알고리즘_4장-Greedy Approach

8/3/2019 _4 -Greedy Approach

1/17

CH4. Greedy Approach Grab data items in sequence,

each time with best choice,

without thinking future

Efficient, Simple!!!But, cannot solve many problems

G.A. : choose locally optimal choice 1-by-1 D.P. : solve smaller problems optimal solution


2/17

CH4. Greedy Approache.g. coin change problem:

Goal: correct change + as few coins as possible

Approach : choose largest possible coin

Locally optimal choice optimal solutionNo reconsideration!!!

Q?: Is the solution really optimal? (proof required!!!)

e.g. U.S. coin system : Greedy Approach O.K. Suppose : 16 ? Dynamic Programming


3/17

Outline of Greedy Approachwhile (there are more coins and the instance is not solved) {

grab the largest remaining coin;// selection procedure

if(adding the coin makes the change exceed target amount)// feasibility check

reject the coin;else

add the coin to the change;

if(the total value of the change equals to the target amount)// solution check

the instance is solved;}


4/17

G.A. vs D.P. - Knapsack Problem 0/1 knapsack problem

S = {item 1, item 2,,item n}

wi= weight of item i

pi= profit of item i

W = maximum weight the knapsack can hold

(wi, pi, W : positive integers)

Determine AS s.t. is maximized underBrute Force Approach : consider all subsets

exponential time algo !!!

A

ip

A

iWw


5/17

G.A. for Knapsack Problem G.A. to 0/1 knapsack problem

: Largest profit first : incorrectEx. (w1, w2, w3) = (25, 10, 10), (P1, P2, P3 ) = (10, 9, 9), W = 30

Greedy(item1) : $10 vs. Optimal(item 2 & 3) : $18

: Lightest item first :

: Largest profit per unit weight first

Ex. (w1, w2, w3) = (5,10,20), (P1, P2, P3 ) = (50,60,140), W = 30profit per unit weight = item1($10), item3($7), item2($6)

Greedy(item1 & 3) : $190 vs. Optimal(item 2 & 3) : $200


6/17

G.A. for Knapsack Problem G.A. to fractional knapsack problem

: optimal solution guaranteed

Ex. $50+$140+5/10($60)=$220

(proof necessary)


7/17

4.3. Scheduling Minimization Total Time in the System

e.g. t1=5, t2=10, t3=4

Schedule Total time in the System

[1, 2, 3] 5 + (5+10) + (5+10+4) = 39[1, 3, 2] 5 + (5+4) + (5+4+10) = 33[2, 1, 3] 10 + (10+5) + (10+5+4) = 44[2, 3, 1] 10 + (10+4) + (10+4+5) = 43

[3, 1, 2] 4 + (4+5) + (4+5+10) = 32[3, 2, 1] 4 + (4+10) + (4+10+5) = 37

Algorithm :Sort in non-decreasing order, and Schedule


8/17

4.2. Single-Source Shortest Paths

All-Pairs Shortest Path Prob.: (n3) Floyds algo (D.P.)

(from each vertex to all other vertices) Single-Source Shortest Path Prob.: Dijkstras algo (G.A.)

(from one particular vertex to all the others)

v5

v4 v3

v2v13

5

1

9

1 2

2

4

3

3

v5

v4 v3

v2v13

5

1

9

1 2

2

4

3

3


9/17

Dijkstras Algorithm for Shortest Paths Start with {v1}; //source

Choose nearest vertex from v1,

Y={v1};

F=; //shortest paths treewhile (not solved) do {

select v from V-Y nearest from v1 using only Y as intermediates;add v to Y;add the edge touching v to F;if

Y=V then Exit;}

Time Complexity ???


10/17

4.1 Minimum Spanning Trees(MST) Connected weighted undirected graph: G=(V,E)

Disconnected graph

Tree : connected acyclic graph

Rooted tree, spanning tree

A conn. weightedundirected G

v1

v3 v4

v5

v21

3 3 64

2 5

A S.T.

1

36

5

1

34

2

An MST

Spanning forest


11/17

Minimum Spanning Tree Weight of a spanning tree T :

M.S.T : a S.T. with min. such weight

Input : Adjacency Matrix W of G=(V,E)

Output : An MST T=(V,F) (FE) Uniqueness? (|F|=|V|-1)

Greedy Approach :

F= //initializationwhile (not solved yet) do {Select an edge (L.O.C) //selection

if create not cycle then add //feasibilityifT=(V,F) is a S.T. then exit //solution check

}


12/17

Prims Algorithm for MST

F=; Y={v1}; //initializationwhile (not solved) do {

select vV-Y nearest to Y; //selection & feasibilityadd v to Y; add the edge to F;ifY=V then exit; //solution check

}

v1

v3 v4

v5

v21

33

6

4

2 5

v1

v3 v4

v5

v21

33

6

4

2 5


13/17

Prims Algorithm for MST Data Structures :

nnadjacency Matrix W nearest[i]=index of vertex in Y nearest to vi distance[i]= weight of edge (vi, nearest[i])

Time Complexity ??? Q1: Spanning tree? : Yes. Easy Q2: Minimum S.T.? Formal Proof required


14/17

Kruskals Algorithm for MST

F=;

create ndisjoint subsets {v1},{v2},,{vn};sort the edges in E;while (not solved) do {

select next edge; //selectionifthe edge connects 2 disjoint subsets { //feasibility check

merge the subsets; add the edge to F;}

ifall the subsets are merged, then exit; //solution check}

v1

v3

v4

v5

v21

33

6

4

2 5

v1

v3

v4

v5

v21

33

6

4

2 5


15/17

Kruskals Algorithm for MST Worst-case time complexity Analysis : (m log m)

Sorting : (mlog m) Initialization : (n)

Total time for find, merge : (m(m, n)) Remaining ops : O(m)

Prims algo : (n2) where n-1 m n(n-1)/2


16/17

4.4 Huffman Code Suppose encoding a text(chars from n-char alphabet)by assigning codeword(sequence of bits) to each charEx. a(45), b(13), c(12), d(16), e(9), f(5)

Fixed-length encodinga(000), b(001), c(010), d(011), e(100), f(101): 300-bits

Variable-length encodinga(0), b(1), c(00), d(01), e(10), f(11): 142-bitsaaba 0010 ce???

Prefix codea(0), b(101), c(100), d(111), e(1101), f(1100): 224-bitsaaba 001010 aaba


17/17

Huffman Code Input: alphabet A, aA, an associated freq f(a) Output: tree T so that B(T)=f(a)dT(a) is minimized

100a(45) 550 25 30c(12) b(13) 14 d(16)100 101 111f(5) e(9)

1100 1101

Documents

알고리즘_4장-Greedy Approach