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### Algorithms: Greedy Algorithms - Fractional Knapsack Problems, Task Scheduling Problem - with C Program source codes

These are two popular examples of Greedy Algorithms.

### 1. Fractional Knapsack Problem

Given the weight of items along with its value and also the maximum weight that we can to take,
we have to maximize the value we can get. Unlike 0-1 knapsack, here we are allowed to take
fractional items. Hence this problem is called fractional knapsack. Fractional knapsack problem
is solvable by greedy strategy.

Greedy algorithm – It obtains the solution of a problem by making a sequence of choices. At
each decision point, the best possible choice is made.

The basic idea is to calculate for each item the ratio of value/weight, and sort them according to
this ratio. Then take the items with the highest ratios and add them until we can’t add the next
item as whole. Finally add as much as you can of the next item.

Properties

If the items are already sorted into decreasing order of vi / wi, then the for-loop takes O(n) time. Where n is the total number of items. Therefore, the total time including the sort is in O(n log n) as quicksort’s average time complexity is O(nlogn).

Complete Tutorial for the Fractional Knapsack Problem :

## Fractional Knapsack Problem - C Program Source Code

#include<stdio.h>
/* Given the weight of items along with its value and also the maximum weight that we can to take,
we have to maximise the value we can get. Unlike 0-1 knapsack, here we are allowed to take fractional
items. Hence this problem is called fractional knapsack */

typedef struct item
{

int weight;

int value;
}item;
int compare(const void *x, const void *y)
{
item
*i1 = (item *)x, *i2 = (item *)y;

double ratio1 = (*i1).value*1.0 / (*i1).weight;

double ratio2 = (*i2).value*1.0 / (*i2).weight;

if(ratio1 < ratio2) return 1;

else if(ratio1 > ratio2) return -1;

else return 0;
}
int main()
{

int items;
scanf
("%d",&items);
item I
[items];

int iter;

for(iter=0;iter<items;iter++)

{
scanf
("%d%d",&I[iter].weight,&I[iter].value);

}
qsort
(I,items,sizeof(item),compare);

int maxWeight;
scanf
("%d",&maxWeight);

double value = 0.0;

int presentWeight = 0;

for(iter=0;iter<items;iter++)

{

if(presentWeight + I[iter].weight <maxWeight)

{
presentWeight
= presentWeight + I[iter].weight ;
value
+= I[iter].value;

}

else

{

int remaining  = maxWeight - presentWeight;
value
+= I[iter].value*remaining *1.0/I[iter].weight;

break;

}

}
printf
("Maximum value that can be attained is %.6lf\n",value);
}

Given a set of events, our goal is to maximize the number of events that we can attend. Let E ={1,2,3…n} be the events we need to attend. Each event has a start time si and a finish time fi, where si < fi. Events i and j are compatible if the intervals [si , fi) and [sj , fj) do not overlap (i.e., i and j are compatible if si ≥ fj or sj ≤ fi. The goal is to select a maximum-size set of mutually compatible events. Greedy algorithm is used for solving this problem. For this we need to arrange the events in order of increasing finish time: f1 ≤ f2 ≤ ……≤ fn.

Properties

If the events are already sorted into increasing order of fi , then the for-loop takes O(n) time. Where n is the total number of events. Therefore, the total time including the sort is in O(n log n) as quicksort’s average time complexity is O(nlogn).

Complete Tutorial for the Task Scheduling Problem :

## Task Scheduling - C Program Source Code

#include<stdio.h>
typedef struct event
{

int start_time;

int end_time;

int event_number;
}event;
int compare(const void *x, const void *y)
{
event
*e1 = (event *)x, *e2 = (event *)y;

return (*e1).end_time - (*e2).end_time;
}
/* Given the list of events, our goal is to maximise the number of events we can attend. */
int main()
{

int number_of_events;
scanf
("%d",&number_of_events);
event T
[number_of_events];

int iter;

for(iter=0;iter<number_of_events;iter++)

{
scanf
("%d%d",&T[iter].start_time,&T[iter].end_time);
T
[iter].event_number = iter;

}

/* Sort the events according to their respective finish time. */
qsort
(T,number_of_events,sizeof(event),compare);

int events[number_of_events]; // This is used to store the event numbers that can be attended.

int possible_events = 0; // To store the number of possible events

events
[possible_events++] = T[0].event_number;

int previous_event = 0;

/* Select the task if it is compatable with the previously selected task*/

for(iter=1;iter<number_of_events;iter++)

{

if(T[iter].start_time >= T[previous_event].end_time)

{
events
[possible_events++] = T[iter].event_number;
previous_event
= iter;

}

}
printf
("Maximum possible events that can be attended are %d. They are\n",possible_events);

for(iter=0;iter<possible_events;iter++)

{
printf
("%d\n",events[iter]);

}

}

Related Tutorials ( Common examples of Greedy Algorithms ) :

 Elementary cases : Fractional Knapsack Problem, Task Scheduling Elementary problems in Greedy algorithms - Fractional Knapsack, Task Scheduling. Along with C Program source code. Data Compression using Huffman Trees Compression using Huffman Trees. A greedy technique for encoding information.

Some Important Data Structures and Algorithms, at a glance:

 Arrays : Popular Sorting and Searching Algorithms Bubble Sort Insertion Sort Selection Sort Shell Sort Merge Sort Quick Sort Heap Sort Binary Search Algorithm Basic Data Structures  and Operations on them Stacks Queues Single Linked List Double Linked List Circular Linked List
 Tree Data Structures Binary Search Trees Heaps Height Balanced Trees Graphs and Graph Algorithms Depth First Search Breadth First Search Minimum Spanning Trees: Kruskal Algorithm Minumum Spanning Trees: Prim's Algorithm Dijkstra Algorithm for Shortest Paths Floyd Warshall Algorithm for Shortest Paths Bellman Ford Algorithm Popular Algorithms in Dynamic Programming Dynamic Programming Integer Knapsack problem Matrix Chain Multiplication Longest Common Subsequence Greedy Algorithms Elementary cases : Fractional Knapsack Problem, Task Scheduling Data Compression using Huffman Trees

Basic Data Structures and Algorithms

Sorting- at a glance