Array¶

Category 3 Subarray problems¶

Type of subarray problem¶

Fixed length subarray indexing

To correctly index an array in solving subarray problems are critical. Here is some tips: 1. To iterate through a subarray of certain size, alwasy using the "one-of-the-end" pattern. Namely, the iteration index i point to the "one-off-the-end" of the subarray. The subarray of size K before the index is started at index i - k. 2. The above convention is especially useful in subarray problems given constrains, such as "the subarray size greater than k", "maximum sum of non-overlapping subarray", etc. 3. Example problems - Maximum Average Subarray II - Maximum Sum of Two Non-Overlapping Subarrays

Two types of prefix sum¶

There are two ways to calculate the prefix sum array. Take which ever conveniece for your when solving a problem.

Option 1: sums.resize(n, 0);

nums: [1, 2, 3,  4,  5,  6,  7,  8,  9]
i               j
sums: [1, 3, 6, 10, 15, 21, 28, 36, 45]


In this case, each element sums[i] in sums represent the cumulative sum for indexes [0, ..., i]. In other words, sum[i] represent cumulative sum up to element i inclusive. When you want to get the range sum rangeSum(i, j), you can get it in the following way:

rangeSum(i, j) = sums[j] - sums[i - 1] // i > 0
rangeSum(i, j) = sums[j]               // i == 0


work with this option is a little complex, to get the rangeSum(i, j):

rangeSum(i, j) = sums[j] - sums[i] + nums[i] // i >= 0


Option 2: sums.resize(n + 1, 0);

In this case, each element sums[i] in sums represent the prefix sum of the first i elements in original array nums. When you want to get the range sum by rangeSum(i, j), you can get it in the following way:

rangeSum(i, j) = sums[j + 1] - sums[i] // i >= 0


Using prefix sum with map¶

One of the core trick in solving the following subarray problems is to build a map from prefix sum to array index for efficient lookup. For example, problems with keywaords "maximum size equal to K", "differ by K", or "differ by multiple of k" are solved using this trick. There are two hints.

Hint

When a map is used, it need to be initialized using <0, -1>. It is useful for handling some of the corner cases such as [-1, 1], -1 in the problem Maximum Size Subarray Sum Equals k.

Hint

It is usually easier to work with these problem when adding dummy element at the beginning of the array. For example: using sums[i] to represent the sum of first i element of array nums.

Maximum Subarray¶

This is a DP solution, it reduced the f array to two variables. Making the problem $O(1)$ in space. Discuss about this solution, where it make use of the idea of global maximum and local maximum.

Note

Why can not compare to f[i - 1] to find the maximum. If comparing to the f[i - 1], it will skip elements, the sum will not from a subarray, but sequence of numbers in the array. This is very similar to problems Longest Common Substring and Longest Common Subsequence. The DP "Choices" here is NOT to choose or not choose A[i], but "Add A[i] to the result of the subproblem or we have to start a subarray from i" because we cannot skip A[i].

Note

The idea of global maximum and local maximum is very useful to solve DP problems. The local maximum is the maximum sum of a continuous subarray, the global maximum is to keep the maximum of the from the local and global maximum.

Prefix sum solution

The ideas is we have array sums, sums[i] = A[0] +, ... + A[i], called prefix sum. With one for loop we can find the maxSum so far and the minSum before it. The difference is the possible results, we collect the maximum of those differences.

related to Jump Game

class Solution {
public:
int maxSubArray(vector<int>& nums) {
int n = nums.size();
int res = INT_MIN;

int f[n + 1] = {0}; // f[i] = maxSubArray of first i elements
f[0] = 0;           // initial value

for (int i = 1; i <= n; i++) {
f[i] = max(f[i - 1] + nums[i - 1], nums[i - 1]);
res = max(f[i], res);
}

return res;
}
};

class Solution {
public:
int maxSubArray(vector<int>& nums) {
int n = nums.size();
int res = INT_MIN;
int curr = 0;

for (int i = 0; i < n; i++) {
curr = max(curr + nums[i], nums[i]);
res = max(curr, res);
}

return res;
}
};

public class Solution {
public int maxSubArray(int[] A) {
if (A == null || A.length == 0){
return 0;
}

int max = Integer.MIN_VALUE, sum = 0, minSum = 0;
for (int i = 0; i < A.length; i++) {
sum += A[i];
max = Math.max(max, sum - minSum);
minSum = Math.min(minSum, sum);
}

return max;
}
};

