Greedy Algorithms&Dynamic Programming

Prof Amir GevaEitan Netzer

Greedy algorithms

Solve local optimization in hope it will bring to global optimization

Often trade off with the best answer possible

Huffman codingHuffman coding is an entropy encoding algorithm used for lossless data compression

Main idea encode repeating symbols in as short word

Input: file containing symbols or charsOutput: compressed file* Optimal!!!

Huffman(C) n<-length(C) % insert the group into priority queue Q<- C for i<-1 to n-1

do z<-allocate-node() x<-left[z]<-extract-min(Q) y<-right[z]<-extract-min(Q) f[z]<-f[x]+f[y] Insert(Q,z)

return extract-min(Q) logT n O n n

this is an example of a huffman tree

Dynamic Programming All the Torah on one leg

dynamic programming is a method for solving complex problems by breaking them down into simpler sub problems.

Solve only onceOften trade off with memory complexity

Guide lines

1.Find recursive method2.Discover that algorithm (top to bottom) is

exponential time3.If cause of exponential is repeating problems

then Dynamic Programming is in order4.Solve sub problems and keep there answer

ExampleFibonacci sequence

function fib(n) return fib(n − 1) + fib(n − 2)

function fib(n) if key n is not in map m

m[n] := fib(n − 1) + fib(n − 2) return m[n]

T n O n

Longest common subsequence

Find Longest common subsequence (not have to be continuous)sequence X: AGCAT(n elements)sequence Y: GAC(m elements)

# combinations 2n

Dynamic

function LCSLength(X[1..m], Y[1..n]) C = array(0..m, 0..n) for i := 0..m

C[i,0] = 0 for j := 0..n

C[0,j] = 0 for i := 1..m

for j := 1..n if X[i] = Y[j]

C[i,j] := C[i-1,j-1] + 1 else

C[i,j] := max(C[i,j-1], C[i-1,j]) return C[m,n] T n O nm

ExampleAGCAT GAC

ExampleAGCAT GAC

ExampleAGCAT GAC