---
comments: true
---
# 14.5 Unbounded Knapsack Problem
In this section, we first solve another common knapsack problem: the unbounded knapsack, and then explore a special case of it: the coin change problem.
## 14.5.1 Unbounded Knapsack Problem
!!! question
Given $n$ items, where the weight of the $i$-th item is $wgt[i-1]$ and its value is $val[i-1]$, and a knapsack with capacity $cap$. **Each item can be selected multiple times**. What is the maximum value that can be placed in the knapsack within the capacity limit? An example is shown in Figure 14-22.
{ class="animation-figure" }
Figure 14-22 Example data for unbounded knapsack problem
### 1. Dynamic Programming Approach
The unbounded knapsack problem is very similar to the 0-1 knapsack problem, **differing only in that there is no limit on the number of times an item can be selected**.
- In the 0-1 knapsack problem, there is only one of each type of item, so after placing item $i$ in the knapsack, we can only choose from the first $i-1$ items.
- In the unbounded knapsack problem, the quantity of each type of item is unlimited, so after placing item $i$ in the knapsack, **we can still choose from the first $i$ items**.
Under the rules of the unbounded knapsack problem, the changes in state $[i, c]$ are divided into two cases.
- **Not putting item $i$**: Same as the 0-1 knapsack problem, transfer to $[i-1, c]$.
- **Putting item $i$**: Different from the 0-1 knapsack problem, transfer to $[i, c-wgt[i-1]]$.
Thus, the state transition equation becomes:
$$
dp[i, c] = \max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1])
$$
### 2. Code Implementation
Comparing the code for the two problems, there is one change in state transition from $i-1$ to $i$, with everything else identical:
=== "Python"
```python title="unbounded_knapsack.py"
def unbounded_knapsack_dp(wgt: list[int], val: list[int], cap: int) -> int:
"""Unbounded knapsack: Dynamic programming"""
n = len(wgt)
# Initialize dp table
dp = [[0] * (cap + 1) for _ in range(n + 1)]
# State transition
for i in range(1, n + 1):
for c in range(1, cap + 1):
if wgt[i - 1] > c:
# If exceeds knapsack capacity, don't select item i
dp[i][c] = dp[i - 1][c]
else:
# The larger value between not selecting and selecting item i
dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1])
return dp[n][cap]
```
=== "C++"
```cpp title="unbounded_knapsack.cpp"
/* Unbounded knapsack: Dynamic programming */
int unboundedKnapsackDP(vector &wgt, vector &val, int cap) {
int n = wgt.size();
// Initialize dp table
vector> dp(n + 1, vector(cap + 1, 0));
// State transition
for (int i = 1; i <= n; i++) {
for (int c = 1; c <= cap; c++) {
if (wgt[i - 1] > c) {
// If exceeds knapsack capacity, don't select item i
dp[i][c] = dp[i - 1][c];
} else {
// The larger value between not selecting and selecting item i
dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);
}
}
}
return dp[n][cap];
}
```
=== "Java"
```java title="unbounded_knapsack.java"
/* Unbounded knapsack: Dynamic programming */
int unboundedKnapsackDP(int[] wgt, int[] val, int cap) {
int n = wgt.length;
// Initialize dp table
int[][] dp = new int[n + 1][cap + 1];
// State transition
for (int i = 1; i <= n; i++) {
for (int c = 1; c <= cap; c++) {
if (wgt[i - 1] > c) {
// If exceeds knapsack capacity, don't select item i
dp[i][c] = dp[i - 1][c];
} else {
// The larger value between not selecting and selecting item i
dp[i][c] = Math.max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);
}
}
}
return dp[n][cap];
}
```
=== "C#"
```csharp title="unbounded_knapsack.cs"
/* Unbounded knapsack: Dynamic programming */
int UnboundedKnapsackDP(int[] wgt, int[] val, int cap) {
int n = wgt.Length;
// Initialize dp table
int[,] dp = new int[n + 1, cap + 1];
// State transition
for (int i = 1; i <= n; i++) {
for (int c = 1; c <= cap; c++) {
if (wgt[i - 1] > c) {
// If exceeds knapsack capacity, don't select item i
dp[i, c] = dp[i - 1, c];
} else {
// The larger value between not selecting and selecting item i
dp[i, c] = Math.Max(dp[i - 1, c], dp[i, c - wgt[i - 1]] + val[i - 1]);
}
}
}
return dp[n, cap];
}
```
=== "Go"
```go title="unbounded_knapsack.go"
/* Unbounded knapsack: Dynamic programming */
func unboundedKnapsackDP(wgt, val []int, cap int) int {
n := len(wgt)
// Initialize dp table
dp := make([][]int, n+1)
for i := 0; i <= n; i++ {
dp[i] = make([]int, cap+1)
}
// State transition
for i := 1; i <= n; i++ {
for c := 1; c <= cap; c++ {
if wgt[i-1] > c {
// If exceeds knapsack capacity, don't select item i
dp[i][c] = dp[i-1][c]
} else {
// The larger value between not selecting and selecting item i
dp[i][c] = int(math.Max(float64(dp[i-1][c]), float64(dp[i][c-wgt[i-1]]+val[i-1])))
}
}
}
return dp[n][cap]
}
```
=== "Swift"
```swift title="unbounded_knapsack.swift"
/* Unbounded knapsack: Dynamic programming */
func unboundedKnapsackDP(wgt: [Int], val: [Int], cap: Int) -> Int {
let n = wgt.count
// Initialize dp table
var dp = Array(repeating: Array(repeating: 0, count: cap + 1), count: n + 1)
// State transition
for i in 1 ... n {
for c in 1 ... cap {
if wgt[i - 1] > c {
// If exceeds knapsack capacity, don't select item i
dp[i][c] = dp[i - 1][c]
} else {
// The larger value between not selecting and selecting item i
dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1])
}
}
}
return dp[n][cap]
}
```
=== "JS"
```javascript title="unbounded_knapsack.js"
/* Unbounded knapsack: Dynamic programming */
function unboundedKnapsackDP(wgt, val, cap) {
const n = wgt.length;
// Initialize dp table
const dp = Array.from({ length: n + 1 }, () =>
Array.from({ length: cap + 1 }, () => 0)
);
// State transition
for (let i = 1; i <= n; i++) {
for (let c = 1; c <= cap; c++) {
if (wgt[i - 1] > c) {
// If exceeds knapsack capacity, don't select item i
dp[i][c] = dp[i - 1][c];
} else {
// The larger value between not selecting and selecting item i
dp[i][c] = Math.max(
dp[i - 1][c],
dp[i][c - wgt[i - 1]] + val[i - 1]
);
}
}
}
return dp[n][cap];
}
```
=== "TS"
```typescript title="unbounded_knapsack.ts"
/* Unbounded knapsack: Dynamic programming */
function unboundedKnapsackDP(
wgt: Array,
val: Array,
cap: number
): number {
const n = wgt.length;
// Initialize dp table
const dp = Array.from({ length: n + 1 }, () =>
Array.from({ length: cap + 1 }, () => 0)
);
// State transition
for (let i = 1; i <= n; i++) {
for (let c = 1; c <= cap; c++) {
if (wgt[i - 1] > c) {
// If exceeds knapsack capacity, don't select item i
dp[i][c] = dp[i - 1][c];
} else {
// The larger value between not selecting and selecting item i
dp[i][c] = Math.max(
dp[i - 1][c],
dp[i][c - wgt[i - 1]] + val[i - 1]
);
}
}
}
return dp[n][cap];
}
```
=== "Dart"
```dart title="unbounded_knapsack.dart"
/* Unbounded knapsack: Dynamic programming */
int unboundedKnapsackDP(List wgt, List val, int cap) {
int n = wgt.length;
// Initialize dp table
List> dp = List.generate(n + 1, (index) => List.filled(cap + 1, 0));
// State transition
for (int i = 1; i <= n; i++) {
for (int c = 1; c <= cap; c++) {
if (wgt[i - 1] > c) {