// why this greedy solution works?
class Solution {
public:
int maxSubArray(vector<int>& nums) {
int n = nums.size();
int sum = 0;
int max = 0;
if (n == 0)
return 0;

max = nums[0];
for (int i = 0; i < n; i++) {
sum += nums[i];
max = sum > max ? sum : max;
sum = sum > 0 ? sum : 0;
}

return max;
}
};


Maximum Subarray II*¶

Given an array of integers, find two non-overlapping subarrays which have the
largest sum. The number in each subarray should be contiguous. Return the largest
sum.

Notice
The subarray should contain at least one number

Example
For given [1, 3, -1, 2, -1, 2], the two subarrays are [1, 3] and [2, -1, 2] or
[1, 3, -1, 2] and [2], they both have the largest sum 7.


Prefix sum solution

class Solution {
public:
/*
* @param nums: A list of integers
* @return: An integer denotes the sum of max two non-overlapping subarrays
*/
int maxTwoSubArrays(vector<int> nums) {
int n = nums.size();
int minSum = 0;
int sums = 0;
int maxSum = INT_MIN;
int left [n] = {0};
int right [n] = {0};

/* calculate the prefix sum */
for (int i = 0; i < n; i++) {
sums += nums[i];
maxSum = max(maxSum, sums - minSum); // minSum is previous calculated
minSum = min(minSum, sums);
left[i] = maxSum;
}

/* calculate the postfix sum */
minSum = 0;
sums = 0;
maxSum = INT_MIN;
for (int i = n - 1; i >= 0; i--) {
sums += nums[i];
maxSum = max(maxSum, sums - minSum);
minSum = min(minSum, sums);
right[i] = maxSum;
}

/* iterate the divider line, left[i] stored the maxSubArraySum
* from nums[0] to nums[i], similar for right[i] */
maxSum = INT_MIN;
for (int i = 0; i < n - 1; i++) {
maxSum = max(maxSum, left[i] + right[i + 1]);
}

return maxSum;
}
};


Warning

Cannot swap the highlighted lines. Because the maximum sum is calculated from current sum minus the previous minSum.

Maximum Subarray III*¶

Given an array of integers and a number k, find k non-overlapping subarrays which
have the largest sum. The number in each subarray should be contiguous.

Return the largest sum.

Notice
The subarray should contain at least one number

Example
Input:
List = [-1,4,-2,3,-2,3]
k = 2
Output: 8
Explanation: 4 + (3 + -2 + 3) = 8


DP solution

Use the idea of global maximum and local maximum from Maximum Subarray. See this artical for detailed explaination of the solution.

class Solution {
public:
/**
* My initial try: O(n^2 k)
* Partitioning DP: f[n][k], maximum K subarrays of first n elements.
* Last partition: A[j] ,... A[n - 1]
* f[i][k] = max_{0 <= j < i}(f[j][k - 1] + MS(A[j] ,... A[i - 1]))
* f[0][0] =
*
* Solution 2, O(nk)
* local[i][k]: Max k subarray sum from "first i elements" that include nums[i]
* global[i][k]: Max k subarray sum from "first i elements" that may not include nums[i]
*
* 2 cases:     nums[i - 1] is kth subarray, nums[i - 1] belongs to kth subarray
* local[i][k] = max(global[i - 1][k - 1], local[i - 1][k]) + nums[i - 1]
*
* 2 cases:     not include nums[i - 1], include nums[i - 1]
* global[i][k] = max(global[i - 1][k], local[i][k])
*/
int maxSubArray(vector<int> nums, int k) {
int n = nums.size();

int local[n + 1][k + 1] = {0};
int global[n + 1][k + 1] = {0};

for (int j = 1; j <= k; j++) {
// first j - 1 elements cannot form j groups, set to INT_MIN.
local[j - 1][j] = INT_MIN;
for (int i = j; i <= n; i++) { // must: i >= k.
local[i][j] = max(global[i - 1][j - 1], local[i - 1][j]) + nums[i - 1];
// the case when we divide k elements into k groups.
if (i == j) {
global[i][j] = local[i][j];
} else {
global[i][j] = max(global[i - 1][j], local[i][j]);
}
}
}

return global[n][k];
}
};


Maximum Subarray Difference*¶

Given an array with integers.
Find two non-overlapping subarrays A and B, which |SUM(A) - SUM(B)| is the largest.

