# Optimal Storage on Tapes

Given programs stored on a computer tape and length of each program is where , find the order in which the programs should be stored in the tape for which the Mean Retrieval Time (MRT given as ) is minimized.

**Example:**

Input : n = 3 L[] = { 5, 3, 10 } Output : Order should be { 3, 5, 10 } with MRT = 29/3

**Prerequisites:** Magnetic Tapes Data Storage

Let us first break down the problem and understand what needs to be done.

A magnetic tape provides only sequential access of data. In an audio tape/cassette, unlike a CD, a fifth song from the tape can’t be just directly played. The length of the first four songs must be traversed to play the fifth song. So in order to access certain data, head of the tape should be positioned accordingly.

Now suppose there are 4 songs in a tape of audio lengths 5, 7, 3 and 2 mins respectively. In order to play the fourth song, we need to traverse an audio length of 5 + 7 + 3 = 15 mins and then position the tape head.

Retrieval time of the data is the time taken to retrieve/access that data in its entirety. Hence retrieval time of the fourth song is 15 + 2 = 17 mins.

Now, considering that all programs in a magnetic tape are retrieved equally often and the tape head points to the front of the tape every time, a new term can be defined called the Mean Retrieval Time (MRT).

Let’s suppose that the retrieval time of program is . Therefore,

MRT is the average of all such . Therefore , or

The sequential access of data in a tape has some limitations. Order must be defined in which the data/programs in a tape are stored so that least MRT can be obtained. Hence the order of storing becomes very important to reduce the data retrieval/access time.

Thus, the task gets reduced – to define the correct order and hence minimize the MRT, i.e. to minimize the term

For e.g. Suppose there are 3 programs of lengths 2, 5 and 4 respectively. So there are total 3! = 6 possible orders of storage.

Order | Total Retrieval Time | Mean Retrieval Time | |
---|---|---|---|

1 | 1 2 3 | 2 + (2 + 5) + (2 + 5 + 4) = 20 | 20/3 |

2 | 1 3 2 | 2 + (2 + 4) + (2 + 4 + 5) = 19 | 19/3 |

3 | 2 1 3 | 5 + (5 + 2) + (5 + 2 + 4) = 23 | 23/3 |

4 | 2 3 1 | 5 + (5 + 4) + (5 + 4 + 2) = 25 | 25/3 |

5 | 3 1 2 | 4 + (4 + 2) + (4 + 2 + 5) = 21 | 21/3 |

6 | 3 2 1 | 4 + (4 + 5) + (4 + 5 + 2) = 24 | 24/3 |

It’s clear that by following the second order in storing the programs, the mean retrieval time is least.

In above example, the first program’s length is added ‘n’ times, the second ‘n-1’ times…and so on till the last program is added only once. So, careful analysis suggests that in order to minimize the MRT, programs having greater lengths should be put towards the end so that the summation is reduced. Or, the lengths of the programs should be sorted in increasing order. That’s the **Greedy Algorithm** in use – at each step we make the immediate choice of putting the program having the least time first, in order to build up the ultimate optimized solution to the problem piece by piece.

Below is the implementation:

## C++

`// CPP Program to find the order ` `// of programs for which MRT is ` `// minimized ` `#include <bits/stdc++.h> ` ` ` `using` `namespace` `std; ` ` ` `// This functions outputs the required ` `// order and Minimum Retrieval Time ` `void` `findOrderMRT(` `int` `L[], ` `int` `n) ` `{ ` ` ` `// Here length of i'th program is L[i] ` ` ` `sort(L, L + n); ` ` ` ` ` `// Lengths of programs sorted according to increasing ` ` ` `// lengths. This is the order in which the programs ` ` ` `// have to be stored on tape for minimum MRT ` ` ` `cout << ` `"Optimal order in which programs are to be"` ` ` `"stored is: "` `; ` ` ` `for` `(` `int` `i = 0; i < n; i++) ` ` ` `cout << L[i] << ` `" "` `; ` ` ` `cout << endl; ` ` ` ` ` `// MRT - Minimum Retrieval Time ` ` ` `double` `MRT = 0; ` ` ` `for` `(` `int` `i = 0; i < n; i++) { ` ` ` `int` `sum = 0; ` ` ` `for` `(` `int` `j = 0; j <= i; j++) ` ` ` `sum += L[j]; ` ` ` `MRT += sum; ` ` ` `} ` ` ` `MRT /= n; ` ` ` `cout << ` `"Minimum Retrieval Time of this"` ` ` `" order is "` `<< MRT; ` `} ` ` ` `// Driver Code to test above function ` `int` `main() ` `{ ` ` ` `int` `L[] = { 2, 5, 4 }; ` ` ` `int` `n = ` `sizeof` `(L) / ` `sizeof` `(L[0]); ` ` ` `findOrderMRT(L, n); ` ` ` `return` `0; ` `} ` |