// If exceeds knapsack capacity, don't select item i
dp[i][c] = dp[i - 1][c];
} else {
// The larger value between not selecting and selecting item i
dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);
}
}
}
return dp[n][cap];
}
```
=== "Rust"
```rust title="unbounded_knapsack.rs"
/* Unbounded knapsack: Dynamic programming */
fn unbounded_knapsack_dp(wgt: &[i32], val: &[i32], cap: usize) -> i32 {
let n = wgt.len();
// Initialize dp table
let mut dp = vec![vec![0; cap + 1]; n + 1];
// State transition
for i in 1..=n {
for c in 1..=cap {
if wgt[i - 1] > c as i32 {
// If exceeds knapsack capacity, don't select item i
dp[i][c] = dp[i - 1][c];
} else {
// The larger value between not selecting and selecting item i
dp[i][c] = std::cmp::max(dp[i - 1][c], dp[i][c - wgt[i - 1] as usize] + val[i - 1]);
}
}
}
return dp[n][cap];
}
```
=== "C"
```c title="unbounded_knapsack.c"
/* Unbounded knapsack: Dynamic programming */
int unboundedKnapsackDP(int wgt[], int val[], int cap, int wgtSize) {
int n = wgtSize;
// Initialize dp table
int **dp = malloc((n + 1) * sizeof(int *));
for (int i = 0; i <= n; i++) {
dp[i] = calloc(cap + 1, sizeof(int));
}
// State transition
for (int i = 1; i <= n; i++) {
for (int c = 1; c <= cap; c++) {
if (wgt[i - 1] > c) {
// If exceeds knapsack capacity, don't select item i
dp[i][c] = dp[i - 1][c];
} else {
// The larger value between not selecting and selecting item i
dp[i][c] = myMax(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);
}
}
}
int res = dp[n][cap];
// Free memory
for (int i = 0; i <= n; i++) {
free(dp[i]);
}
return res;
}
```
=== "Kotlin"
```kotlin title="unbounded_knapsack.kt"
/* Unbounded knapsack: Dynamic programming */
fun unboundedKnapsackDP(wgt: IntArray, _val: IntArray, cap: Int): Int {
val n = wgt.size
// Initialize dp table
val dp = Array(n + 1) { IntArray(cap + 1) }
// State transition
for (i in 1..n) {
for (c in 1..cap) {
if (wgt[i - 1] > c) {
// If exceeds knapsack capacity, don't select item i
dp[i][c] = dp[i - 1][c]
} else {
// The larger value between not selecting and selecting item i
dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + _val[i - 1])
}
}
}
return dp[n][cap]
}
```
=== "Ruby"
```ruby title="unbounded_knapsack.rb"
### Unbounded knapsack: dynamic programming ###
def unbounded_knapsack_dp(wgt, val, cap)
n = wgt.length
# Initialize dp table
dp = Array.new(n + 1) { Array.new(cap + 1, 0) }
# State transition
for i in 1...(n + 1)
for c in 1...(cap + 1)
if wgt[i - 1] > c
# If exceeds knapsack capacity, don't select item i
dp[i][c] = dp[i - 1][c]
else
# The larger value between not selecting and selecting item i
dp[i][c] = [dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]].max
end
end
end
dp[n][cap]
end
```
### 3. Space Optimization
Since the current state is transferred from states on the left and above, **after space optimization, each row in the $dp$ table should be traversed in forward order**.
This traversal order is exactly opposite to the 0-1 knapsack. Please refer to Figure 14-23 to understand the difference between the two.
=== "<1>"
{ class="animation-figure" }
=== "<2>"
{ class="animation-figure" }
=== "<3>"
{ class="animation-figure" }
=== "<4>"
{ class="animation-figure" }
=== "<5>"
{ class="animation-figure" }
=== "<6>"
{ class="animation-figure" }
Figure 14-23 Space-optimized dynamic programming process for unbounded knapsack problem
The code implementation is relatively simple, just delete the first dimension of the array `dp`:
=== "Python"
```python title="unbounded_knapsack.py"
def unbounded_knapsack_dp_comp(wgt: list[int], val: list[int], cap: int) -> int:
"""Unbounded knapsack: Space-optimized dynamic programming"""
n = len(wgt)
# Initialize dp table
dp = [0] * (cap + 1)
# State transition
for i in range(1, n + 1):
# Traverse in forward order
for c in range(1, cap + 1):
if wgt[i - 1] > c:
# If exceeds knapsack capacity, don't select item i
dp[c] = dp[c]
else:
# The larger value between not selecting and selecting item i
dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])
return dp[cap]
```
=== "C++"
```cpp title="unbounded_knapsack.cpp"
/* Unbounded knapsack: Space-optimized dynamic programming */
int unboundedKnapsackDPComp(vector &wgt, vector &val, int cap) {
int n = wgt.size();
// Initialize dp table
vector dp(cap + 1, 0);
// State transition
for (int i = 1; i <= n; i++) {
for (int c = 1; c <= cap; c++) {
if (wgt[i - 1] > c) {
// If exceeds knapsack capacity, don't select item i
dp[c] = dp[c];
} else {
// The larger value between not selecting and selecting item i
dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);
}
}
}
return dp[cap];
}
```
=== "Java"
```java title="unbounded_knapsack.java"
/* Unbounded knapsack: Space-optimized dynamic programming */
int unboundedKnapsackDPComp(int[] wgt, int[] val, int cap) {
int n = wgt.length;
// Initialize dp table
int[] dp = new int[cap + 1];
// State transition
for (int i = 1; i <= n; i++) {
for (int c = 1; c <= cap; c++) {
if (wgt[i - 1] > c) {
// If exceeds knapsack capacity, don't select item i
dp[c] = dp[c];
} else {
// The larger value between not selecting and selecting item i
dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);
}
}
}
return dp[cap];
}
```
=== "C#"
```csharp title="unbounded_knapsack.cs"
/* Unbounded knapsack: Space-optimized dynamic programming */
int UnboundedKnapsackDPComp(int[] wgt, int[] val, int cap) {
int n = wgt.Length;
// Initialize dp table
int[] dp = new int[cap + 1];
// State transition
for (int i = 1; i <= n; i++) {
for (int c = 1; c <= cap; c++) {
if (wgt[i - 1] > c) {
// If exceeds knapsack capacity, don't select item i
dp[c] = dp[c];
} else {
// The larger value between not selecting and selecting item i
dp[c] = Math.Max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);
}
}
}
return dp[cap];
}
```
=== "Go"
```go title="unbounded_knapsack.go"
/* Unbounded knapsack: Space-optimized dynamic programming */
func unboundedKnapsackDPComp(wgt, val []int, cap int) int {
n := len(wgt)
// Initialize dp table
dp := make([]int, cap+1)
// State transition
for i := 1; i <= n; i++ {
for c := 1; c <= cap; c++ {
if wgt[i-1] > c {
// If exceeds knapsack capacity, don't select item i
dp[c] = dp[c]
} else {
// The larger value between not selecting and selecting item i
dp[c] = int(math.Max(float64(dp[c]), float64(dp[c-wgt[i-1]]+val[i-1])))
}
}
}
return dp[cap]
}
```
=== "Swift"
```swift title="unbounded_knapsack.swift"
/* Unbounded knapsack: Space-optimized dynamic programming */
func unboundedKnapsackDPComp(wgt: [Int], val: [Int], cap: Int) -> Int {
let n = wgt.count
// Initialize dp table
var dp = Array(repeating: 0, count: cap + 1)
// State transition
for i in 1 ... n {
for c in 1 ... cap {
if wgt[i - 1] > c {
// If exceeds knapsack capacity, don't select item i
dp[c] = dp[c]
} else {
// The larger value between not selecting and selecting item i
dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])
}
}
}
return dp[cap]
}
```
=== "JS"
```javascript title="unbounded_knapsack.js"