Return the largest difference.

Notice
The subarray should contain at least one number

Example
For [1, 2, -3, 1], return 6.


Prefix sum solution

We use the similar idea for problem Maximum Subarray II. We have to maintain four arrays. from forward maximum and minimum subarray sum and backward maximum and minimum subarray sum.

class Solution {
public:
/*
* @param nums: A list of integers
* @return: value of maximum difference between two subarrays
*/
int maxDiffSubArrays(vector<int> nums) {
int n = nums.size();
int minSum = 0;
int sums = 0;
int maxSum = INT_MIN;

int left_max[n] = {0};
int left_min[n] = {0};
int right_max[n] = {0};
int right_min[n] = {0};

for (int i = 0; i < n; i++) {
sums += nums[i];
maxSum = max(maxSum, sums - minSum);
minSum = min(minSum, sums);
left_max[i] = maxSum;
//left_min[i] = minSum;
}

minSum = INT_MAX;
sums = 0;
maxSum = 0;
for (int i = 0; i < n; i++) {
sums += nums[i];
minSum = min(minSum, sums - maxSum);
maxSum = max(maxSum, sums);
//left_max[i] = maxSum;
left_min[i] = minSum;
}

minSum = 0;
sums = 0;
maxSum = INT_MIN;
for (int i = n - 1; i >= 0; i--) {
sums += nums[i];
maxSum = max(maxSum, sums - minSum);
minSum = min(minSum, sums);
right_max[i] = maxSum;
//right_min[i] = minSum;
}

minSum = INT_MAX;
sums = 0;
maxSum = 0;
for (int i = n - 1; i >= 0; i--) {
sums += nums[i];
minSum = min(minSum, sums - maxSum);
maxSum = max(maxSum, sums);
//right_max[i] = maxSum;
right_min[i] = minSum;
}

int diff = INT_MIN;
for (int i = 0; i < n - 1; i++) {
diff = max(left_max[i] - right_min[i + 1], diff);
diff = max(right_max[i + 1] - left_min[i], diff);
}

return diff;
}
};


Maximum Product Subarray¶

DP solution

• It is similar to the problem Maximum Subarray. Notice the negative number, min multiply a minus number could become the largest product.
class Solution {
public:
int maxProduct(vector<int>& nums) {
int n = nums.size();
int max_pro[n] = {0};
int min_pro[n] = {0};
int result = nums[0];

max_pro[0] = nums[0];
min_pro[0] = nums[0];

for (int i = 1; i < n; i++) {
if (nums[i] > 0) {
max_pro[i] = max(max_pro[i - 1] * nums[i], nums[i]);
min_pro[i] = min(min_pro[i - 1] * nums[i], nums[i]);
} else {
max_pro[i] = max(min_pro[i - 1] * nums[i], nums[i]);
min_pro[i] = min(max_pro[i - 1] * nums[i], nums[i]);
}

result = max(result, max_pro[i]);
}

return result;
}
};


Constant space solution

Without need to check whether nums[i] is positive is negative, we can just find the maximum or minium of three cases.

class Solution {
public:
/*
* @param nums: An array of integers
* @return: An integer
*/
int maxProduct(vector<int> nums) {
int n = nums.size();
int res = nums[0];
int cur_max = nums[0];
int cur_min = nums[0];

for (int i = 1; i < n; i++) {
int tmp = cur_max;
cur_max = max(max(cur_max * nums[i], nums[i]), cur_min * nums[i]);
cur_min = min(min(cur_min * nums[i], nums[i]), tmp * nums[i]);
res = max(res, cur_max);
}

return res;
}
};


Subarray Sum*¶

Given an integer array, find a subarray where the sum of numbers is zero.
Your code should return the index of the first number and the index of the last
number.

Notice
There is at least one subarray that it's sum equals to zero.

Example
Given [-3, 1, 2, -3, 4], return [0, 2] or [1, 3].