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## Java

`// Java Program to find the order ` `// of programs for which MRT is ` `// minimized ` `import` `java.io.*; ` `import` `java .util.*; ` ` ` `class` `GFG ` `{ ` ` ` `// This functions outputs ` `// the required order and ` `// Minimum Retrieval Time ` `static` `void` `findOrderMRT(` `int` `[]L, ` ` ` `int` `n) ` `{ ` ` ` `// Here length of ` ` ` `// i'th program is L[i] ` ` ` `Arrays.sort(L); ` ` ` ` ` `// Lengths of programs sorted ` ` ` `// according to increasing lengths. ` ` ` `// This is the order in which ` ` ` `// the programs have to be stored ` ` ` `// on tape for minimum MRT ` ` ` `System.out.print(` `"Optimal order in which "` `+ ` ` ` `"programs are to be stored is: "` `); ` ` ` `for` `(` `int` `i = ` `0` `; i < n; i++) ` ` ` `System.out.print(L[i] + ` `" "` `); ` ` ` `System.out.println(); ` ` ` ` ` `// MRT - Minimum Retrieval Time ` ` ` `double` `MRT = ` `0` `; ` ` ` `for` `(` `int` `i = ` `0` `; i < n; i++) ` ` ` `{ ` ` ` `int` `sum = ` `0` `; ` ` ` `for` `(` `int` `j = ` `0` `; j <= i; j++) ` ` ` `sum += L[j]; ` ` ` `MRT += sum; ` ` ` `} ` ` ` `MRT /= n; ` ` ` `System.out.print( ` `"Minimum Retrieval Time"` `+ ` ` ` `" of this order is "` `+ MRT); ` `} ` ` ` `// Driver Code ` `public` `static` `void` `main (String[] args) ` `{ ` ` ` `int` `[]L = { ` `2` `, ` `5` `, ` `4` `}; ` ` ` `int` `n = L.length; ` ` ` `findOrderMRT(L, n); ` `} ` `} ` ` ` `// This code is contributed ` `// by anuj_67. ` |

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## C#

`// C# Program to find the ` `// order of programs for ` `// which MRT is minimized ` `using` `System; ` ` ` `class` `GFG ` `{ ` ` ` `// This functions outputs ` `// the required order and ` `// Minimum Retrieval Time ` `static` `void` `findOrderMRT(` `int` `[]L, ` ` ` `int` `n) ` `{ ` ` ` `// Here length of ` ` ` `// i'th program is L[i] ` ` ` `Array.Sort(L); ` ` ` ` ` `// Lengths of programs sorted ` ` ` `// according to increasing lengths. ` ` ` `// This is the order in which ` ` ` `// the programs have to be stored ` ` ` `// on tape for minimum MRT ` ` ` `Console.Write(` `"Optimal order in "` `+ ` ` ` `"which programs are"` `+ ` ` ` `" to be stored is: "` `); ` ` ` `for` `(` `int` `i = 0; i < n; i++) ` ` ` `Console.Write(L[i] + ` `" "` `); ` ` ` `Console.WriteLine(); ` ` ` ` ` `// MRT - Minimum Retrieval Time ` ` ` `double` `MRT = 0; ` ` ` `for` `(` `int` `i = 0; i < n; i++) ` ` ` `{ ` ` ` `int` `sum = 0; ` ` ` `for` `(` `int` `j = 0; j <= i; j++) ` ` ` `sum += L[j]; ` ` ` `MRT += sum; ` ` ` `} ` ` ` `MRT /= n; ` ` ` `Console.WriteLine(` `"Minimum Retrieval "` `+ ` ` ` `"Time of this order is "` `+ ` ` ` `MRT); ` `} ` ` ` `// Driver Code ` `public` `static` `void` `Main () ` `{ ` ` ` `int` `[]L = { 2, 5, 4 }; ` ` ` `int` `n = L.Length; ` ` ` `findOrderMRT(L, n); ` `} ` `} ` ` ` `// This code is contributed ` `// by anuj_67. ` |

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**Output:**

Optimal order in which programs are to be stored are: 2 4 5 Minimum Retrieval Time of this order is 6.33333

Time complexity of the above program is the time complexity for sorting, that is (Since std::sort() operates in ) If you use bubble sort instead of std::sort(), it will take

You may think that the time complexity for this particular above code should be due to both the loops in ‘mrt’ calculation, that is,, but do remember that intuitively, the for loops used can also be coded in this manner to avoid two loops :

for (int i = 0; i < n; i++) MRT += (n - i) * L[i];

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