we try them all:

• $\displaystyle c(i, j) = \begin{cases} & 0, \text{if }S_{ij} = \emptyset \\ & \max_{a_k \in S_{ij}}\{c(i,k) + 1 + c(k,j)\}, \text{if }S_{ij} \ne \emptyset \end{cases}$

• We could implement this top down or bottom up. What would the table look like?
• The DP solution tries all subproblems. Can we find a Greedy solution?

### A Greedy Solution

• Don't need dynamic programming - greedy solution works

• Greedy Approach: choose activity to add to optimal solution before solving subproblems!

• Which activity to add: the one that leaves the most time for others
• Which leaves the most time: the first to finish!
• ie a1

• After choosing first to finish, only one subproblem remains
• And it is solved by the same method

• Algorithm:
• Choose activity that finishes first
• Throw out activities that start before the chosen one finishes
• Repeat until done

• Let's try it:
•             :a5-----.
:a4-----------.
:a2---.       :a7-.   :a9---.
:a1-. :a3---. :a6-. :a8---.
+ + + + + + + + + + + + + + + +
1-2-3-4-5-6-7-8-9-0-1-2-3-4-5-6

• Add a1, throw out a2, a4
• Add a3, throw out a5
• Add a6, throw out a7
• Add a8, throw out a9

### Formalizing the Greedy Approach

• Simpler notation for subproblem: Sk = {ai ∈ S |fk ≤ si }
• In other words, Sk is the set of activities that finish when or after activity ak finishes

• After choosing ak to add to solution, we must solve Sk

• If ak is the first to finish in Sij, can we guarantee that ak is part of an optimal solution to Sij (ie ak ∈ Aij for some optimal solution Aij):
• Let ak be the earliest finisher in Sij
• Let am ∈ Aij be the earliest finisher in Aij
• If k=m then ak is part of an optimal solution, and we are done.
• If k ≠ m then
• Simply replace am with a1 in the optimal solution
• This must be possible (because both start after ai, and ak ends at or before am)
• We get a new optimal of the same size!
• Thus choosing ak can lead to an optimal solution

• We can extend this to all of the Sk

### Recursive Greedy Solution

• Define a0 with f0 = 0 and thus S0 = S
• Code:
•     -- s and f are start and finish arrays
-- n activities in original problem
-- k index of current subproblem
-- Finds maximal set of activies that start after activity k finishes
-- Call: RAS(s, f, 0, n)
Rec-Activity-Selector(s, f, k, n)
m = k + 1

-- Find first activity that starts when or after k finishes
while m ≤ n and s(m) < f(k) loop
m := m + 1;
end loop

if m ≤ n then
return {am} ∪ Rec-Activity-Selector(s, f, m, n)
else
return empty-set
end if;

• Time: Θ(n), if s and f are sorted

### Iterative Greedy Algorithm

• Code is easy:
•     Greedy-Activity-Selector(s, f)
n := s.length
A := {a1}  -- Put first activity in maximal
k := 1
-- Find next activity to finish
for m in 2 .. n loop
if s(m) ≥ f(k) then
A := A ∪ (am}
k := m
end if
end loop
return A

• Time: Θ(n), if s and f are sorted

### Greedy vs Dynamic Programming (1)

• DP:
• Choice at each step depends on solutions to subproblems
• Work on subproblems from bottom up
• A memoized recursive solution effectively works from bottom up

• Many subproblems are repeated in solving larger problems
• Example: solving rod cuttin for length 3 uses the solutions for lengths 2, and 1. Solving it for length 4 uses solutions for 3, 2, and 1. Thus, the solutions for 2 and 1 are reused in solving every value larger than 2.
• This repetition results in great savings when the computation is bottom up.

• Greedy:
• Make best choice at current time, then work on subproblems
• Best choice does not depend on solutions to subproblems
• Best choice does depend on choices so far

• Problems solved by both exhibit optimal substructure
• Optimal Substructure: solution to problem contains within it optimal solutions to subproblems

• Key idea: do you have to compare solutions that contain and don't contain the item
• 0-1 Knapsack: to determine whether to include item i for a given size, must consider best solution, at that size, with and without item i
• Fractional knapsack: at each step, choose item with highest ratio

• Proof needed: must show that optimal solution contains greedy solution

### Greedy vs Dynamic Programming (2)

• We can characterize greedy and dynamic programming solutions, as follows
• Dynamic programming - to find max value for problem P:
•     T - a Table of the values of the best solutions of problems of sizes Smallest upto P

for i in Smallest subproblem to P loop
T(i) := MAX of:
T(j) + cost of choice that changes subproblem j into problem i
T(k) + cost of choice that changes subproblem k into problem i
... as many subproblems as needed
end loop
Result is T(P)

• T has as many dimensions as needed
• Number of dimensions determined by recursive equation
• Each dimension needs a loop
• Think of T as cached solutions to smaller problems
• We fill T with solutions first to small problems, then to large problems

• Greedy Algorithm - to find maximum value for problem P:
•     tempP = P    -- tempP is the remaining subproblem
while tempP not empty loop
in subproblem tempP, decide greedy choice C
Add value of C to solution
tempP := subproblem tempP reduced based on choice C
end loop