Candy

class Solution {
public:
    int sumOfNaturalNumbers(int n) { return (n * (n + 1)) / 2; }
    int distributeCandies(vector<int>& ratings) {
        if (ratings.size() <= 1) {
            return ratings.size();
        }
        int totalCandies = 0;
        int ascending = 0;
        int descending = 0;
        int previousSlope = 0;
        for (int i = 1; i < ratings.size(); i++) {
            int currentSlope = (ratings[i] > ratings[i - 1])
                               ? 1
                               : (ratings[i] < ratings[i - 1] ? -1 : 0);
            
            if ((previousSlope > 0 && currentSlope == 0) || (previousSlope < 0 && currentSlope >= 0)) {
                totalCandies += sumOfNaturalNumbers(ascending) + sumOfNaturalNumbers(descending) + max(ascending, descending);
                ascending = 0;
                descending = 0;
            }
            
            if (currentSlope > 0) {
                ascending++;
            } else if (currentSlope < 0) {
                descending++;
            } else {
                totalCandies++;
            }
            previousSlope = currentSlope;
        }
        totalCandies += sumOfNaturalNumbers(ascending) + sumOfNaturalNumbers(descending) + max(ascending, descending) + 1;
        return totalCandies;
    }
};

The provided C++ code implements a function to distribute candies based on an array of child ratings. Each child must receive at least one candy, and children with a higher rating than their immediate neighbors should receive more candies. The logic centers on maintaining and adjusting a count of candies distributed as you iterate through the ratings array.

• The code starts by checking if the size of the ratings vector is less than or equal to 1, returning the size directly as each child (or no child) gets exactly 1 candy.
• Variables totalCandies, ascending, descending, and previousSlope are initialized. ascending and descending track the length of consecutive sequences where ratings increase or decrease. previousSlope helps determine changes in rating trends.
• A loop goes through the ratings vector, computing the slope —whether it's ascending, descending, or neutral— between consecutive ratings. This helps in determining if sequences of increasing or decreasing ratings end, which is crucial for candy distribution.
  • If a rating slope change (from ascending to neutral/descending or descending to ascending/neutral) is detected, candies are distributed based on the sums of natural numbers up to the lengths of the sequences (ascending and descending), plus the greater of the two to ensure fairness around peaks.
  • Reset ascending and descending accordingly.
• After iterating through the vector, wrap up by calculating candies for any remaining sequences.
• The function uses an auxiliary function sumOfNaturalNumbers to calculate the sum of the first n natural numbers, which is central to distributing candies according to consecutive sequences of higher or lower ratings.

This solution efficiently handles the complex criteria for candy distribution in (O(n)) time, where (n) is the number of ratings, thus ensuring each child receives the correct amount of candy according to their relative ratings.

public class Solution {
    public int sum_natural_numbers(int x) {
        return (x * (x + 1)) / 2;
    }

    public int distribute_candies(int[] scores) {
        if (scores.length <= 1) {
            return scores.length;
        }
        int totalCandies = 0;
        int increasing = 0;
        int decreasing = 0;
        int previousSlope = 0;
        for (int j = 1; j < scores.length; j++) {
            int currentSlope = (scores[j] > scores[j - 1])
                ? 1
                : (scores[j] < scores[j - 1] ? -1 : 0);

            if (
                (previousSlope > 0 && currentSlope == 0) ||
                (previousSlope < 0 && currentSlope >= 0)
            ) {
                totalCandies += sum_natural_numbers(increasing) + sum_natural_numbers(decreasing) + Math.max(increasing, decreasing);
                increasing = 0;
                decreasing = 0;
            }
            if (currentSlope > 0) {
                increasing++;
            } else if (currentSlope < 0) {
                decreasing++;
            } else {
                totalCandies++;
            }

            previousSlope = currentSlope;
        }
        totalCandies += sum_natural_numbers(increasing) + sum_natural_numbers(decreasing) + Math.max(increasing, decreasing) + 1;
        return totalCandies;
    }
}

Implement the solution for candy distribution based on the scores using the Java programming language through two main functions in the Solution class.

• sum_natural_numbers(int x)

  • Calculate the sum of the first x natural numbers. Utilize the mathematical formula (x * (x + 1)) / 2. This function supports efficient calculation of the total candies to give during increasing or decreasing streaks of scores.

