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This method involves using a 2D array to keep track of number of ways to make up a certain amount using coins up to a particular denomination. The rows in the array represent the number of coins available (up to that index), and the columns represent each possible amount up to the target.
Define dp[i][j] as the number of ways to make amount j using the first i types of coins. Initialize dp[i][0] to 1 since there is one way to make amount 0 (choosing no coin). For each coin, iterate over all amounts and accumulate the number of ways by including the current coin.
Time complexity is O(n*m) where n is the amount and m is the number of coins. Space complexity is also O(n*m).
Time Complexity: O(n*m), Space Complexity: O(n*m) where `n` is the `amount` and `m` is the number of coins.
1def change(amount, coins):
2 dp = [[0] * (amount + 1) for _ in range(len(coins) + 1)
In Python, the 2D list `dp` allows us to handle subproblem results for dynamic programming. We consider each coin and all amounts achievable with prior denominations, accumulating solutions from previously calculated states.
This method optimizes space usage by employing a 1D array to store results. It iterates over each coin and updates the number of ways for each possible amount.
Define dp[j] as the number of ways to make amount j. Initially, set dp[0] to 1 (to denote the empty set that sums up to zero). Iterate through each coin, then iterate over possible amounts in nested for-loops from current coin's value to amount, updating the dp array by adding ways from previous calculations.
Time complexity remains O(n*m), but space complexity improves to O(n).
Time Complexity: O(n*m), Space Complexity: O(n) where `n` is the `amount` and `m` is the number of coins.
The C representation here manages a 1D array `dp` after consolidating lesser used states, promoting space efficiency while preserving the calculated subproblems on a single line at every stage of iteration.