Move weighting scale alternate under given constraints

Given a weighting scale and an array of different positive weights where we have an infinite supply of each weight. Our task is to put weights on left and right pans of scale one by one in such a way that pans move to that side where weight is put i.e. each time, pans of scale moves to alternate sides.

  • We are given another integer ‘steps’, times which we need to perform this operation.
  • Another constraint is that we can’t put same weight consecutively i.e. if weight w is taken then in next step while putting the weight on opposite pan we can’t take w again.
  • Examples:

    Let weight array is [7, 11]  and steps = 3 
    then 7, 11, 7 is the sequence in which 
    weights should be kept in order to move
    scale alternatively.
    
    Let another weight array is [2, 3, 5, 6] 
    and steps = 10 then, 3, 2, 3, 5, 6, 5, 3, 
    2, 3 is the sequence in which weights should
    be kept in order to move scale alternatively.
    

    This problem can be solved by doing DFS among scale states.

    1. We traverse among various DFS states for the solution where each DFS state will correspond to actual difference value between left and right pans and current step count.
    2. Instead of storing weights of both pans, we just store the difference residue value and each time chosen weight value should be greater than this difference and shouldn’t be equal to previously chosen value of weight.
    3. If it is, then we call the DFS method recursively with new difference value and one more step.

    Please see below code for better understanding,

    C++

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    // C++ program to print weights for alternating
    // the weighting scale
    #include <bits/stdc++.h>
    using namespace std;
      
    // DFS method to traverse among states of weighting scales
    bool dfs(int residue, int curStep, int wt[], int arr[],
             int N, int steps)
    {
        // If we reach to more than required steps,
        // return true
        if (curStep > steps)
            return true;
      
        // Try all possible weights and choose one which
        // returns 1 afterwards
        for (int i = 0; i < N; i++)
        {
            /* Try this weight only if it is greater than
               current residueand not same as previous chosen
               weight */
            if (arr[i] > residue && arr[i] != wt[curStep - 1])
            {
                // assign this weight to array and recur for
                // next state
                wt[curStep] = arr[i];
                if (dfs(arr[i] - residue, curStep + 1, wt,
                        arr, N, steps))
                    return true;
            }
        }
      
        //    if any weight is not possible, return false
        return false;
    }
      
    // method prints weights for alternating scale and if
    // not possible prints 'not possible'
    void printWeightsOnScale(int arr[], int N, int steps)
    {
        int wt[steps];
      
        // call dfs with current residue as 0 and current
        // steps as 0
        if (dfs(0, 0, wt, arr, N, steps))
        {
            for (int i = 0; i < steps; i++)
                cout << wt[i] << " ";
            cout << endl;
        }
        else
            cout << "Not possible\n";
    }
      
    //    Driver code to test above methods
    int main()
    {
        int arr[] = {2, 3, 5, 6};
        int N = sizeof(arr) / sizeof(int);
      
        int steps = 10;
        printWeightsOnScale(arr, N, steps);
        return 0;
    }

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    Java

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    // Java program to print weights for alternating 
    // the weighting scale
    class GFG 
    {
      
        // DFS method to traverse among 
        // states of weighting scales
        static boolean dfs(int residue, int curStep, 
                           int[] wt, int[] arr,
                           int N, int steps) 
        {
      
            // If we reach to more than required steps,
            // return true
            if (curStep >= steps)
                return true;
      
            // Try all possible weights and 
            // choose one which returns 1 afterwards
            for (int i = 0; i < N; i++) 
            {
      
                /*
                * Try this weight only if it is 
                * greater than current residue 
                * and not same as previous chosen weight
                */
                if (curStep - 1 < 0 || 
                   (arr[i] > residue &&
                    arr[i] != wt[curStep - 1]))
                {
      
                    // assign this weight to array and 
                    // recur for next state
                    wt[curStep] = arr[i];
                    if (dfs(arr[i] - residue, curStep + 1,
                                       wt, arr, N, steps))
                        return true;
                }
            }
      
            // if any weight is not possible,
            // return false
            return false;
        }
      
        // method prints weights for alternating scale 
        // and if not possible prints 'not possible'
        static void printWeightOnScale(int[] arr, 
                                       int N, int steps) 
        {
            int[] wt = new int[steps];
      
            // call dfs with current residue as 0 
            // and current steps as 0
            if (dfs(0, 0, wt, arr, N, steps)) 
            {
                for (int i = 0; i < steps; i++)
                    System.out.print(wt[i] + " ");
                System.out.println();
            
            else
                System.out.println("Not Possible");
        }
      
        // Driver Code
        public static void main(String[] args)
        {
            int[] arr = { 2, 3, 5, 6 };
            int N = arr.length;
            int steps = 10;
            printWeightOnScale(arr, N, steps);
        }
    }
      
    // This code is contributed by
    // sanjeev2552

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    Python3

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    # Python3 program to print weights for  
    # alternating the weighting scale 
      
    # DFS method to traverse among states 
    # of weighting scales 
    def dfs(residue, curStep, wt, arr, N, steps):
          
        # If we reach to more than required 
        # steps, return true 
        if (curStep >= steps): 
            return True
      
        # Try all possible weights and choose 
        # one which returns 1 afterwards
        for i in range(N):
              
            # Try this weight only if it is greater 
            # than current residueand not same as 
            # previous chosen weight 
            if (arr[i] > residue and 
                arr[i] != wt[curStep - 1]):
                  
                # assign this weight to array and 
                # recur for next state 
                wt[curStep] = arr[i] 
                if (dfs(arr[i] - residue, curStep + 1,
                                  wt, arr, N, steps)): 
                    return True
      
        # if any weight is not possible,
        # return false 
        return False
      
    # method prints weights for alternating scale 
    # and if not possible prints 'not possible' 
    def printWeightsOnScale(arr, N, steps):
        wt = [0] * (steps) 
      
        # call dfs with current residue as 0 
        # and current steps as 0 
        if (dfs(0, 0, wt, arr, N, steps)):
            for i in range(steps):
                print(wt[i], end = " ")
        else:
            print("Not possible")
      
    # Driver Code
    if __name__ == '__main__':
      
        arr = [2, 3, 5, 6
        N = len(arr)
      
        steps = 10
        printWeightsOnScale(arr, N, steps)
      
    # This code is contributed by PranchalK

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    Output:

    2 3 2 3 5 6 5 3 2 3
    

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