Problem Description

The problem provides an array called coins which contains integers representing the denominations of different coins. In addition, you are given an integer amount which signifies the total amount of money you need to create using these coins. The question is to find out the total number of distinct combinations of coins that can add up to exactly that amount. If it is not possible to get to that amount using the given coins, the function should return 0. Notably, it is assumed that you have an unlimited supply of each coin denomination in the array coins.

The answer needs to be such that it can fit within a 32-bit signed integer, effectively suggesting that the solution does not need to handle extremely large numbers that exceed this size.

Intuition

The solution requires a methodical approach to count the combinations without having to consider each one explicitly, which would be inefficient. This is where dynamic programming becomes useful. Dynamic programming is a strategy for solving complex problems by breaking them down into simpler subproblems, solving each subproblem just once, and storing their solutions – ideally, using a memory-based data structure.

The intuition for the solution arises from realizing that the problem resembles the classic "Complete Knapsack Problem". With the knapsack problem, imagine you have a bag (knapsack) and you aim to fill it with items (in this case, coins) up to a certain weight limit (the amount), and you are interested in counting the number of ways to reach exactly that limit.

1. We use a list dp to store the number of ways to reach every amount from 0 to amount. So, dp[x] will indicate how many distinct combinations of coins can sum up to the value x.

2. Initialize the dp list with zeros since initially, we have no coin, so there are zero ways to reach any amount except for 0. For an amount of 0, there is exactly one way to reach it – by choosing no coins. This means dp[0] is initialized to 1.

3. Iterate over each coin in coins, considering it as a potential candidate for making up the amount. For each coin, update the dp list.

4. For each value from the coin's denomination up to amount, increment dp[j] by dp[j - coin]. This represents that the number of combinations for an amount j includes all the combinations we had for amount (j - coin) plus this coin.

By filling the dp list iteratively and using the previously computed values, we build up the solution without redundant calculations and eventually dp[amount] gives us the total number of combinations to form amount using the given denominations.

Learn more about Dynamic Programming patterns.

Solution Approach

The implementation of the solution is grounded in the dynamic programming approach. To better understand the solution, let's take a detailed look at each step that is performed in the provided code:

1. A dynamic programming array dp is initiated with a length of one more than the amount, because we need to store ways to reach amounts ranging from 0 to amount inclusively. Every entry in dp is set to 0 to start.

   dp = [0] * (amount + 1)

2. To establish our base case, dp[0] is set to 1, because there is always exactly one way to reach an amount of 0 — by not using any coins.

   dp[0] = 1

3. The primary loop iterates over each coin in the coins array. This loop is necessary to consider each coin denomination for making up different amounts.

   for coin in coins:
       # loop body

4. Within this loop, a nested loop runs from coin (the current coin's value) up to amount + 1. This loop updates dp[j] where j is the current target amount being considered. It is important we start at coin because that is the smallest amount that can be made with the current coin denomination, and previous amounts would have been evaluated already with the other coins.

   for j in range(coin, amount + 1):
       dp[j] += dp[j - coin]

5. With each iteration of the inner loop, dp[j] is increased by dp[j - coin]. The reasoning behind this is the essence of dynamic programming where we use previously computed results to construct the current result. In this aspect, if dp[j - coin] represents the number of ways to reach j - coin amount, then we are effectively saying if you add the current coin to all those combinations, you now reach j. This way we are adding to the count of reaching j with all the ways that existed to reach j - coin.

Following these steps, dp[amount] eventually holds the total number of combinations that can be used to reach the amount, and thus, is returned as the final answer:

return dp[-1]

This dynamic programming table (dp array) allows us to avoid re-calculating combinations for smaller amounts, making the algorithm efficient and scalable for large inputs. By relying on the iterative addition of combinations, the algorithm effectively builds the solution from the ground up, solving smaller sub-problems before piecing them together to yield the solution for the larger problem.

Example Walkthrough

Let's illustrate the solution approach with a small example. Suppose we're given coins = [1, 2, 5] and amount = 5. We want to find the total number of distinct combinations that can add up to the amount.

1. Initialize the dynamic programming array dp: Create a dp array of size amount + 1, which in this case is 6 because our target amount is 5. Initially, it will look like this after initialization:

   dp = [0, 0, 0, 0, 0, 0]

2. Establish the base case: Since there's only one way to reach 0 (using no coins), we set dp[0] to 1:

   dp = [1, 0, 0, 0, 0, 0]

3. Iterate over each coin in coins:

   • Start with the first coin 1. For each amount from 1 to 5 (amount + 1), we add the number of combinations without using the coin (already stored in dp) to the number of ways to make the current amount using this coin.

   For coin = 1:
   dp = [1, 1+dp[0], 1+dp[1], 1+dp[2], 1+dp[3], 1+dp[4]]
   After iteration: dp = [1, 1, 1, 1, 1, 1]

   • For coin 2, we start at amount 2 and update the dp array similarly.

   For coin = 2:
   dp = [1, 1, 1+dp[0], 1+dp[1], 1+dp[2], 1+dp[3]]
   After iteration: dp = [1, 1, 2, 2, 3, 3]

   • Lastly, for 5, we start at amount 5, as it is the only value that can be updated by a coin of denomination 5.

   For coin = 5:
   dp = [1, 1, 2, 2, 3, 3+dp[0]]
   After iteration: dp = [1, 1, 2, 2, 3, 4]

4. Result: The final dp array represents the number of ways to make each amount up to 5. dp[5] is our answer because it's the number of combinations that make up the amount 5.

   dp = [1, 1, 2, 2, 3, 4]

So, there are 4 distinct combinations that can add up to 5 using the denominations [1, 2, 5]. These combinations are:

• 1 + 1 + 1 + 1 + 1 (five 1's)
• 1 + 1 + 1 + 2 (three 1's and one 2)
• 1 + 2 + 2 (one 1 and two 2's)
• 5 (one 5)

In Python, the final result would be obtained by returning the last element of the dp array:

return dp[-1]

Solution Implementation

Time and Space Complexity

The given code is a dynamic programming solution that calculates the number of ways to make up a certain amount using the given denominations of coins.

Time Complexity:

To determine the time complexity, we need to consider the two nested loops in the code:

• The outer loop runs once for each coin in the coins list. Let's assume the number of coins is n.
• The inner loop runs for each value from coin up to amount. In the worst case, this will run from 1 to amount, which we represent as m.

For each combination of a coin and an amount, the algorithm performs a constant amount of work by updating the dp array. Therefore, the total number of operations is proportional to the number of times the inner loop runs times the number of coins, which is O(n*m).

So, the time complexity is O(n*m) where n is the number of coins, and m is the amount.

Space Complexity:

The space complexity is determined by the size of the data structures used in the code. Here, there is one primary data structure, the dp array, which has a size of amount + 1. No other data structures depend on n or m.

Therefore, the space complexity is O(m), where m is the amount.

Learn more about how to find time and space complexity quickly using problem constraints.