/* Unbounded knapsack: Space-optimized dynamic programming */
function unboundedKnapsackDPComp(wgt, val, cap) {
const n = wgt.length;
// Initialize dp table
const dp = Array.from({ length: cap + 1 }, () => 0);
// State transition
for (let i = 1; i <= n; i++) {
for (let c = 1; c <= cap; c++) {
if (wgt[i - 1] > c) {
// If exceeds knapsack capacity, don't select item i
dp[c] = dp[c];
} else {
// The larger value between not selecting and selecting item i
dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);
}
}
}
return dp[cap];
}
```
=== "TS"
```typescript title="unbounded_knapsack.ts"
/* Unbounded knapsack: Space-optimized dynamic programming */
function unboundedKnapsackDPComp(
wgt: Array,
val: Array,
cap: number
): number {
const n = wgt.length;
// Initialize dp table
const dp = Array.from({ length: cap + 1 }, () => 0);
// State transition
for (let i = 1; i <= n; i++) {
for (let c = 1; c <= cap; c++) {
if (wgt[i - 1] > c) {
// If exceeds knapsack capacity, don't select item i
dp[c] = dp[c];
} else {
// The larger value between not selecting and selecting item i
dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);
}
}
}
return dp[cap];
}
```
=== "Dart"
```dart title="unbounded_knapsack.dart"
/* Unbounded knapsack: Space-optimized dynamic programming */
int unboundedKnapsackDPComp(List wgt, List val, int cap) {
int n = wgt.length;
// Initialize dp table
List dp = List.filled(cap + 1, 0);
// State transition
for (int i = 1; i <= n; i++) {
for (int c = 1; c <= cap; c++) {
if (wgt[i - 1] > c) {
// If exceeds knapsack capacity, don't select item i
dp[c] = dp[c];
} else {
// The larger value between not selecting and selecting item i
dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);
}
}
}
return dp[cap];
}
```
=== "Rust"
```rust title="unbounded_knapsack.rs"
/* Unbounded knapsack: Space-optimized dynamic programming */
fn unbounded_knapsack_dp_comp(wgt: &[i32], val: &[i32], cap: usize) -> i32 {
let n = wgt.len();
// Initialize dp table
let mut dp = vec![0; cap + 1];
// State transition
for i in 1..=n {
for c in 1..=cap {
if wgt[i - 1] > c as i32 {
// If exceeds knapsack capacity, don't select item i
dp[c] = dp[c];
} else {
// The larger value between not selecting and selecting item i
dp[c] = std::cmp::max(dp[c], dp[c - wgt[i - 1] as usize] + val[i - 1]);
}
}
}
dp[cap]
}
```
=== "C"
```c title="unbounded_knapsack.c"
/* Unbounded knapsack: Space-optimized dynamic programming */
int unboundedKnapsackDPComp(int wgt[], int val[], int cap, int wgtSize) {
int n = wgtSize;
// Initialize dp table
int *dp = calloc(cap + 1, sizeof(int));
// State transition
for (int i = 1; i <= n; i++) {
for (int c = 1; c <= cap; c++) {
if (wgt[i - 1] > c) {
// If exceeds knapsack capacity, don't select item i
dp[c] = dp[c];
} else {
// The larger value between not selecting and selecting item i
dp[c] = myMax(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);
}
}
}
int res = dp[cap];
// Free memory
free(dp);
return res;
}
```
=== "Kotlin"
```kotlin title="unbounded_knapsack.kt"
/* Unbounded knapsack: Space-optimized dynamic programming */
fun unboundedKnapsackDPComp(
wgt: IntArray,
_val: IntArray,
cap: Int
): Int {
val n = wgt.size
// Initialize dp table
val dp = IntArray(cap + 1)
// State transition
for (i in 1..n) {
for (c in 1..cap) {
if (wgt[i - 1] > c) {
// If exceeds knapsack capacity, don't select item i
dp[c] = dp[c]
} else {
// The larger value between not selecting and selecting item i
dp[c] = max(dp[c], dp[c - wgt[i - 1]] + _val[i - 1])
}
}
}
return dp[cap]
}
```
=== "Ruby"
```ruby title="unbounded_knapsack.rb"
### Unbounded knapsack: space-optimized DP ###
def unbounded_knapsack_dp_comp(wgt, val, cap)
n = wgt.length
# Initialize dp table
dp = Array.new(cap + 1, 0)
# State transition
for i in 1...(n + 1)
# Traverse in forward order
for c in 1...(cap + 1)
if wgt[i -1] > c
# If exceeds knapsack capacity, don't select item i
dp[c] = dp[c]
else
# The larger value between not selecting and selecting item i
dp[c] = [dp[c], dp[c - wgt[i - 1]] + val[i - 1]].max
end
end
end
dp[cap]
end
```
## 14.5.2 Coin Change Problem
The knapsack problem represents a large class of dynamic programming problems and has many variants, such as the coin change problem.
!!! question
Given $n$ types of coins, where the denomination of the $i$-th type of coin is $coins[i - 1]$, and the target amount is $amt$. **Each type of coin can be selected multiple times**. What is the minimum number of coins needed to make up the target amount? If it is impossible to make up the target amount, return $-1$. An example is shown in Figure 14-24.
{ class="animation-figure" }
Figure 14-24 Example data for coin change problem
### 1. Dynamic Programming Approach
**The coin change problem can be viewed as a special case of the unbounded knapsack problem**, with the following connections and differences.
- The two problems can be converted to each other: "item" corresponds to "coin", "item weight" corresponds to "coin denomination", and "knapsack capacity" corresponds to "target amount".
- The optimization goals are opposite: the unbounded knapsack problem aims to maximize item value, while the coin change problem aims to minimize the number of coins.
- The unbounded knapsack problem seeks solutions "not exceeding" the knapsack capacity, while the coin change problem seeks solutions that "exactly" make up the target amount.
**Step 1: Think about the decisions in each round, define the state, and thus obtain the $dp$ table**
State $[i, a]$ corresponds to the subproblem: **the minimum number of coins among the first $i$ types of coins that can make up amount $a$**, denoted as $dp[i, a]$.
The two-dimensional $dp$ table has size $(n+1) \times (amt+1)$.
**Step 2: Identify the optimal substructure, and then derive the state transition equation**
This problem differs from the unbounded knapsack problem in the following two aspects regarding the state transition equation.
- This problem seeks the minimum value, so the operator $\max()$ needs to be changed to $\min()$.
- The optimization target is the number of coins rather than item value, so when a coin is selected, simply add $1$.
$$
dp[i, a] = \min(dp[i-1, a], dp[i, a - coins[i-1]] + 1)
$$
**Step 3: Determine boundary conditions and state transition order**
When the target amount is $0$, the minimum number of coins needed to make it up is $0$, so all $dp[i, 0]$ in the first column equal $0$.
When there are no coins, **it is impossible to make up any amount $> 0$**, which is an invalid solution. To enable the $\min()$ function in the state transition equation to identify and filter out invalid solutions, we consider using $+ \infty$ to represent them, i.e., set all $dp[0, a]$ in the first row to $+ \infty$.
### 2. Code Implementation
Most programming languages do not provide a $+ \infty$ variable, and can only use the maximum value of integer type `int` as a substitute. However, this can lead to integer overflow: the $+ 1$ operation in the state transition equation may cause overflow.
