[FEATURE] Addition of Weighted Interval Scheduling in C++ (#3014)

dynamic_programming/weighted_interval_scheduling.cpp

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+/**
+ * @file
+ * @brief Implementation of [Weighted Interval Scheduling Problem]
+ * (https://en.wikipedia.org/wiki/Interval_scheduling)
+ *
+ * @details
+ * The Weighted Interval Scheduling problem is a classic optimization problem
+ * where we have n intervals, each with a start time, end time, and profit.
+ * The goal is to select a subset of non-overlapping intervals such that the
+ * total profit is maximized.
+ *
+ * ### Problem Statement:
+ * Given n intervals where each interval i has:
+ * - start[i]: start time of interval i
+ * - end[i]: end time of interval i  
+ * - profit[i]: profit gained by selecting interval i
+ *
+ * Find the maximum profit that can be obtained by selecting non-overlapping
+ * intervals.
+ *
+ * ### Algorithm:
+ * 1. Sort intervals by their end times
+ * 2. For each interval, use binary search to find the latest non-overlapping
+ *    interval
+ * 3. Use dynamic programming: dp[i] = max(profit[i] + dp[j], dp[i-1])
+ *    where j is the latest non-overlapping interval with interval i
+ *
+ * Time Complexity: O(n log n) - due to sorting and binary search
+ * Space Complexity: O(n) - for the DP array
+ *
+ * @author [Your Name]
+ */
+
+#include <algorithm>  /// for std::sort, std::max
+#include <cassert>    /// for assert
+#include <iostream>   /// for IO operations
+#include <vector>     /// for std::vector
+
+/**
+ * @namespace dynamic_programming
+ * @brief Dynamic Programming algorithms
+ */
+namespace dynamic_programming {
+
+/**
+ * @namespace weighted_interval_scheduling
+ * @brief Implementation of Weighted Interval Scheduling problem
+ */
+namespace weighted_interval_scheduling {
+
+/**
+ * @brief Structure to represent an interval with start time, end time, and profit
+ */
+struct Interval {
+    int start;   ///< start time of the interval
+    int end;     ///< end time of the interval
+    int profit;  ///< profit gained by selecting this interval
+
+    /**
+     * @brief Constructor for Interval
+     * @param s start time
+     * @param e end time  
+     * @param p profit
+     */
+    Interval(int s, int e, int p) : start(s), end(e), profit(p) {}
+};
+
+/**
+ * @brief Comparator to sort intervals by their end times
+ * @param a first interval
+ * @param b second interval
+ * @return true if a should come before b in sorted order
+ */
+bool compareByEndTime(const Interval &a, const Interval &b) {
+    return a.end < b.end;
+}
+
+/**
+ * @brief Find the latest non-overlapping interval using binary search
+ * @param intervals vector of sorted intervals (by end time)
+ * @param index current interval index
+ * @return index of latest non-overlapping interval, or -1 if none exists
+ */
+int findLatestNonOverlapping(const std::vector<Interval> &intervals, int index) {
+    int left = 0, right = index - 1;
+    int result = -1;
+
+    // Binary search for the latest interval that ends before current interval starts
+    while (left <= right) {
+        int mid = left + (right - left) / 2;
+
+        if (intervals[mid].end <= intervals[index].start) {
+            result = mid;
+            left = mid + 1;
+        } else {
+            right = mid - 1;
+        }
+    }
+
+    return result;
+}
+
+/**
+ * @brief Solve the weighted interval scheduling problem
+ * @param intervals vector of intervals with start time, end time, and profit
+ * @return maximum profit that can be obtained
+ */
+int solveWeightedIntervalScheduling(std::vector<Interval> intervals) {
+    if (intervals.empty()) {
+        return 0;
+    }
+
+    int n = intervals.size();
+
+    // Sort intervals by their end times
+    std::sort(intervals.begin(), intervals.end(), compareByEndTime);
+
+    // dp[i] represents maximum profit using intervals 0 to i
+    std::vector<int> dp(n);
+    dp[0] = intervals[0].profit;
+
+    for (int i = 1; i < n; i++) {
+        // Option 1: Don't include current interval
+        int excludeProfit = dp[i - 1];
+
+        // Option 2: Include current interval
+        int includeProfit = intervals[i].profit;
+        int latestNonOverlapping = findLatestNonOverlapping(intervals, i);
+
+        if (latestNonOverlapping != -1) {
+            includeProfit += dp[latestNonOverlapping];
+        }
+
+        // Take maximum of both options
+        dp[i] = std::max(includeProfit, excludeProfit);
+    }
+
+    return dp[n - 1];
+}
+
+/**
+ * @brief Solve and return the selected intervals (bonus: solution reconstruction)
+ * @param intervals vector of intervals with start time, end time, and profit
+ * @return pair containing maximum profit and vector of selected intervals
+ */
+std::pair<int, std::vector<Interval>> solveWithSolution(std::vector<Interval> intervals) {
