Optimal read list for given number of days - GeeksforGeeks

Optimal read list for given number of days - GeeksforGeeks
A person is determined to finish the book in 'k' days but he never wants to stop a chapter in between. Find the optimal assignment of chapters, such that the person doesn't read too many extra/less pages overall.

Consider the example 2 above where a book has 4 chapters with pages 8, 5, 6 and 12. User wishes to finish it in 3 days. The graphical representation of the above scenario is,

In the above graph vertex represents the chapter and an edge e(u, v) represents number of pages to be read to reach ‘v ‘ from ‘u ‘. Sink node is added to symbolize the end of book.

First, calculate the average number of pages to read in a day (here 31/3 roughly 10). New edge weight e ‘(u, v) would be the mean difference |avg – e(u, v)|. Modified graph for the above problem would be,

The idea is to start from chapter 1 and do a DFS to find sink with count of edges being ‘k ‘. Keep storing the visited vertices in an array say ‘path[]’. If we reach the destination vertex, and path sum is less than the optimal path update the optimal assignment optimal_path[]. Note, that the graph is DAG thus there is no need to take care of cycles during DFS.
Following, is the C++ implementation of the same, adjacency matrix is used to represent the graph. The following program has mainly 4 phases.
1) Construct a directed acyclic graph.
2) Find the optimal path using DFS.
3) Print the found optimal path.

// A DFS based recursive function to store the optimal path

// in path[] of size path_len.  The variable sum stores sum of

// of all edges on current path.  k is number of days spent so

// far.

void assignChapters(int u, int* path, int path_len, int sum, int k)

{

    static int min = INT_MAX;

    // Ignore the assignment which requires more than required days

    if (k < 0)

        return;

    // Current assignment of chapters to days

    path[path_len] = u;

    path_len++;

    // Update the optimal assignment if necessary

    if (k == 0 && u == CHAPTERS)

    {

        if (sum < min)

        {

            updateAssignment(path, path_len);

            min = sum;

        }

    }

    // DFS - Depth First Search for sink

    for (int v = u+1; v <= CHAPTERS; v++)

    {

        sum += DAG[u][v];

        assignChapters(v, path, path_len, sum, k-1);

        sum -= DAG[u][v];

    }

}

// This function finds and prints optimal read list.  It first creates a

// graph, then calls assignChapters().

void minAssignment(int pages[])

{

    // 1) ............CONSTRUCT GRAPH.................

    // Partial sum array construction S[i] = total pages

    // till ith chapter

    int avg_pages = 0, sum = 0, S[CHAPTERS+1], path[DAYS+1];

    S[0] = 0;

    for (int i = 0; i < CHAPTERS; i++)

    {

        sum += pages[i];

        S[i+1] = sum;

    }

    // Average pages to be read in a day

    avg_pages = round(sum/DAYS);

    /* DAG construction vertices being chapter name &

     * Edge weight being |avg_pages - pages in a chapter|

     * Adjacency matrix representation  */

    for (int i = 0; i <= CHAPTERS; i++)

    {

        for (int j = 0; j <= CHAPTERS; j++)

        {

            if (j <= i)

                DAG[i][j] = NOLINK;

            else

            {

                sum = abs(avg_pages - (S[j] - S[i]));

                DAG[i][j] = sum;

            }

        }

    }

    // 2) ............FIND OPTIMAL PATH................

    assignChapters(0, path, 0, 0, DAYS);

    // 3) ..PRINT OPTIMAL READ LIST USING OPTIMAL PATH....

    cout << "Optimal Chapter Assignment :" << endl;

    int ch;

    for (int i = 0; i < DAYS; i++)

    {

        ch = optimal_path[i];

        cout << "Day" << i+1 << ": " << ch << " ";

        ch++;

        while ( (i < DAYS-1 && ch < optimal_path[i+1]) ||

                (i == DAYS-1 && ch <= CHAPTERS))

        {

           cout <<  ch << " ";

           ch++;

        }

        cout << endl;

    }

}

void updateAssignment(int* path, int path_len)

{

    for (int i = 0; i < path_len; i++)

        optimal_path[i] = path[i] + 1;

}

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