For this reason, we use the number $amt + 1$ to represent invalid solutions, because the maximum number of coins needed to make up $amt$ is at most $amt$. Before returning, check whether $dp[n, amt]$ equals $amt + 1$; if so, return $-1$, indicating that the target amount cannot be made up. The code is as follows:
=== "Python"
```python title="coin_change.py"
def coin_change_dp(coins: list[int], amt: int) -> int:
"""Coin change: Dynamic programming"""
n = len(coins)
MAX = amt + 1
# Initialize dp table
dp = [[0] * (amt + 1) for _ in range(n + 1)]
# State transition: first row and first column
for a in range(1, amt + 1):
dp[0][a] = MAX
# State transition: rest of the rows and columns
for i in range(1, n + 1):
for a in range(1, amt + 1):
if coins[i - 1] > a:
# If exceeds target amount, don't select coin i
dp[i][a] = dp[i - 1][a]
else:
# The smaller value between not selecting and selecting coin i
dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1)
return dp[n][amt] if dp[n][amt] != MAX else -1
```
=== "C++"
```cpp title="coin_change.cpp"
/* Coin change: Dynamic programming */
int coinChangeDP(vector &coins, int amt) {
int n = coins.size();
int MAX = amt + 1;
// Initialize dp table
vector> dp(n + 1, vector(amt + 1, 0));
// State transition: first row and first column
for (int a = 1; a <= amt; a++) {
dp[0][a] = MAX;
}
// State transition: rest of the rows and columns
for (int i = 1; i <= n; i++) {
for (int a = 1; a <= amt; a++) {
if (coins[i - 1] > a) {
// If exceeds target amount, don't select coin i
dp[i][a] = dp[i - 1][a];
} else {
// The smaller value between not selecting and selecting coin i
dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);
}
}
}
return dp[n][amt] != MAX ? dp[n][amt] : -1;
}
```
=== "Java"
```java title="coin_change.java"
/* Coin change: Dynamic programming */
int coinChangeDP(int[] coins, int amt) {
int n = coins.length;
int MAX = amt + 1;
// Initialize dp table
int[][] dp = new int[n + 1][amt + 1];
// State transition: first row and first column
for (int a = 1; a <= amt; a++) {
dp[0][a] = MAX;
}
// State transition: rest of the rows and columns
for (int i = 1; i <= n; i++) {
for (int a = 1; a <= amt; a++) {
if (coins[i - 1] > a) {
// If exceeds target amount, don't select coin i
dp[i][a] = dp[i - 1][a];
} else {
// The smaller value between not selecting and selecting coin i
dp[i][a] = Math.min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);
}
}
}
return dp[n][amt] != MAX ? dp[n][amt] : -1;
}
```
=== "C#"
```csharp title="coin_change.cs"
/* Coin change: Dynamic programming */
int CoinChangeDP(int[] coins, int amt) {
int n = coins.Length;
int MAX = amt + 1;
// Initialize dp table
int[,] dp = new int[n + 1, amt + 1];
// State transition: first row and first column
for (int a = 1; a <= amt; a++) {
dp[0, a] = MAX;
}
// State transition: rest of the rows and columns
for (int i = 1; i <= n; i++) {
for (int a = 1; a <= amt; a++) {
if (coins[i - 1] > a) {
// If exceeds target amount, don't select coin i
dp[i, a] = dp[i - 1, a];
} else {
// The smaller value between not selecting and selecting coin i
dp[i, a] = Math.Min(dp[i - 1, a], dp[i, a - coins[i - 1]] + 1);
}
}
}
return dp[n, amt] != MAX ? dp[n, amt] : -1;
}
```
=== "Go"
```go title="coin_change.go"
/* Coin change: Dynamic programming */
func coinChangeDP(coins []int, amt int) int {
n := len(coins)
max := amt + 1
// Initialize dp table
dp := make([][]int, n+1)
for i := 0; i <= n; i++ {
dp[i] = make([]int, amt+1)
}
// State transition: first row and first column
for a := 1; a <= amt; a++ {
dp[0][a] = max
}
// State transition: rest of the rows and columns
for i := 1; i <= n; i++ {
for a := 1; a <= amt; a++ {
if coins[i-1] > a {
// If exceeds target amount, don't select coin i
dp[i][a] = dp[i-1][a]
} else {
// The smaller value between not selecting and selecting coin i
dp[i][a] = int(math.Min(float64(dp[i-1][a]), float64(dp[i][a-coins[i-1]]+1)))
}
}
}
if dp[n][amt] != max {
return dp[n][amt]
}
return -1
}
```
=== "Swift"
```swift title="coin_change.swift"
/* Coin change: Dynamic programming */
func coinChangeDP(coins: [Int], amt: Int) -> Int {
let n = coins.count
let MAX = amt + 1
// Initialize dp table
var dp = Array(repeating: Array(repeating: 0, count: amt + 1), count: n + 1)
// State transition: first row and first column
for a in 1 ... amt {
dp[0][a] = MAX
}
// State transition: rest of the rows and columns
for i in 1 ... n {
for a in 1 ... amt {
if coins[i - 1] > a {
// If exceeds target amount, don't select coin i
dp[i][a] = dp[i - 1][a]
} else {
// The smaller value between not selecting and selecting coin i
dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1)
}
}
}
return dp[n][amt] != MAX ? dp[n][amt] : -1
}
```
=== "JS"
```javascript title="coin_change.js"
/* Coin change: Dynamic programming */
function coinChangeDP(coins, amt) {
const n = coins.length;
const MAX = amt + 1;
// Initialize dp table
const dp = Array.from({ length: n + 1 }, () =>
Array.from({ length: amt + 1 }, () => 0)
);
// State transition: first row and first column
for (let a = 1; a <= amt; a++) {
dp[0][a] = MAX;
}
// State transition: rest of the rows and columns
for (let i = 1; i <= n; i++) {
for (let a = 1; a <= amt; a++) {
if (coins[i - 1] > a) {
// If exceeds target amount, don't select coin i
dp[i][a] = dp[i - 1][a];
} else {
// The smaller value between not selecting and selecting coin i
dp[i][a] = Math.min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);
}
}
}
return dp[n][amt] !== MAX ? dp[n][amt] : -1;
}
```
=== "TS"
```typescript title="coin_change.ts"
/* Coin change: Dynamic programming */
function coinChangeDP(coins: Array, amt: number): number {
const n = coins.length;
const MAX = amt + 1;
// Initialize dp table
const dp = Array.from({ length: n + 1 }, () =>
Array.from({ length: amt + 1 }, () => 0)
);
// State transition: first row and first column
for (let a = 1; a <= amt; a++) {
dp[0][a] = MAX;
}
// State transition: rest of the rows and columns
for (let i = 1; i <= n; i++) {
for (let a = 1; a <= amt; a++) {
if (coins[i - 1] > a) {
// If exceeds target amount, don't select coin i
dp[i][a] = dp[i - 1][a];
} else {
// The smaller value between not selecting and selecting coin i
dp[i][a] = Math.min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);
}
}
}
return dp[n][amt] !== MAX ? dp[n][amt] : -1;
}
```
=== "Dart"
```dart title="coin_change.dart"
/* Coin change: Dynamic programming */
int coinChangeDP(List coins, int amt) {
int n = coins.length;
int MAX = amt + 1;
// Initialize dp table
List> dp = List.generate(n + 1, (index) => List.filled(amt + 1, 0));
// State transition: first row and first column
for (int a = 1; a <= amt; a++) {
dp[0][a] = MAX;
}
// State transition: rest of the rows and columns
for (int i = 1; i <= n; i++) {
for (int a = 1; a <= amt; a++) {
if (coins[i - 1] > a) {
// If exceeds target amount, don't select coin i
dp[i][a] = dp[i - 1][a];
} else {
// The smaller value between not selecting and selecting coin i
dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);
}
}
}
return dp[n][amt] != MAX ? dp[n][amt] : -1;
}
```
=== "Rust"
```rust title="coin_change.rs"
/* Coin change: Dynamic programming */
fn coin_change_dp(coins: &[i32], amt: usize) -> i32 {
let n = coins.len();
let max = amt + 1;
// Initialize dp table
let mut dp = vec![vec![0; amt + 1]; n + 1];
// State transition: first row and first column
for a in 1..=amt {
dp[0][a] = max;
}
// State transition: rest of the rows and columns
for i in 1..=n {
for a in 1..=amt {
if coins[i - 1] > a as i32 {
// If exceeds target amount, don't select coin i
dp[i][a] = dp[i - 1][a];
} else {
// The smaller value between not selecting and selecting coin i
dp[i][a] = std::cmp::min(dp[i - 1][a], dp[i][a - coins[i - 1] as usize] + 1);
}
}
}
if dp[n][amt] != max {
return dp[n][amt] as i32;
} else {
-1
}
}
```
=== "C"
```c title="coin_change.c"
/* Coin change: Dynamic programming */
int coinChangeDP(int coins[], int amt, int coinsSize) {
int n = coinsSize;
int MAX = amt + 1;
// Initialize dp table
int **dp = malloc((n + 1) * sizeof(int *));
for (int i = 0; i <= n; i++) {
dp[i] = calloc(amt + 1, sizeof(int));
}
// State transition: first row and first column
for (int a = 1; a <= amt; a++) {
dp[0][a] = MAX;
}
// State transition: rest of the rows and columns
for (int i = 1; i <= n; i++) {
for (int a = 1; a <= amt; a++) {
if (coins[i - 1] > a) {
// If exceeds target amount, don't select coin i
dp[i][a] = dp[i - 1][a];
} else {
// The smaller value between not selecting and selecting coin i
dp[i][a] = myMin(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);
}
}
}
int res = dp[n][amt] != MAX ? dp[n][amt] : -1;
// Free memory
for (int i = 0; i <= n; i++) {
free(dp[i]);
}
free(dp);
return res;
}
```
=== "Kotlin"
```kotlin title="coin_change.kt"
/* Coin change: Dynamic programming */
fun coinChangeDP(coins: IntArray, amt: Int): Int {
val n = coins.size
val MAX = amt + 1
// Initialize dp table
val dp = Array(n + 1) { IntArray(amt + 1) }
// State transition: first row and first column
for (a in 1..amt) {
dp[0][a] = MAX
}
// State transition: rest of the rows and columns
for (i in 1..n) {
for (a in 1..amt) {
if (coins[i - 1] > a) {
// If exceeds target amount, don't select coin i
dp[i][a] = dp[i - 1][a]
} else {
// The smaller value between not selecting and selecting coin i
dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1)
}
}
}
return if (dp[n][amt] != MAX) dp[n][amt] else -1
}
```
=== "Ruby"
```ruby title="coin_change.rb"
### Coin change: dynamic programming ###
def coin_change_dp(coins, amt)
n = coins.length
_MAX = amt + 1
# Initialize dp table
dp = Array.new(n + 1) { Array.new(amt + 1, 0) }
# State transition: first row and first column
(1...(amt + 1)).each { |a| dp[0][a] = _MAX }
# State transition: rest of the rows and columns
for i in 1...(n + 1)
for a in 1...(amt + 1)
if coins[i - 1] > a
# If exceeds target amount, don't select coin i
dp[i][a] = dp[i - 1][a]
else
# The smaller value between not selecting and selecting coin i
dp[i][a] = [dp[i - 1][a], dp[i][a - coins[i - 1]] + 1].min
end
end
end
dp[n][amt] != _MAX ? dp[n][amt] : -1
end
```
Figure 14-25 shows the dynamic programming process for coin change, which is very similar to the unbounded knapsack problem.
