Advanced Algorithm Analysis

Dynamic Programming

Problem having multiple solution

Dynamic Programming used to solve Optimization Problem

Dynamic Programming is used to find solution with optimal value.

Dynamic Programming

Solve the problem by combining the solution of sub-problems

Divide and Conquer approach re-compute the solution on each step

Dynamic Programming solve sub-problem only once

Developing DP Algorithm

It has 4-steps1. Characterize the structure of Optimal

Solution2. Recursively define the value of an

Optimal Solution3. Compute the value of Optimal Solution,

typically in bottom-up fashion4. Construct the Optimal Solution from

computed information

Developing DP Algorithm

Step 1-3 are the basis of dynamic programming solution to a problem.

› It gives the optimal value only

Step 4 is used to construct the Optimal Solution

Assembly Line Scheduling

Step-1: Characterize the Optimal Solution

If we look at a fastest way through station S1,j, it must go through station j-1 on either line 1 or line 2. Thus the fastest way through station S1,j is either › The fastest way through S1,j-1 and then

directly through station S1,j or › The fastest way through station S2,j-1, a

transfer from line 2 to line 1, and then through station S1,j.

Step-1: Characterize the Optimal Solution

Using symmetric reasoning, the fastest way through stations S2,j is either › The fastest way through S2,j-1 and then

directly through station S2,j or › The fastest way through station S1,j-1, a

transfer from line 1 to line 2, and then through station S2,j

Step 2: Recursive Solution

f* = min(f1[n] + x1, f2[n] + x2) f1[1] = e1 + a1,1

f2[1] = e2 + a2,1

f1[j] = e1 + a1,1 if j = 1 = min(f1[j-1] + a1,j, f2[j-1] + t2,j-1 + a1,j)

if j ≥ 2 f2[j] = e2 + a2,1 if j = 1 = min(f2[j-1] + a2,j, f1[j-1] + t1,j-1 + a2,j)

if j ≥ 2

Step 3: Computing Fastest Time

FASTEST-WAY(a, t, e, x, n) 1 f1[1] ← e1 + a1,1

2 f2[1] ←e2 + a2,1

3 for j ← 2 to n 4 do if f1[j - 1] + a1,j ≤ f2[j - 1] + t2,j-1 + a1,j

5 then f1[j] ← f1[j - 1] + a1, j

6 l1[j] ← 1 7 else f1[j] ← f2[j - 1] + t2,j-1 + a1,j

8 l1[j] ← 2 9 if f2[j - 1] + a2,j ≤ f1[j - 1] + t1,j-1 + a2,j

10 then f2[j] ← f2[j - 1] + a2,j 11 l2[j] ← 2 12 else f2[j] ∞ f1[j - 1] + t1,j-1 + a2,j

13 l2[j] ← 1 14 if f1[n] + x1 ≤ f2[n] + x2

15 then f* = f1[n] + x1

16 l* = 1 17 else f* = f2[n] + x2

18 l* = 2  

Step 4: Construct the Fastest Way

PRINT-STATIONS(l, n) 1 i ← l* 2 print "line " i ", station " n 3 for j ← n downto 2 4 do i ← li[j] 5 print "line " i ", station " j - 1

Problem Specification

1 2 3 4 5 6

S1,i 7 9 3 4 8 4

S2,i 8 5 6 4 5 7

t1,i 2 3 1 3 4 -

t2,i 2 1 2 2 1 -

ei 2 4

xi 3 2

Solution- Step 01

Solution-Step 02

f* = min(f1[n] + x1, f2[n] + x2) f1[1] = e1 + a1,1

f2[1] = e2 + a2,1

f1[j] = e1 + a1,1 if j = 1 = min(f1[j-1] + a1,j, f2[j-1] + t2,j-1 + a1,j)

if j ≥ 2 f2[j] = e2 + a2,1 if j = 1 = min(f2[j-1] + a2,j, f1[j-1] + t1,j-1 + a2,j)