• distribute_candies(int[] scores)

  • Allocate candies to participants based on their scores, ensuring that participants with higher scores than their neighboring participant(s) receive more candies.
  • Initialize variables to manage the total candies distributed, the length of increasing and decreasing sequences of scores, and tracking the previous comparison state between two continuous scores.
  • Iterate through the scores array to update the direction of the slope (increasing, decreasing, or flat) based on consecutive scores.
  • Adjust candy calculation when a transition between increasing and decreasing sequences is detected, ensuring that sequences are appropriately rewarded with candies by making use of the sum_natural_numbers function.
  • Apply the formula for an unfinished sequence at the end of the loop to ensure all candies are counted.
  • The method returns the computed total candies, effectively ensuring fairness and meeting the condition that a higher score receives more candies relative to its immediate neighbors.

This structured approach ensures clarity in distributing candies according to a given scoring pattern while maintaining time efficiency with direct calculations for sequences of scores.

int sum_natural_numbers(int num) { return (num * (num + 1)) / 2; }
int maximum(int x, int y) { return ((x) > (y) ? (x) : (y)); }
int distribute_candies(int* scores, int size) {
    if (size <= 1) {
        return size;
    }
    int total_candies = 0;
    int ascent = 0;
    int descent = 0;
    int previous_slope = 0;
    for (int i = 1; i < size; i++) {
        int current_slope = (scores[i] > scores[i - 1])
                       ? 1
                       : ((scores[i] < scores[i - 1]) ? -1 : 0);
        if ((previous_slope > 0 && current_slope == 0) ||
            (previous_slope < 0 && current_slope >= 0)) {
            total_candies += sum_natural_numbers(ascent) + sum_natural_numbers(descent) + maximum(ascent, descent);
            ascent = 0;
            descent = 0;
        }
        if (current_slope > 0) {
            ascent++;
        }
        else if (current_slope < 0) {
            descent++;
        }
        else {
            total_candies++;
        }
        previous_slope = current_slope;
    }
    total_candies += sum_natural_numbers(ascent) + sum_natural_numbers(descent) + maximum(ascent, descent) + 1;
    return total_candies;
}

The provided C program is designed to distribute candies to a list of students based on their scores in a way that students with higher scores than their immediate neighbors receive more candies. The solution computes the total amount of candies needed through these steps:

1. Define helper functions:

   • A function, sum_natural_numbers(int num), computes the sum of the first num natural numbers. This is useful for calculating the total candies given to players on a upward or downward streak (ascent or descent).
   • Another function, maximum(int x, int y), returns the maximum of two integers. It's used to determine which slope, ascent or descent, should contribute an extra candy when transitioning.

2. The main logic in distribute_candies(int* scores, int size) function:

   • Edge cases handle scenarios with size <= 1. If there's one or zero students, the candies needed are equal to the number of students.
   • Iterate through scores using a for loop from the second score to analyze the relationship between consecutive scores:
     • Determine the current slope (increase, decrease, or no change) between successive scores.
     • Handle changes in slope (from ascent to descent and vice versa). Upon each change:
       • Calculate candies for the previous streaks by summing up natural numbers up to the lengths of ascent and descent and add the maximum of the two streak lengths to ensure the highest peak gets an adequate number of candies.
       • Reset both ascent and descent.
     • Update total_candies directly for unchanged scores.
     • Maintain a count of ascent and descent based on the current slope.
   • After looping, add to total_candies any remaining candies based on the final streak lengths using similar calculations outside the loop.

3. Return the total candies calculated which ensures a fair distribution where each student only gets more candies than their closest lesser-scored neighbor.

By dissecting the code in this manner, gain control over candy distribution according to varying score sequences among students, adapting dynamically to any score pattern. The implementation strives for optimal space and time complexity while ensuring clarity and efficiency in execution.

var sum_natural_numbers = function (n) {
    return (n * (n + 1)) / 2;
};

var distribute_candies = function (ratings) {
    if (ratings.length <= 1) {
        return ratings.length;
    }
    var total_candies = 0;
    var ascend = 0;
    var descend = 0;
    var previous_slope = 0;
    for (var index = 1; index < ratings.length; index++) {
        var current_slope =
            ratings[index] > ratings[index - 1]
                ? 1
                : ratings[index] < ratings[index - 1]
                  ? -1
                  : 0;

        if (
            (previous_slope > 0 && current_slope == 0) ||
            (previous_slope < 0 && current_slope >= 0)
        ) {
            total_candies += sum_natural_numbers(ascend) + sum_natural_numbers(descend) + Math.max(ascend, descend);
            ascend = 0;
            descend = 0;
        }

        if (current_slope > 0) {
            ascend++;
        }
        else if (current_slope < 0) {
            descend++;
        }
        else {
            total_candies++;
        }

        previous_slope = current_slope;
    }
    total_candies += sum_natural_numbers(ascend) + sum_natural_numbers(descend) + Math.max(ascend, descend) + 1;
    return total_candies;
};

