560. Subarray Sum Equals K (Medium)

Brute force (TLE)

• We can find all subarrays and count subarrays with sum equal to k.
• TC: O(n^3)
• SC: O(1)

class Solution {
public:
    int subarraySum(vector<int>& nums, int k)
    {
        int n = nums.size();
        int cnt = 0;
        for (int i = 0; i < n; i++) {
            for (int j = i + 1; j <= n; j++) {
                int sm = 0;
                for (int k = i; k < j; k++)
                    sm += nums[k];
                if (sm == k)
                    cnt++;
            }
        }
        return cnt;
    }
};

prefix sum (TLE)

• Better Brute Force.
• In the 3rd loop we are finding the sum of the subarray.
• We can optimize 3rd loop by prefix sum.
• Store the prefix sum of array elements.
• While traversing array if(prefix[j] - prefix[i]==k) increment the count.
• return the count.
• TC: O(n^2)
• SC: O(n)

class Solution {
public:
    int subarraySum(vector<int>& nums, int k)
    {
        int n = nums.size();
        // prefix sum
        vector<int> prefix(n + 1, 0);
        for (int i = 0; i < n; i++)
            prefix[i + 1] = prefix[i] + nums[i];
        int ans = 0;
        for (int i = 0; i < n; i++) {
            for (int j = i + 1; j <= n; j++) {
                if (prefix[j] - prefix[i] == k)
                    ans++;
            }
        }
        return ans;
    }
};

Running sum (TLE)

• Instead of using prefix array to store the prefix sum, we can find the running sum.
• If sum=k increment the count.
• Return the count.
• TC: O(n^2)
• SC: O(1)

class Solution {
public:
    int subarraySum(vector<int>& nums, int k)
    {
        int n = nums.size();
        int ans = 0;
        for (int i = 0; i < n; i++) {
            int sum = 0;
            for (int j = i; j < n; j++) {
                sum += nums[j];
                if (sum == k)
                    ans++;
            }
        }
        return ans;
    }
};

Using hashmap (AC)

• Below explanation is from LeetCode Discuss.

• Consider the below example: array : 3 4 7 -2 2 1 4 2 presum : 3 7 14 12 14 15 19 21 index : 0 1 2 3 4 5 6 7 k = 7 Map should look like : (0,1) , (3,1), (7,1), (14,2) , (12,1) , (15,1) , (19,1) , (21,1)

• Consider 21(presum) now what we do is sum-k that is 21-7 = 14. Now we will check our map if we go by just count++ logic we will just increment the count once and here is where we go wrong.

• When we search for 14 in presum array we find it on 2 and 4 index. The logic here is that 14 + 7 = 21 so the array between indexes -> 3 to 7 (-2 2 1 4 2) -> 5 to 7 both have sum 7 ( 1 4 2) The sum is still 7 in this case because there are negative numbers that balance up for. So if we consider count++ we will have one count less as we will consider only array 5 to 7 but now we know that 14 sum occurred earlier too so even that needs to be added up so map.get(sum-k).

• Another way of understanding this is that if there is increase of k in the presum array we have found a new subarray so that is why we look for currentSum-k.
• TC: O(n)
• SC: O(n)

class Solution {
public:
    int subarraySum(vector<int>& nums, int k)
    {
        int n = nums.size();
        int sm = 0;
        int ans = 0;
        unordered_map<int, int> mp;
        mp[0] = 1; // getting 0 sum is always possible
        for (int i = 0; i < n; ++i) {
            sm += nums[i];
            if (mp.count(sm - k)) ans += mp[sm - k];
            mp[sm]++;
        }
        return ans;
    }
};