Given N jobs where every job is represented by following three elements of it.
1. Start Time
2. Finish Time
3. Profit or Value Associated
Find the subset of jobs associated with maximum profit such that no two jobs in the subset overlap.
Examples:
Input:
Number of Jobs n = 4
Job Details {Start Time, Finish Time, Profit}
Job 1: {1, 2, 50}
Job 2: {3, 5, 20}
Job 3: {6, 19, 100}
Job 4: {2, 100, 200}
Output:
Jobs involved in maximum profit are
Job 1: {1, 2, 50}
Job 4: {2, 100, 200}
In previous post, we have discussed about Weighted Job Scheduling problem. However, the post only covered code related to finding maximum profit. In this post, we will also print the jobs invloved in maximum profit.
Let arr[0..n-1] be the input array of Jobs. We define an array DP[] such that DP[i] stores Jobs involved to achieve maximum profit of array arr[0..i]. i.e. DP[i] stores solution to subproblem arr[0..i]. The rest of algorithm remains same as discussed in previous post.
Below is its C++ implementation –
// C++ program for weighted job scheduling using Dynamic // Programming and Binary Search #include <bits/stdc++.h> using namespace std; // A job has start time, finish time and profit. struct Job { int start, finish, profit; }; // to store subset of jobs struct weightedJob { // vector of Jobs vector<Job> job; // find profit associated with included Jobs int value; }; // A utility function that is used for sorting events // according to finish time bool jobComparator(Job s1, Job s2) { return (s1.finish < s2.finish); } // A Binary Search based function to find the latest job // (before current job) that doesn't conflict with current // job. "index" is index of the current job. This function // returns -1 if all jobs before index conflict with it. The // array jobs[] is sorted in increasing order of finish time int latestNonConflict(Job jobs[], int index) { // Initialize 'lo' and 'hi' for Binary Search int lo = 0, hi = index - 1; // Perform binary Search iteratively while (lo <= hi) { int mid = (lo + hi) / 2; if (jobs[mid].finish <= jobs[index].start) { if (jobs[mid + 1].finish <= jobs[index].start) lo = mid + 1; else return mid; } else hi = mid - 1; } return -1; } // The main function that finds the subset of jobs // associated with maximum profit such that no two // jobs in the subset overlap. int findMaxProfit(Job arr[], int n) { // Sort jobs according to finish time sort(arr, arr + n, jobComparator); // Create an array to store solutions of subproblems. // DP[i] stores the Jobs involved and their total profit // till arr[i] (including arr[i]) weightedJob DP[n]; // initialize DP[0] to arr[0] DP[0].value = arr[0].profit; DP[0].job.push_back(arr[0]); // Fill entries in DP[] using recursive property for (int i = 1; i < n; i++) { // Find profit including the current job int inclProf = arr[i].profit; int l = latestNonConflict(arr, i); if (l != - 1) inclProf += DP[l].value; // Store maximum of including and excluding if (inclProf > DP[i - 1].value) { DP[i].value = inclProf; // including previous jobs and current job DP[i].job = DP[l].job; DP[i].job.push_back(arr[i]); } else // excluding the current job DP[i] = DP[i - 1]; } // DP[n - 1] stores the result cout << "Optimal Jobs for maximum profits are\n" ; for (int i=0; i<DP[n-1].job.size(); i++) { Job j = DP[n-1].job[i]; cout << "(" << j.start << ", " << j.finish << ", " << j.profit << ")" << endl; } cout << "\nTotal Optimal profit is " << DP[n - 1].value; } // Driver program int main() { Job arr[] = {{3, 5, 20}, {1, 2, 50}, {6, 19, 100}, {2, 100, 200} }; int n = sizeof(arr)/sizeof(arr[0]); findMaxProfit(arr, n); return 0; } |
Output:
Optimal Jobs for maximum profits are (1, 2, 50) (2, 100, 200) Total Optimal profit is 250
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