=== "<1>"
{ class="animation-figure" }
=== "<2>"
{ class="animation-figure" }
=== "<3>"
{ class="animation-figure" }
=== "<4>"
{ class="animation-figure" }
=== "<5>"
{ class="animation-figure" }
=== "<6>"
{ class="animation-figure" }
=== "<7>"
{ class="animation-figure" }
=== "<8>"
{ class="animation-figure" }
=== "<9>"
{ class="animation-figure" }
=== "<10>"
{ class="animation-figure" }
=== "<11>"
{ class="animation-figure" }
=== "<12>"
{ class="animation-figure" }
=== "<13>"
{ class="animation-figure" }
=== "<14>"
{ class="animation-figure" }
=== "<15>"
{ class="animation-figure" }
Figure 14-25 Dynamic programming process for coin change problem
### 3. Space Optimization
The space optimization for the coin change problem is handled in the same way as the unbounded knapsack problem:
=== "Python"
```python title="coin_change.py"
def coin_change_dp_comp(coins: list[int], amt: int) -> int:
"""Coin change: Space-optimized dynamic programming"""
n = len(coins)
MAX = amt + 1
# Initialize dp table
dp = [MAX] * (amt + 1)
dp[0] = 0
# State transition
for i in range(1, n + 1):
# Traverse in forward order
for a in range(1, amt + 1):
if coins[i - 1] > a:
# If exceeds target amount, don't select coin i
dp[a] = dp[a]
else:
# The smaller value between not selecting and selecting coin i
dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1)
return dp[amt] if dp[amt] != MAX else -1
```
=== "C++"
```cpp title="coin_change.cpp"
/* Coin change: Space-optimized dynamic programming */
int coinChangeDPComp(vector &coins, int amt) {
int n = coins.size();
int MAX = amt + 1;
// Initialize dp table
vector dp(amt + 1, MAX);
dp[0] = 0;
// State transition
for (int i = 1; i <= n; i++) {
for (int a = 1; a <= amt; a++) {
if (coins[i - 1] > a) {
// If exceeds target amount, don't select coin i
dp[a] = dp[a];
} else {
// The smaller value between not selecting and selecting coin i
dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1);
}
}
}
return dp[amt] != MAX ? dp[amt] : -1;
}
```
=== "Java"
```java title="coin_change.java"
/* Coin change: Space-optimized dynamic programming */
int coinChangeDPComp(int[] coins, int amt) {
int n = coins.length;
int MAX = amt + 1;
// Initialize dp table
int[] dp = new int[amt + 1];
Arrays.fill(dp, MAX);
dp[0] = 0;
// State transition
for (int i = 1; i <= n; i++) {
for (int a = 1; a <= amt; a++) {
if (coins[i - 1] > a) {
// If exceeds target amount, don't select coin i
dp[a] = dp[a];
} else {
// The smaller value between not selecting and selecting coin i
dp[a] = Math.min(dp[a], dp[a - coins[i - 1]] + 1);
}
}
}
return dp[amt] != MAX ? dp[amt] : -1;
}
```
=== "C#"
```csharp title="coin_change.cs"
/* Coin change: Space-optimized dynamic programming */
int CoinChangeDPComp(int[] coins, int amt) {
int n = coins.Length;
int MAX = amt + 1;
// Initialize dp table
int[] dp = new int[amt + 1];
Array.Fill(dp, MAX);
dp[0] = 0;
// State transition
for (int i = 1; i <= n; i++) {
for (int a = 1; a <= amt; a++) {
if (coins[i - 1] > a) {
// If exceeds target amount, don't select coin i
dp[a] = dp[a];
} else {
// The smaller value between not selecting and selecting coin i
dp[a] = Math.Min(dp[a], dp[a - coins[i - 1]] + 1);
}
}
}
return dp[amt] != MAX ? dp[amt] : -1;
}
```
=== "Go"
```go title="coin_change.go"
/* Coin change: Dynamic programming */
func coinChangeDPComp(coins []int, amt int) int {
n := len(coins)
max := amt + 1
// Initialize dp table
dp := make([]int, amt+1)
for i := 1; i <= amt; i++ {
dp[i] = max
}
// State transition
for i := 1; i <= n; i++ {
// Traverse in forward order
for a := 1; a <= amt; a++ {
if coins[i-1] > a {
// If exceeds target amount, don't select coin i
dp[a] = dp[a]
} else {
// The smaller value between not selecting and selecting coin i
dp[a] = int(math.Min(float64(dp[a]), float64(dp[a-coins[i-1]]+1)))
}
}
}
if dp[amt] != max {
return dp[amt]
}
return -1
}
```
=== "Swift"
```swift title="coin_change.swift"
/* Coin change: Space-optimized dynamic programming */
func coinChangeDPComp(coins: [Int], amt: Int) -> Int {
let n = coins.count
let MAX = amt + 1
// Initialize dp table
var dp = Array(repeating: MAX, count: amt + 1)
dp[0] = 0
// State transition
for i in 1 ... n {
for a in 1 ... amt {
if coins[i - 1] > a {
// If exceeds target amount, don't select coin i
dp[a] = dp[a]
} else {
// The smaller value between not selecting and selecting coin i
dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1)
}
}
}
return dp[amt] != MAX ? dp[amt] : -1
}
```
=== "JS"
```javascript title="coin_change.js"
/* Coin change: Space-optimized dynamic programming */
function coinChangeDPComp(coins, amt) {
const n = coins.length;
const MAX = amt + 1;
// Initialize dp table
const dp = Array.from({ length: amt + 1 }, () => MAX);
dp[0] = 0;
// State transition
for (let i = 1; i <= n; i++) {
for (let a = 1; a <= amt; a++) {
if (coins[i - 1] > a) {
// If exceeds target amount, don't select coin i
dp[a] = dp[a];
} else {
// The smaller value between not selecting and selecting coin i
dp[a] = Math.min(dp[a], dp[a - coins[i - 1]] + 1);
}
}
}
