(Latest Revision: Nov 20, 2022)

Weighted Intervals Problem

EXAMPLE:

Suppose we have the following weighted intervals. 
start   finish   weight
  12      14      7
  17      20     10
  22      25      5
  21      27      8
  19      24      4
  10      13      6
  15      23      5
  11      16      8
To solve the problem, we sort the intervals by increasing finish time, and compute the p-values.

This gives us the following list. (I include a pictorial depiction of the intervals.) 
# start   finish   weight    10     11     12     13     14     15     16     17     18     19     20     21     22     23     24     25     26     27
1  10       13       6        --------------------- p=0
2  12       14       7                      -------------- p=0
3  11       16       8               ----------------------------------- p=0
4  17       20      10                                                         --------------------- p=3
5  15       23       5                                           -------------------------------------------------------- p=2
6  19       24       4                                                                       ----------------------------------- p=3
7  22       25       5                                                                                            --------------------- p=4
8  21       27       8                                                                                      ----------------------------------------- p=4
We then use the recursive relation to compute the OPT values from row 1 to row 8.

(As an aid to implementing the base case of the algorithm, we start with a "sentinel interval 0" that has just row of zeros for the start & finish times, the weight, the p-value, and the OPT value.) 
#   strt-fnsh    w    p   OPT
0      0-0       0    0    0
1     10-13      6    0    6 = max {OPT(0), w1+OPT(p(1))} = max { 0,  6 +  0} =  6
2     12-14      7    0    7 = max {OPT(1), w2+OPT(p(2))} = max { 6,  7 +  0} =  7
3     11-16      8    0    8 = max {OPT(2), w3+OPT(p(3))} = max { 7,  8 +  0} =  8
4     17-20     10    3   18 = max {OPT(3), w4+OPT(p(4))} = max { 8, 10 +  8} = 18
5     15-23      5    2   18 = max {OPT(4), w5+OPT(p(5))} = max {18,  5 +  7} = 18
6     19-24      4    3   18 = max {OPT(5), w6+OPT(p(6))} = max {18,  4 +  8} = 18
7     22-25      5    4   23 = max {OPT(6), w7+OPT(p(7))} = max {18,  5 + 18} = 23
8     21-27      8    4   26 = max {OPT(7), w8+OPT(p(8))} = max {23,  8 + 18} = 26
Then we do the backtracking.

26=OPT(8) > OPT(7)=23, so we need interval 8, and we resume backtracking at interval p(8), which is interval 4.
18=OPT(4) > OPT(3)=8, so we need interval 4, and we resume backtracking at interval p(4), which is interval 3.
8=OPT(3) > OPT(2)=7, so we need interval 3, and we resume backtracking at interval p(3), which is interval 0, (but backtracking at interval 0 is the sentinel, so we stop).

The work above shows that intervals 8, 4, and 3 comprise an optimal solution, having combined weight of 8+10+8=26.

Note: Generally, when backtracking, if OPT(j) = OPT(j-1), it may or may not be possible to use interval j in an optimal solution. (It is only possible when w(j)+OPT(p(j))=OPT(j-1).) However, when OPT(j) = OPT(j-1), we never need interval j to get an optimal solution. Rather than check to see if w(j)+OPT(p(j))=OPT(j-1), it's less work just to resume backtracking at row j-1. We will always find an optimal solution this way.

The discussion above shows how to work this kind of problem. Your assignment is to do the following problem, and to produce a corresponding table and backtracking results, according to the technique illustrated in the example.

YOUR ASSIGNED PROBLEM: 
start   finish   weight
  10      18        9
  22      28       10
  12      15        5
  26      29        8
  11      14        7
  19      23        6
  17      21        6
  16      20       10
Let me know if you have questions about how such problems are solved. The algorithm is covered in sections 6.1 and 6.2.

Directions For Submitting Solution:

You will e-mail me your homework solution.
Below is a sample of the information I want in your e-mail. It shows what the information would be for the example problem at the beginning of this document. You must give me the same kind of information, in the same format, for your assigned problem. 
-------------------------------------------
#   strt-fnsh   w   p   OPT
1     10-13     6   0    6
2     12-14     7   0    7
3     11-16     8   0    8
4     17-20    10   3   18
5     15-23     5   2   18
6     19-24     4   3   18
7     22-25     5   4   23
8     21-27     8   4   26

Backtracking finds that intervals 8, 4, and 3 
form an optimal solution, with total weight 26. 
-------------------------------------------