## Solution

#### Approach #1 (Brute Force) [Time Limit Exceeded]

Each time sumRange is called, we use a for loop to sum each individual element from index to .

private int[] data;

public NumArray(int[] nums) {
data = nums;
}

public int sumRange(int i, int j) {
int sum = 0;
for (int k = i; k <= j; k++) {
sum += data[k];
}
return sum;
}


Complexity analysis:

• Time complexity : time per query. Each sumRange query takes time.

• Space complexity : . Note that data is a reference to nums and is not a copy of it.

#### Approach #2 (Caching) [Accepted]

Imagine that sumRange is called one thousand times with the exact same arguments. How could we speed that up?

We could trade in extra space for speed. By pre-computing all range sum possibilities and store its results in a hash table, we can speed up the query to constant time.

private Map<Pair<Integer, Integer>, Integer> map = new HashMap<>();

public NumArray(int[] nums) {
for (int i = 0; i < nums.length; i++) {
int sum = 0;
for (int j = i; j < nums.length; j++) {
sum += nums[j];
map.put(Pair.create(i, j), sum);
}
}
}

public int sumRange(int i, int j) {
return map.get(Pair.create(i, j));
}


Complexity analysis

• Time complexity : time per query, time pre-computation. The pre-computation done in the constructor takes time. Each sumRange query's time complexity is as the hash table's look up operation is constant time.

• Space complexity : . The extra space required is as there are candidates for both and .

#### Approach #3 (Caching) [Accepted]

The above approach takes a lot of space, could we optimize it?

Imagine that we pre-compute the cummulative sum from index to . Could we use this information to derive ?

Let us define as the cumulative sum for (inclusive):

Now, we can calculate sumRange as following:

private int[] sum;

public NumArray(int[] nums) {
sum = new int[nums.length + 1];
for (int i = 0; i < nums.length; i++) {
sum[i + 1] = sum[i] + nums[i];
}
}

public int sumRange(int i, int j) {
return sum[j + 1] - sum[i];
}


Notice in the code above we inserted a dummy 0 as the first element in the sum array. This trick saves us from an extra conditional check in sumRange function.

Complexity analysis

• Time complexity : time per query, time pre-computation. Since the cumulative sum is cached, each sumRange query can be calculated in time.

• Space complexity : .