+    if (intervals.empty()) {
+        return {0, {}};
+    }
+
+    int n = intervals.size();
+    std::sort(intervals.begin(), intervals.end(), compareByEndTime);
+
+    std::vector<int> dp(n);
+    dp[0] = intervals[0].profit;
+
+    for (int i = 1; i < n; i++) {
+        int excludeProfit = dp[i - 1];
+        int includeProfit = intervals[i].profit;
+        int latestNonOverlapping = findLatestNonOverlapping(intervals, i);
+
+        if (latestNonOverlapping != -1) {
+            includeProfit += dp[latestNonOverlapping];
+        }
+
+        dp[i] = std::max(includeProfit, excludeProfit);
+    }
+
+    // Reconstruct solution
+    std::vector<Interval> selectedIntervals;
+    int i = n - 1;
+
+    while (i >= 0) {
+        if (i == 0 || dp[i] != dp[i - 1]) {
+            // Current interval is included
+            selectedIntervals.push_back(intervals[i]);
+            int latestNonOverlapping = findLatestNonOverlapping(intervals, i);
+            i = latestNonOverlapping;
+        } else {
+            // Current interval is not included
+            i--;
+        }
+    }
+
+    // Reverse to get chronological order
+    std::reverse(selectedIntervals.begin(), selectedIntervals.end());
+
+    return {dp[n - 1], selectedIntervals};
+}
+
+}  // namespace weighted_interval_scheduling
+}  // namespace dynamic_programming
+
+/**
+ * @brief Self-test implementations
+ * @returns void
+ */
+static void test() {
+    using namespace dynamic_programming::weighted_interval_scheduling;
+
+    // Test case 1: Basic example
+    std::vector<Interval> intervals1 = {
+        Interval(1, 4, 2),   // [1,4] profit=2
+        Interval(3, 5, 4),   // [3,5] profit=4  
+        Interval(0, 6, 1),   // [0,6] profit=1
+        Interval(5, 7, 6),   // [5,7] profit=6
+        Interval(8, 9, 5),   // [8,9] profit=5
+        Interval(5, 9, 2)    // [5,9] profit=2
+    };
+
+    int result1 = solveWeightedIntervalScheduling(intervals1);
+    // Let me trace through this step by step:
+    // After sorting by end time: [1,4,2], [3,5,4], [0,6,1], [5,7,6], [8,9,5], [5,9,2]
+    // Optimal solution: [3,5] (profit=4), [5,7] (profit=6), [8,9] (profit=5) = 15 total
+    assert(result1 == 15);
+
+
+    // Test case 1: Simple non-overlapping case  
+    std::vector<Interval> simpleTest = {
+        Interval(1, 2, 10),
+        Interval(3, 4, 20), 
+        Interval(5, 6, 30)
+    };
+    int simpleResult = solveWeightedIntervalScheduling(simpleTest);
+    assert(simpleResult == 60); // All intervals can be selected: 10+20+30=60
+
+    // Test case 2: Overlapping intervals where we need to choose
+    std::vector<Interval> overlapTest = {
+        Interval(0, 2, 1),   // [0,2] profit=1
+        Interval(1, 3, 10),  // [1,3] profit=10 (overlaps with first)
+        Interval(2, 4, 5)    // [2,4] profit=5 (overlaps with second)
+    };
+    int overlapResult = solveWeightedIntervalScheduling(overlapTest);
+    assert(overlapResult == 10); // Should select [1,3] for maximum profit
+
+    // Test case 3: Empty input
+    std::vector<Interval> emptyTest = {};
+    int emptyResult = solveWeightedIntervalScheduling(emptyTest);
+    assert(emptyResult == 0);
+
+    // Test solution reconstruction
+    auto [maxProfit, selectedIntervals] = solveWithSolution(simpleTest);
+    assert(maxProfit == 60);
+    assert(selectedIntervals.size() == 3);
+
+    std::cout << "All test cases passed!" << std::endl;
+}
+
+/**
+ * @brief Main function
+ * @returns 0 on exit
+ */
+int main() {
+    using namespace dynamic_programming::weighted_interval_scheduling;
+
+    int n;
+    std::cout << "Enter number of intervals: ";
+    std::cin >> n;
+
+    if (n <= 0) {
+        std::cout << "Number of intervals must be positive." << std::endl;
+        return 1;
+    }
+
+    std::vector<Interval> intervals;
+    intervals.reserve(n);
+
+    std::cout << "Enter intervals (start_time end_time profit):" << std::endl;
+    for (int i = 0; i < n; i++) {
+        int start, end, profit;
+        std::cin >> start >> end >> profit;
+
+        if (start >= end) {
+            std::cout << "Invalid interval: start time must be less than end time." << std::endl;
+            return 1;
+        }
+
+        intervals.emplace_back(start, end, profit);
+    }
+
+    int maxProfit = solveWeightedIntervalScheduling(intervals);
+    std::cout << "\nMaximum profit: " << maxProfit << std::endl;
+
+    // Also show the solution
+    auto [profit, selectedIntervals] = solveWithSolution(intervals);
+    std::cout << "\nSelected intervals:" << std::endl;
+    for (const auto& interval : selectedIntervals) {
+        std::cout << "[" << interval.start << ", " << interval.end 
+                  << "] profit=" << interval.profit << std::endl;
+    }
+
+    test();  // run self-test implementations
+
+    return 0;
+}