return dp[amt] !== MAX ? dp[amt] : -1;
}
```
=== "TS"
```typescript title="coin_change.ts"
/* Coin change: Space-optimized dynamic programming */
function coinChangeDPComp(coins: Array, amt: number): number {
const n = coins.length;
const MAX = amt + 1;
// Initialize dp table
const dp = Array.from({ length: amt + 1 }, () => MAX);
dp[0] = 0;
// State transition
for (let i = 1; i <= n; i++) {
for (let a = 1; a <= amt; a++) {
if (coins[i - 1] > a) {
// If exceeds target amount, don't select coin i
dp[a] = dp[a];
} else {
// The smaller value between not selecting and selecting coin i
dp[a] = Math.min(dp[a], dp[a - coins[i - 1]] + 1);
}
}
}
return dp[amt] !== MAX ? dp[amt] : -1;
}
```
=== "Dart"
```dart title="coin_change.dart"
/* Coin change: Space-optimized dynamic programming */
int coinChangeDPComp(List coins, int amt) {
int n = coins.length;
int MAX = amt + 1;
// Initialize dp table
List dp = List.filled(amt + 1, MAX);
dp[0] = 0;
// State transition
for (int i = 1; i <= n; i++) {
for (int a = 1; a <= amt; a++) {
if (coins[i - 1] > a) {
// If exceeds target amount, don't select coin i
dp[a] = dp[a];
} else {
// The smaller value between not selecting and selecting coin i
dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1);
}
}
}
return dp[amt] != MAX ? dp[amt] : -1;
}
```
=== "Rust"
```rust title="coin_change.rs"
/* Coin change: Space-optimized dynamic programming */
fn coin_change_dp_comp(coins: &[i32], amt: usize) -> i32 {
let n = coins.len();
let max = amt + 1;
// Initialize dp table
let mut dp = vec![0; amt + 1];
dp.fill(max);
dp[0] = 0;
// State transition
for i in 1..=n {
for a in 1..=amt {
if coins[i - 1] > a as i32 {
// If exceeds target amount, don't select coin i
dp[a] = dp[a];
} else {
// The smaller value between not selecting and selecting coin i
dp[a] = std::cmp::min(dp[a], dp[a - coins[i - 1] as usize] + 1);
}
}
}
if dp[amt] != max {
return dp[amt] as i32;
} else {
-1
}
}
```
=== "C"
```c title="coin_change.c"
/* Coin change: Space-optimized dynamic programming */
int coinChangeDPComp(int coins[], int amt, int coinsSize) {
int n = coinsSize;
int MAX = amt + 1;
// Initialize dp table
int *dp = malloc((amt + 1) * sizeof(int));
for (int j = 1; j <= amt; j++) {
dp[j] = MAX;
}
dp[0] = 0;
// State transition
for (int i = 1; i <= n; i++) {
for (int a = 1; a <= amt; a++) {
if (coins[i - 1] > a) {
// If exceeds target amount, don't select coin i
dp[a] = dp[a];
} else {
// The smaller value between not selecting and selecting coin i
dp[a] = myMin(dp[a], dp[a - coins[i - 1]] + 1);
}
}
}
int res = dp[amt] != MAX ? dp[amt] : -1;
// Free memory
free(dp);
return res;
}
```
=== "Kotlin"
```kotlin title="coin_change.kt"
/* Coin change: Space-optimized dynamic programming */
fun coinChangeDPComp(coins: IntArray, amt: Int): Int {
val n = coins.size
val MAX = amt + 1
// Initialize dp table
val dp = IntArray(amt + 1)
dp.fill(MAX)
dp[0] = 0
// State transition
for (i in 1..n) {
for (a in 1..amt) {
if (coins[i - 1] > a) {
// If exceeds target amount, don't select coin i
dp[a] = dp[a]
} else {
// The smaller value between not selecting and selecting coin i
dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1)
}
}
}
return if (dp[amt] != MAX) dp[amt] else -1
}
```
=== "Ruby"
```ruby title="coin_change.rb"
### Coin change: space-optimized DP ###
def coin_change_dp_comp(coins, amt)
n = coins.length
_MAX = amt + 1
# Initialize dp table
dp = Array.new(amt + 1, _MAX)
dp[0] = 0
# State transition
for i in 1...(n + 1)
# Traverse in forward order
for a in 1...(amt + 1)
if coins[i - 1] > a
# If exceeds target amount, don't select coin i
dp[a] = dp[a]
else
# The smaller value between not selecting and selecting coin i
dp[a] = [dp[a], dp[a - coins[i - 1]] + 1].min
end
end
end
dp[amt] != _MAX ? dp[amt] : -1
end
```
## 14.5.3 Coin Change Problem II
!!! question
Given $n$ types of coins, where the denomination of the $i$-th type of coin is $coins[i - 1]$, and the target amount is $amt$. Each type of coin can be selected multiple times. **What is the number of coin combinations that can make up the target amount?** An example is shown in Figure 14-26.
{ class="animation-figure" }
Figure 14-26 Example data for coin change problem II
### 1. Dynamic Programming Approach
Compared to the previous problem, this problem's goal is to find the number of combinations, so the subproblem becomes: **the number of combinations among the first $i$ types of coins that can make up amount $a$**. The $dp$ table remains a two-dimensional matrix of size $(n+1) \times (amt + 1)$.
The number of combinations for the current state equals the sum of the combinations from not selecting the current coin and selecting the current coin. The state transition equation is:
$$
dp[i, a] = dp[i-1, a] + dp[i, a - coins[i-1]]
$$
When the target amount is $0$, no coins need to be selected to make up the target amount, so all $dp[i, 0]$ in the first column should be initialized to $1$. When there are no coins, it is impossible to make up any amount $>0$, so all $dp[0, a]$ in the first row equal $0$.
