Re: (Attock VU Group) mera aj ka paper rah gaya hai kia karo plz advise me plzzzzzzzzzz

Example: Let us go through an example. We want to find the optimal multiplication order for

A1

(5×4) · A2

(4×6) · A3

(6×2) · A4

(2×7) · A5

(7×3)

We will compute the entries of the m matrix starting with the base condition. We first fill that main

diagonal:

0

0

0

0

0

Next, we compute the entries in the first super diagonal, i.e., the diagonal above the main diagonal:

m[1, 2] = m[1, 1] +m[2, 2] + p0 · p1 · p2 = 0 + 0 + 5 · 4 · 6 = 120

m[2, 3] = m[2, 2] +m[3, 3] + p1 · p2 · p3 = 0 + 0 + 4 · 6 · 2 = 48

m[3, 4] = m[3, 3] +m[4, 4] + p2 · p3 · p4 = 0 + 0 + 6 · 2 · 7 = 84

m[4, 5] = m[4, 4] +m[5, 5] + p3 · p4 · p5 = 0 + 0 + 2 · 7 · 3 = 42

The matrix m now looks as follows:

0 120

0 48

0 84

0 42

0

We now proceed to the second super diagonal. This time, however, we will need to try two possible

values for k. For example, there are two possible splits for computing m[1, 3]; we will choose the split

that yields the minimum:

m[1, 3] = m[1, 1] +m[2, 3] + p0 · p1 · p3 == 0 + 48 + 5 · 4 · 2 = 88

m[1, 3] = m[1, 2] +m[3, 3] + p0 · p2 · p3 = 120 + 0 + 5 · 6 · 2 = 180

the minimum m[1, 3] = 88 occurs with k = 1

88 CHAPTER 6. DYNAMIC PROGRAMMING

Similarly, for m[2, 4] and m[3, 5]:

m[2, 4] = m[2, 2] +m[3, 4] + p1 · p2 · p4 = 0 + 84 + 4 · 6 · 7 = 252

m[2, 4] = m[2, 3] +m[4, 4] + p1 · p3 · p4 = 48 + 0 + 4 · 2 · 7 = 104

minimum m[2, 4] = 104 at k = 3

m[3, 5] = m[3, 3] +m[4, 5] + p2 · p3 · p5 = 0 + 42 + 6 · 2 · 3 = 78

m[3, 5] = m[3, 4] +m[5, 5] + p2 · p4 · p5 = 84 + 0 + 6 · 7 · 3 = 210

minimum m[3, 5] = 78 at k = 3

With the second super diagonal computed, the m matrix looks as follow:

0 120 88

0 48 104

0 84 78

0 42

0

We repeat the process for the remaining diagonals. However, the number of possible splits (values of k)

increases:

m[1, 4] = m[1, 1] +m[2, 4] + p0 · p1 · p4 = 0 + 104 + 5 · 4 · 7 = 244

m[1, 4] = m[1, 2] +m[3, 4] + p0 · p2 · p4 = 120 + 84 + 5 · 6 · 7 = 414

m[1, 4] = m[1, 3] +m[4, 4] + p0 · p3 · p4 = 88 + 0 + 5 · 2 · 7 = 158

minimum m[1, 4] = 158 at k = 3

m[2, 5] = m[2, 2] +m[3, 5] + p1 · p2 · p5 = 0 + 78 + 4 · 6 · 3 = 150

m[2, 5] = m[2, 3] +m[4, 5] + p1 · p3 · p5 = 48 + 42 + 4 · 2 · 3 = 114

m[2, 5] = m[2, 4] +m[5, 5] + p1 · p4 · p5 = 104 + 0 + 4 · 7 · 3 = 188

minimum m[2, 5] = 114 at k = 3

The matrix m at this stage is:

6.4. CHAIN MATRIX MULTIPLY 89

0 120 88 158

0 48 104 114

0 84 78

0 42

0

That leaves the m[1, 5] which can now be computed:

m[1, 5] = m[1, 1] +m[2, 5] + p0 · p1 · p5 = 0 + 114 + 5 · 4 · 3 = 174

m[1, 5] = m[1, 2] +m[3, 5] + p0 · p2 · p5 = 120 + 78 + 5 · 6 · 3 = 288

m[1, 5] = m[1, 3] +m[4, 5] + p0 · p3 · p5 = 88 + 42 + 5 · 2 · 3 = 160

m[1, 5] = m[1, 4] +m[5, 5] + p0 · p4 · p5 = 158 + 0 + 5 · 7 · 3 = 263

minimum m[1, 5] = 160 at k = 3

We thus have the final cost matrix.

0 120 88 158 160

0 0 48 104 114

0 0 0 84 78

0 0 0 0 42

0 0 0 0 0

Here is the order in which m entries are calculated

0 1 5 8 10

0 0 2 6 9

0 0 0 3 7

0 0 0 0 4

0 0 0 0 0

and the split k values that led to a minimum m[i, j] value

0 1 1 3 3

0 2 3 3

0 3 3

0 4

0

90 CHAPTER 6. DYNAMIC PROGRAMMING

Based on the computation, the minimum cost for multiplying the five matrices is 160 and the optimal

order for multiplication is

((A1(A2A3))(A4A5))

This can be represented as a binary tree

A1

A2 A3

2

1

A4 A5

4

3

ap mujhy ye example samjha sakti hain kia

mujhy is main  ye values kaisy maloom hoti hain in ki samjh nahi aarahi hy

 m[1,2]=m[1,1]+m[2,2]+po.p1.p2=0+0+5.4.6=120

is terha ki example ko samjha dain

On Sun, Nov 27, 2011 at 11:13 AM, Shazia <shazia277@gmail.com> wrote:

sure sis........see attachment.......

On Sun, Nov 27, 2011 at 10:00 PM, sara raza <sararazameer@gmail.com> wrote:

sis shazi ap mujhy cs502 ki handout mail ker sakti hoo 

mujhy chey thii

thanks

On Sun, Nov 27, 2011 at 10:57 AM, Shazia <shazia277@gmail.com> wrote:

wasalam.......
sis send mail regarding it to exams department n also to date sheet id........


DateSheet@vu.edu.pk
Conduct@vu.edu.pk

On Sun, Nov 27, 2011 at 9:53 PM, mc110200312 Nabila Hassnain <mc110200312@vu.edu.pk> wrote:

AoA

 

dear fellows mera aj ka paper rah gaya hai plz mujy batien mai kai karu kai koi chance hai k mai dobara paper day saku?

plzzzzzzzzzzzz reply

mai buht pershan ho

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