Algorithm

Getting Started with AlgorithmWhat is an Algorithm?

Characteristics of Algorithm1 Topic

Analysis Framework

Performance Analysis3 Topics

Mathematical Analysis2 Topics

Sorting AlgorithmSorting Algorithm10 Topics

Searching Algorithm6 Topics

Fundamental of Data StructuresStacks

Queues

Graphs

Trees

Sets

Dictionaries

Divide and ConquerGeneral Method

Binary Search

Recurrence Equation for Divide and Conquer

Finding the Maximum and Minimum

Merge Sort

Quick Sort

Stassen’s Matrix Multiplication

Advantages and Disadvantages of Divide and Conquer

Decrease and ConquerInsertion Sort

Topological Sort

Greedy MethodGeneral Method

Coin Change Problem

Knapsack Problem

Job Sequencing with Deadlines

Minimum Cost Spanning Trees2 Topics

Single Source Shortest Paths1 Topic

Optimal Tree Problem1 Topic

Transform and Conquer Approach1 Topic

Dynamic ProgrammingGeneral Method with Examples

Multistage Graphs

Transitive Closure1 Topic

All Pairs Shortest Paths6 Topics

BacktrackingGeneral Method

NQueens Problem

Sum of Subsets problem

Graph Coloring

Hamiltonian Cycles

Branch and Bound2 Topics

0/1 Knapsack problem2 Topics

NPComplete and NPHard Problems1 Topic
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Coin Change Problem
If you are not very familiar with a greedy algorithm, here is the gist: At every step of the algorithm, you take the best available option and hope that everything turns optimal at the end which usually does. The problem at hand is coin change problem, which goes like given coins of denominations 1,5,10,25,100; find out a way to give a customer an amount with the fewest number of coins. For example, if I ask you to return me change for 30, there are more than two ways to do so like
Amount: 30
Solutions : 3 X 10 ( 3 coins )
6 X 5 ( 6 coins )
1 X 25 + 5 X 1 ( 6 coins )
1 X 25 + 1 X 5 ( 2 coins )
The last solution is the optimal one as it gives us a change of amount only with 2 coins, where as all other solutions provide it in more than two coins.
Solution for coin change problem using greedy algorithm is very intuitive. Basic principle is : At every iteration in search of a coin, take the largest coin which can fit into remaining amount we need change for at the instance. At the end you will have optimal solution.
Coin change problem : Algorithm
1. Sort n denomination coins in increasing order of value.
2. Initialize set of coins as empty. S = {}
3. While amount is not zero:
3.1 Ck is largest coin such that amount > Ck
3.1.1 If there is no such coin return “no viable solution”
3.1.2 Else include the coin in the solution S.
3.1.3 Decrease the remaining amount = amount – Ck
Coin change problem : implementation
#include <stdio.h>
int coins[] = { 1,5,10,25,100 };
int findMaxCoin(int amount, int size){
for(int i=0; i<size; i++){
if(amount < coins[i]) return i1;
}
return 1;
}
int findMinimumCoinsForAmount(int amount, int change[]){
int numOfCoins = sizeof(coins)/sizeof(coins[0]);
int count = 0;
while(amount){
int k = findMaxCoin(amount, numOfCoins);
if(k == 1)
printf("No viable solution");
else{
amount= coins[k];
change[count++] = coins[k];
}
}
return count;
}
int main(void) {
int change[10]; // This needs to be dynamic
int amount = 34;
int count = findMinimumCoinsForAmount(amount, change);
printf("\n Number of coins for change of %d : %d", amount, count);
printf("\n Coins : ");
for(int i=0; i<count; i++){
printf("%d ", change[i]);
}
return 0;
}
What will the time complexity of the implementation? First of all, we are sorting the array of coins of size n, hence complexity with O(nlogn). While loop, the worst case is O(amount). If all we have is the coin with 1 denomination. Overall complexity for coin change problem becomes O(n log n) + O(amount).
Will this algorithm work for all sort of denominations? The answer is no. It will not give any solution if there is no coin with denomination 1. So be careful while applying this algorithm.