### 2. Code Implementation
=== "Python"
```python title="coin_change_ii.py"
def coin_change_ii_dp(coins: list[int], amt: int) -> int:
"""Coin change II: Dynamic programming"""
n = len(coins)
# Initialize dp table
dp = [[0] * (amt + 1) for _ in range(n + 1)]
# Initialize first column
for i in range(n + 1):
dp[i][0] = 1
# State transition
for i in range(1, n + 1):
for a in range(1, amt + 1):
if coins[i - 1] > a:
# If exceeds target amount, don't select coin i
dp[i][a] = dp[i - 1][a]
else:
# Sum of the two options: not selecting and selecting coin i
dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]]
return dp[n][amt]
```
=== "C++"
```cpp title="coin_change_ii.cpp"
/* Coin change II: Dynamic programming */
int coinChangeIIDP(vector &coins, int amt) {
int n = coins.size();
// Initialize dp table
vector> dp(n + 1, vector(amt + 1, 0));
// Initialize first column
for (int i = 0; i <= n; i++) {
dp[i][0] = 1;
}
// State transition
for (int i = 1; i <= n; i++) {
for (int a = 1; a <= amt; a++) {
if (coins[i - 1] > a) {
// If exceeds target amount, don't select coin i
dp[i][a] = dp[i - 1][a];
} else {
// Sum of the two options: not selecting and selecting coin i
dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];
}
}
}
return dp[n][amt];
}
```
=== "Java"
```java title="coin_change_ii.java"
/* Coin change II: Dynamic programming */
int coinChangeIIDP(int[] coins, int amt) {
int n = coins.length;
// Initialize dp table
int[][] dp = new int[n + 1][amt + 1];
// Initialize first column
for (int i = 0; i <= n; i++) {
dp[i][0] = 1;
}
// State transition
for (int i = 1; i <= n; i++) {
for (int a = 1; a <= amt; a++) {
if (coins[i - 1] > a) {
// If exceeds target amount, don't select coin i
dp[i][a] = dp[i - 1][a];
} else {
// Sum of the two options: not selecting and selecting coin i
dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];
}
}
}
return dp[n][amt];
}
```
=== "C#"
```csharp title="coin_change_ii.cs"
/* Coin change II: Dynamic programming */
int CoinChangeIIDP(int[] coins, int amt) {
int n = coins.Length;
// Initialize dp table
int[,] dp = new int[n + 1, amt + 1];
// Initialize first column
for (int i = 0; i <= n; i++) {
dp[i, 0] = 1;
}
// State transition
for (int i = 1; i <= n; i++) {
for (int a = 1; a <= amt; a++) {
if (coins[i - 1] > a) {
// If exceeds target amount, don't select coin i
dp[i, a] = dp[i - 1, a];
} else {
// Sum of the two options: not selecting and selecting coin i
dp[i, a] = dp[i - 1, a] + dp[i, a - coins[i - 1]];
}
}
}
return dp[n, amt];
}
```
=== "Go"
```go title="coin_change_ii.go"
/* Coin change II: Dynamic programming */
func coinChangeIIDP(coins []int, amt int) int {
n := len(coins)
// Initialize dp table
dp := make([][]int, n+1)
for i := 0; i <= n; i++ {
dp[i] = make([]int, amt+1)
}
// Initialize first column
for i := 0; i <= n; i++ {
dp[i][0] = 1
}
// State transition: rest of the rows and columns
for i := 1; i <= n; i++ {
for a := 1; a <= amt; a++ {
if coins[i-1] > a {
// If exceeds target amount, don't select coin i
dp[i][a] = dp[i-1][a]
} else {
// Sum of the two options: not selecting and selecting coin i
dp[i][a] = dp[i-1][a] + dp[i][a-coins[i-1]]
}
}
}
return dp[n][amt]
}
```
=== "Swift"
```swift title="coin_change_ii.swift"
/* Coin change II: Dynamic programming */
func coinChangeIIDP(coins: [Int], amt: Int) -> Int {
let n = coins.count
// Initialize dp table
var dp = Array(repeating: Array(repeating: 0, count: amt + 1), count: n + 1)
// Initialize first column
for i in 0 ... n {
dp[i][0] = 1
}
// State transition
for i in 1 ... n {
for a in 1 ... amt {
if coins[i - 1] > a {
// If exceeds target amount, don't select coin i
dp[i][a] = dp[i - 1][a]
} else {
// Sum of the two options: not selecting and selecting coin i
dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]]
}
}
}
return dp[n][amt]
}
```
=== "JS"
```javascript title="coin_change_ii.js"
/* Coin change II: Dynamic programming */
function coinChangeIIDP(coins, amt) {
const n = coins.length;
// Initialize dp table
const dp = Array.from({ length: n + 1 }, () =>
Array.from({ length: amt + 1 }, () => 0)
);
// Initialize first column
for (let i = 0; i <= n; i++) {
dp[i][0] = 1;
}
// State transition
for (let i = 1; i <= n; i++) {
for (let a = 1; a <= amt; a++) {
if (coins[i - 1] > a) {
// If exceeds target amount, don't select coin i
dp[i][a] = dp[i - 1][a];
} else {
// Sum of the two options: not selecting and selecting coin i
dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];
}
}
}
return dp[n][amt];
}
```
=== "TS"
```typescript title="coin_change_ii.ts"
/* Coin change II: Dynamic programming */
function coinChangeIIDP(coins: Array, amt: number): number {
const n = coins.length;
// Initialize dp table
const dp = Array.from({ length: n + 1 }, () =>
Array.from({ length: amt + 1 }, () => 0)
);
// Initialize first column
for (let i = 0; i <= n; i++) {
dp[i][0] = 1;
}
// State transition
for (let i = 1; i <= n; i++) {
for (let a = 1; a <= amt; a++) {
if (coins[i - 1] > a) {
// If exceeds target amount, don't select coin i
dp[i][a] = dp[i - 1][a];
} else {
// Sum of the two options: not selecting and selecting coin i
dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];
}
}
}
return dp[n][amt];
}
```
=== "Dart"
```dart title="coin_change_ii.dart"
/* Coin change II: Dynamic programming */
int coinChangeIIDP(List coins, int amt) {
int n = coins.length;
// Initialize dp table
List> dp = List.generate(n + 1, (index) => List.filled(amt + 1, 0));
// Initialize first column
for (int i = 0; i <= n; i++) {
dp[i][0] = 1;
}
// State transition
for (int i = 1; i <= n; i++) {
for (int a = 1; a <= amt; a++) {
if (coins[i - 1] > a) {
// If exceeds target amount, don't select coin i
dp[i][a] = dp[i - 1][a];
} else {
// Sum of the two options: not selecting and selecting coin i
dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];
}
}
}
return dp[n][amt];
}
```
=== "Rust"
```rust title="coin_change_ii.rs"
/* Coin change II: Dynamic programming */
fn coin_change_ii_dp(coins: &[i32], amt: usize) -> i32 {
let n = coins.len();
// Initialize dp table
let mut dp = vec![vec![0; amt + 1]; n + 1];
// Initialize first column
for i in 0..=n {
dp[i][0] = 1;
}
// State transition
for i in 1..=n {
for a in 1..=amt {
if coins[i - 1] > a as i32 {
// If exceeds target amount, don't select coin i
dp[i][a] = dp[i - 1][a];
} else {
// Sum of the two options: not selecting and selecting coin i
dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1] as usize];
}
}
}
dp[n][amt]
}
```
=== "C"
```c title="coin_change_ii.c"
/* Coin change II: Dynamic programming */
int coinChangeIIDP(int coins[], int amt, int coinsSize) {
int n = coinsSize;
// Initialize dp table
int **dp = malloc((n + 1) * sizeof(int *));
for (int i = 0; i <= n; i++) {
dp[i] = calloc(amt + 1, sizeof(int));
}
// Initialize first column
for (int i = 0; i <= n; i++) {
dp[i][0] = 1;
}
// State transition
for (int i = 1; i <= n; i++) {
for (int a = 1; a <= amt; a++) {
if (coins[i - 1] > a) {
// If exceeds target amount, don't select coin i
dp[i][a] = dp[i - 1][a];
} else {
// Sum of the two options: not selecting and selecting coin i
dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];
}
}
}
int res = dp[n][amt];
// Free memory
for (int i = 0; i <= n; i++) {
free(dp[i]);
}
free(dp);
return res;
}
```
=== "Kotlin"
```kotlin title="coin_change_ii.kt"
/* Coin change II: Dynamic programming */
fun coinChangeIIDP(coins: IntArray, amt: Int): Int {
val n = coins.size
// Initialize dp table
val dp = Array(n + 1) { IntArray(amt + 1) }
// Initialize first column
for (i in 0..n) {
dp[i][0] = 1
}
// State transition
for (i in 1..n) {
for (a in 1..amt) {
if (coins[i - 1] > a) {
// If exceeds target amount, don't select coin i
dp[i][a] = dp[i - 1][a]
} else {
// Sum of the two options: not selecting and selecting coin i
dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]]
}
}
}
