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COMP 731 - Data Struc & Algorithms - Lecture 26 - Tuesday, September 5, 2000
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Today:
Dynamic Programming, continued
o Thief's problem
Today I tried to lead the students to solve the following
problem together:
Thief's problem:
The thief has broken into your house, where he finds N items,
item i having value v_i (dollars) and weight w_i (kilograms)
The thief can only carry B kilograms.
What items should he take so as to maximize the value of his load?
Formally:
Given: N; v_1, ..., v_N; w_N, ..., w_N; B
Find: A subset T of {1..N} such that
sum v_i
i in T
but subject to:
sum w_i <= B
i in T
First step: Generalize:
Let A[n,b] = the maximal value using only items 1 ... n
having weight <= b
So we are interested in A[N,B], but we will calculate all the
other values, too.
Now write a recurrence relation:
/ 0 if n=0
A[n,b] =| A[n-1,b] if n>=1, w_n>b
\ max { A[n-1,b-w_n]+v_n , A[n-1, b]} if n>=1 and w_n <= b
take the item do not take the item
Example:
item 1 2 3 4 5
value 5 8 6 4 3 thief can carry 10
weight 4 6 4 2 2
Fill in the table!
0 1 2 3 4 5 6 7 8 9 10
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0 | | | | | | | | | | | |
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1 | | | | | | | | | | | |
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2 | | | | | | | | | | | |
---------------------------------------------
3 | | | | | | | | | | | |
---------------------------------------------
4 | | | | | | | | | | | |
---------------------------------------------
5 | | | | | | | | | | | |
---------------------------------------------
Writing the program:
int A[N+1][B+1], w[N+1], v[N+1], n, b, B
// get values for w,v,B;
// compute the table
for (b=0; b<=B; b++) A[0][b]=0;
for (n=1; n<=N; n++)
for (b=0; b<=B; b++)
if (w[n]>b) A[n][b]=A[n-1][b];
else A[n][b]=max(A[n-1][b],A[n-1][b-w[n]]+v[n];
// recover the answer!
cout << "Answer is " << A[N][B] << endl;
b = B;
for (n=N; n; n--)
if (A[n][b]==A[n-1][b]) cout << "Don't take item " << n << endl;
else {cout << "Take item " << n << endl; b -= w[n];}