The given JavaScript solution addresses the problem of candy distribution based on ratings where each student must have more candies than an adjacent student with a higher rating. The solution comprises two primary functions:

• sum_natural_numbers: This function efficiently calculates the sum of the first n natural numbers using the formula n * (n + 1) / 2. This is used to determine the total number of candies during increasing or decreasing rating sequences.

• distribute_candies: This function is designed to handle the logic for distributing candies according to the ratings array provided. The algorithm iteratively evaluates the ratings to determine increases and decreases in the value:

  • Track current slopes (increase or decrease) using the ascend and descend counters.
  • Adjust candy totals when transitions between slopes occur.
  • Use the sum_natural_numbers function to add up the candies for sequences of ascending or descending ratings.
  • Ensure that the peak between an ascending and descending sequence gets the maximum possible candies from either side of the sequence.

Handle edge cases where:

• The ratings array has length 0 or 1, returning 0 or 1 respectively.
• When ratings are not strictly changing, ensuring each student in a flat portion of ratings receives only one candy.

In conclusion, the algorithm effectively adjusts the candy distribution using slope analysis and accumulates the total candies needed, making sure to optimize candy distribution during transitions in the student rating slope. This method guarantees that all conditions for the candy distribution problem are met with optimal efficiency.

class Solution:
    # Implements logic to find the total candy distribution count
    def sum_natural(self, x):
        return (x * (x + 1)) // 2

    def distribute_candy(self, scores):
        if len(scores) <= 1:
            return len(scores)
        total_candies = 0
        ascend = 0
        descend = 0
        previous_slope = 0
        for index in range(1, len(scores)):
            current_slope = (
                1
                if scores[index] > scores[index - 1]
                else (-1 if scores[index] < scores[index - 1] else 0)
            )
            if (previous_slope > 0 and current_slope == 0) or (
                previous_slope < 0 and current_slope >= 0
            ):
                total_candies += self.sum_natural(ascend) + self.sum_natural(descend) + max(ascend, descend)
                ascend = 0
                descend = 0
            if current_slope > 0:
                ascend += 1
            elif current_slope < 0:
                descend += 1
            else:
                total_candies += 1
            previous_slope = current_slope
        total_candies += self.sum_natural(ascend) + self.sum_natural(descend) + max(ascend, descend) + 1
        return total_candies

The Python solution for the "Candy" problem determines the minimum number of candies required for distribution based on an array of scores, where each score represents a child. The algorithm aims to ensure every child has at least one candy, and any child with a higher score than an adjacent child receives more candies than that child.

Key Steps in the Solution:

1. Define an auxiliary function sum_natural(x) which calculates the sum of the first x natural numbers. This helps in summing up the total candies for a group of children either ascending or descending in their scores.

2. In the primary function distribute_candy(scores), check if the scores length is 1 or less, returning the length directly if true, as each child gets at least one candy.

3. Initialize variables ascend, descend, previous_slope, and total_candies to manage candy distribution among ascending and descending sequences of scores, as well as tracking previous score comparisons.

4. Iterate through each score from the second child to the last. Determine the slope (increasing, decreasing, or neutral) by comparing the current score with the previous one.

5. Adjust total_candies based on the changes in sequences (from ascending to descending or at the end of a sequence). The candies are calculated using the natural sum of ascent or descent streak lengths, also considering the maximum of these values for the crucial point where the sequence changes.

6. At the end of the loop, include the remaining candies calculation for the last sequence of children.

7. The resulting total_candies represents the minimum candies required following the constraints provided by the scores array.

Considerations:

• Iterable through scores is necessary for comparing each child to its adjacent counterpart, crucial for relative candy distribution.
• The solution effectively handles varying patterns in the score sequences using the previous_slope and adjusting the candy distribution each time the pattern switches.
• The algorithm ensures fairness in candy distribution with a linear time complexity relative to the number of children.

By following these steps, ensure an efficient distribution of candies such that the requirements of more candies for higher scores in neighbors are met.