return dp[n][amt]
}
```
=== "Ruby"
```ruby title="coin_change_ii.rb"
### Coin change II: dynamic programming ###
def coin_change_ii_dp(coins, amt)
n = coins.length
# Initialize dp table
dp = Array.new(n + 1) { Array.new(amt + 1, 0) }
# Initialize first column
(0...(n + 1)).each { |i| dp[i][0] = 1 }
# State transition
for i in 1...(n + 1)
for a in 1...(amt + 1)
if coins[i - 1] > a
# If exceeds target amount, don't select coin i
dp[i][a] = dp[i - 1][a]
else
# Sum of the two options: not selecting and selecting coin i
dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]]
end
end
end
dp[n][amt]
end
```
### 3. Space Optimization
The space optimization is handled in the same way, just delete the coin dimension:
=== "Python"
```python title="coin_change_ii.py"
def coin_change_ii_dp_comp(coins: list[int], amt: int) -> int:
"""Coin change II: Space-optimized dynamic programming"""
n = len(coins)
# Initialize dp table
dp = [0] * (amt + 1)
dp[0] = 1
# State transition
for i in range(1, n + 1):
# Traverse in forward order
for a in range(1, amt + 1):
if coins[i - 1] > a:
# If exceeds target amount, don't select coin i
dp[a] = dp[a]
else:
# Sum of the two options: not selecting and selecting coin i
dp[a] = dp[a] + dp[a - coins[i - 1]]
return dp[amt]
```
=== "C++"
```cpp title="coin_change_ii.cpp"
/* Coin change II: Space-optimized dynamic programming */
int coinChangeIIDPComp(vector &coins, int amt) {
int n = coins.size();
// Initialize dp table
vector dp(amt + 1, 0);
dp[0] = 1;
// State transition
for (int i = 1; i <= n; i++) {
for (int a = 1; a <= amt; a++) {
if (coins[i - 1] > a) {
// If exceeds target amount, don't select coin i
dp[a] = dp[a];
} else {
// Sum of the two options: not selecting and selecting coin i
dp[a] = dp[a] + dp[a - coins[i - 1]];
}
}
}
return dp[amt];
}
```
=== "Java"
```java title="coin_change_ii.java"
/* Coin change II: Space-optimized dynamic programming */
int coinChangeIIDPComp(int[] coins, int amt) {
int n = coins.length;
// Initialize dp table
int[] dp = new int[amt + 1];
dp[0] = 1;
// State transition
for (int i = 1; i <= n; i++) {
for (int a = 1; a <= amt; a++) {
if (coins[i - 1] > a) {
// If exceeds target amount, don't select coin i
dp[a] = dp[a];
} else {
// Sum of the two options: not selecting and selecting coin i
dp[a] = dp[a] + dp[a - coins[i - 1]];
}
}
}
return dp[amt];
}
```
=== "C#"
```csharp title="coin_change_ii.cs"
/* Coin change II: Space-optimized dynamic programming */
int CoinChangeIIDPComp(int[] coins, int amt) {
int n = coins.Length;
// Initialize dp table
int[] dp = new int[amt + 1];
dp[0] = 1;
// State transition
for (int i = 1; i <= n; i++) {
for (int a = 1; a <= amt; a++) {
if (coins[i - 1] > a) {
// If exceeds target amount, don't select coin i
dp[a] = dp[a];
} else {
// Sum of the two options: not selecting and selecting coin i
dp[a] = dp[a] + dp[a - coins[i - 1]];
}
}
}
return dp[amt];
}
```
=== "Go"
```go title="coin_change_ii.go"
/* Coin change II: Space-optimized dynamic programming */
func coinChangeIIDPComp(coins []int, amt int) int {
n := len(coins)
// Initialize dp table
dp := make([]int, amt+1)
dp[0] = 1
// State transition
for i := 1; i <= n; i++ {
// Traverse in forward order
for a := 1; a <= amt; a++ {
if coins[i-1] > a {
// If exceeds target amount, don't select coin i
dp[a] = dp[a]
} else {
// Sum of the two options: not selecting and selecting coin i
dp[a] = dp[a] + dp[a-coins[i-1]]
}
}
}
return dp[amt]
}
```
=== "Swift"
```swift title="coin_change_ii.swift"
/* Coin change II: Space-optimized dynamic programming */
func coinChangeIIDPComp(coins: [Int], amt: Int) -> Int {
let n = coins.count
// Initialize dp table
var dp = Array(repeating: 0, count: amt + 1)
dp[0] = 1
// State transition
for i in 1 ... n {
for a in 1 ... amt {
if coins[i - 1] > a {
// If exceeds target amount, don't select coin i
dp[a] = dp[a]
} else {
// Sum of the two options: not selecting and selecting coin i
dp[a] = dp[a] + dp[a - coins[i - 1]]
}
}
}
return dp[amt]
}
```
=== "JS"
```javascript title="coin_change_ii.js"
/* Coin change II: Space-optimized dynamic programming */
function coinChangeIIDPComp(coins, amt) {
const n = coins.length;
// Initialize dp table
const dp = Array.from({ length: amt + 1 }, () => 0);
dp[0] = 1;
// State transition
for (let i = 1; i <= n; i++) {
for (let a = 1; a <= amt; a++) {
if (coins[i - 1] > a) {
// If exceeds target amount, don't select coin i
dp[a] = dp[a];
} else {
// Sum of the two options: not selecting and selecting coin i
dp[a] = dp[a] + dp[a - coins[i - 1]];
}
}
}
return dp[amt];
}
```
=== "TS"
```typescript title="coin_change_ii.ts"
/* Coin change II: Space-optimized dynamic programming */
function coinChangeIIDPComp(coins: Array, amt: number): number {
const n = coins.length;
// Initialize dp table
const dp = Array.from({ length: amt + 1 }, () => 0);
dp[0] = 1;
// State transition
for (let i = 1; i <= n; i++) {
for (let a = 1; a <= amt; a++) {
if (coins[i - 1] > a) {
// If exceeds target amount, don't select coin i
dp[a] = dp[a];
} else {
// Sum of the two options: not selecting and selecting coin i
dp[a] = dp[a] + dp[a - coins[i - 1]];
}
}
}
return dp[amt];
}
```
=== "Dart"
```dart title="coin_change_ii.dart"
/* Coin change II: Space-optimized dynamic programming */
int coinChangeIIDPComp(List coins, int amt) {
int n = coins.length;
// Initialize dp table
List dp = List.filled(amt + 1, 0);
dp[0] = 1;
// State transition
for (int i = 1; i <= n; i++) {
for (int a = 1; a <= amt; a++) {
if (coins[i - 1] > a) {
// If exceeds target amount, don't select coin i
dp[a] = dp[a];
} else {
// Sum of the two options: not selecting and selecting coin i
dp[a] = dp[a] + dp[a - coins[i - 1]];
}
}
}
return dp[amt];
}
```
=== "Rust"
```rust title="coin_change_ii.rs"
/* Coin change II: Space-optimized dynamic programming */
fn coin_change_ii_dp_comp(coins: &[i32], amt: usize) -> i32 {
let n = coins.len();
// Initialize dp table
let mut dp = vec![0; amt + 1];
dp[0] = 1;
// State transition
for i in 1..=n {
for a in 1..=amt {
if coins[i - 1] > a as i32 {
// If exceeds target amount, don't select coin i
dp[a] = dp[a];
} else {
// Sum of the two options: not selecting and selecting coin i
dp[a] = dp[a] + dp[a - coins[i - 1] as usize];
}
}
}
dp[amt]
}
```
=== "C"
```c title="coin_change_ii.c"
/* Coin change II: Space-optimized dynamic programming */
int coinChangeIIDPComp(int coins[], int amt, int coinsSize) {
int n = coinsSize;
// Initialize dp table
int *dp = calloc(amt + 1, sizeof(int));
dp[0] = 1;
// State transition
for (int i = 1; i <= n; i++) {
for (int a = 1; a <= amt; a++) {
if (coins[i - 1] > a) {
// If exceeds target amount, don't select coin i
dp[a] = dp[a];
} else {
// Sum of the two options: not selecting and selecting coin i
dp[a] = dp[a] + dp[a - coins[i - 1]];
}
}
}
int res = dp[amt];
// Free memory
free(dp);
return res;
}
```
=== "Kotlin"
```kotlin title="coin_change_ii.kt"
/* Coin change II: Space-optimized dynamic programming */
fun coinChangeIIDPComp(coins: IntArray, amt: Int): Int {
val n = coins.size
// Initialize dp table
val dp = IntArray(amt + 1)
dp[0] = 1
// State transition
for (i in 1..n) {
for (a in 1..amt) {
if (coins[i - 1] > a) {
// If exceeds target amount, don't select coin i
dp[a] = dp[a]
} else {
// Sum of the two options: not selecting and selecting coin i
dp[a] = dp[a] + dp[a - coins[i - 1]]
}
}
}
return dp[amt]
}
```
=== "Ruby"
```ruby title="coin_change_ii.rb"
### Coin change II: space-optimized DP ###
def coin_change_ii_dp_comp(coins, amt)
n = coins.length
# Initialize dp table
dp = Array.new(amt + 1, 0)
dp[0] = 1
# State transition
for i in 1...(n + 1)
# Traverse in forward order
for a in 1...(amt + 1)
if coins[i - 1] > a
# If exceeds target amount, don't select coin i
dp[a] = dp[a]
else
# Sum of the two options: not selecting and selecting coin i
dp[a] = dp[a] + dp[a - coins[i - 1]]
end
end
end
dp[amt]
end
```