if j ≥ 2

Step 3: Computing Fastest Time FASTEST-WAY(a, t, e, x, n) 1 f1[1] ← e1 + a1,1

2 f2[1] ←e2 + a2,1

3 for j ← 2 to n 4 do if f1[j - 1] + a1,j ≤ f2[j - 1] + t2,j-1 + a1,j

5 then f1[j] ← f1[j - 1] + a1, j

6 l1[j] ← 1 7 else f1[j] ← f2[j - 1] + t2,j-1 + a1,j

8 l1[j] ← 2 9 if f2[j - 1] + a2,j ≤ f1[j - 1] + t1,j-1 + a2,j

10 then f2[j] ← f2[j - 1] + a2,j 11 l2[j] ← 2 12 else f2[j] ∞ f1[j - 1] + t1,j-1 + a2,j

13 l2[j] ← 1 14 if f1[n] + x1 ≤ f2[n] + x2

15 then f* = f1[n] + x1

16 l* = 1 17 else f* = f2[n] + x2

18 l* = 2  

Solution-Step 03

Cost and LineJ 1 2 3 4 5 6

f1[j] 9 18 20 24 32 35

f2[j] 12 16 22 25 30 37

J 2 3 4 5 6

l1[j] 1 2 1 1 2

l2[j] 1 2 1 2 2

Solution-Step 03

Optimal Value

› f* = 38

› l* = 1

Step 4: Construct the Fastest Way

PRINT-STATIONS(l, n) 1 i ← l* 2 print "line " i ", station " n 3 for j ← n downto 2 4 do i ← li[j] 5 print "line " i ", station " j - 1

J 2 3 4 5 6

l1[j] 1 2 1 1 2

l2[j] 1 2 1 2 2

Solution-Step 04

line 1, station 6 line 2, station 5 line 2, station 4 line 1, station 3 line 2, station 2 line 1, station 1

J 1 2 3 4

a1,I 3 29 3 8

a2,I 6 5 6 5

t1,I 2 3 6 4

t2,I 2 1 12 5

ei 3 1

Xi 4 9

Example 1

J 1 2 3 4 5

f1[j] 6 35 15 23 27

f2[j] 7 11 17 22 72

J 2 3 4 5

l1[j]

1 2 1 1

l2[j]

2 2 2 2

F* = 31 l*=1

J 1 2 3 4

a1,I 6 7 15 3

a2,I 9 4 8 7

t1,I 3 5 2

t2,I 1 2 1

ei 2 3

Xi 1 4

J 1 2 3 4

f1[j] 8 15 20 27

f2[j] 12 15 23 30

J 2 3 4

l1[j] 1 1 2

l2[j] 1 2 2

F* 28 l*=1S

J 1 2 3 4 5 6

a1,I 7 9 3 4 8 4

a2,I 8 5 6 4 5 7

t1,I 2 3 1 3 4

t2,I 2 1 2 2 1

ei 2 4

Xi 3 3

Example 3

J 1 2 3 4 5 6

f1[j] 9 18 20 24 32 35

f2[j] 12 16 22 25 30 37

J 2 3 4 5 6

l1[j] 1 2 1 1 2

l2[j] 1 `2 1 2 2

F* 38 l*=1

Line 1 station 6Line 2 station 5Line 2 station 4Line 1 station3Line 2 station 2Line 1 station 1

J 1 2 3 4

a1,I 9 3 4 8

a2,I 5 6 5 7

t1,I 3 1 4

t2,I 1 2 1

ei 2 4

Xi 3 2

Example with four station

J 1 2 3 4

f1[j] 11 13 17 25

f2[j] 9 15 19 26

J 2 3 4

l1[j]

2 1 1

l2[j]

2 1 2

F* = 28 l*=2

Line 2 station 4Line 2 station3Line 1 station 2Line 2 station 1