Hash solution

use a hash table to keep the prefix sum. Once we see another prefix sum that exists in the hash table, we discovered the subarray that sums to zero. However, pay attention to the indexing, because it requires to return the original array's index.

class Solution {
public:
/**
* @param nums: A list of integers
* @return: A list of integers includes the index of the first number
*          and the index of the last number
*/
vector<int> subarraySum(vector<int> nums){
int n = nums.size();
vector<int> res(2, 0);
int sum = 0;
unordered_map<int, int> map;

map[0] = -1; //important
for (int i = 0; i < n; i++) {
sum += nums[i];
if (map.count(sum) != 0) {
res[0] = map[sum] + 1;
res[1] = i;
break;
}

map[sum] = i;
}

return res;
}
};
// test cases
//    [-1, 2, 3, -3] A
//  [0,-1, 1, 4,  1] sum
//         i      j


Note

Pay attention to the initial value and initializethe map[0] = -1; This can be validated with an edge case. The time complexity is O(n)

Prefix sum solution

Calculate the prefix sum first and then use the prefix sum to find the subarray. This solution is $O(n^2)$

class Solution {
public:
vector<int> subarraySum(vector<int> nums){
int n = nums.size();
vector<int> res(2, 0);
vector<int> sum(n + 1, 0);
sum[0] = 0;

for (int i = 1; i <= n; i++) {
sum[i] = sum[i - 1] + nums[i - 1];
}

for (int i = 0; i < n; i++) {
for (int j = i; j <= n; j++) {
if (j > 1 && sum[j] - sum[i] == 0) {
res[0] = i;
res[1] = j - 1;
break;
}
}
}

return res;
}
};


Minimum Size Subarray Sum¶

Accumulative sum solution

Using accumulative sum and another moving pointer to check both the sum and the length of the subarray.

class Solution {
public:
int minSubArrayLen(int s, vector<int>& nums) {
int n = nums.size();
if (n == 0)
return 0;

int sum = 0;
int res = INT_MAX;
int left = 0;

for (int i = 0; i < n; i++) {
sum += nums[i];
while (sum >= s) {
res = min(res, i - left + 1);
sum -= nums[left++];
}
}

return res != INT_MAX ? res : 0;
}
};


Maximum Size Subarray Sum Equals k¶

Similar to Continuous Subarray Sum

Hash solution

Use a hash table to keep <sums, i> entries. Look up using sum - k. We only add to the hash table for the first time a value is appeared. It ensures the length of the found subarray is the largest. Notice you also have to initialize the hash with value <0, -1> to handle the edge case.

class Solution {
public:
int maxSubArrayLen(vector<int>& nums, int k) {
int n = nums.size();
if (n == 0)
return 0;

unordered_map<int, int> map;
int sum = 0;
int left = 0;
int res = INT_MIN;

map[0] = -1;
for (int i = 0; i < n; i++) {
sum += nums[i];
if (map.count(sum - k) != 0) {
left = map[sum - k] + 1;
res = max(res, (i - left + 1));
}

if (map.count(sum) == 0) {
map[sum] = i;
}
}

return res != INT_MIN ? res : 0;
}
};
/* test cases:
1. [-1, 1], -1 如果没有初始化hash，这个case就会错误
[-1, 0] sums
2. [-1], 0
3. [-1], -1
4. if return the result will be 1 if there is no res variable
[1, 1, 0], 1
[1, 2, 2], 1
*/


Subarray Sum Equals K¶

Prefix sum solution

Use prefix sum to find the subarray sum. Two pointer to check all the possible subarray sum.

public class Solution {
public int subarraySum(int[] nums, int k) {
int count = 0;
int[] sum = new int[nums.length + 1];
sum[0] = 0;

for (int i = 1; i <= nums.length; i++)
sum[i] = sum[i - 1] + nums[i - 1];

for (int start = 0; start < nums.length; start++) {
for (int end = start + 1; end <= nums.length; end++) {
if (sum[end] - sum[start] == k)
count++;
}
}
return count;
}
}


Hash solution

Use a map to store the prefix sum and a counter. The idea is while calculating prefix sums, if we find an sums - k exist in the map, we found one of target subarray. The subtilty is for a particular prefix sum, there might be multiple earlier prefix sums differ from it by k. We should take this into account. Compare to the hash solution for problem Subarray Sum.

/*
k =  2
i =  1,  2,  3
sum =  1,  2,  3
cnt =  0,  1,  2
key =  1,  2,  3
val =  1,  1,  1
the reason that the cnt += map[sum - k], not cnt += 1 is that
the prefix sum "sum - k" has been shown up for total of map[sum - k] times.
All those prefix sum could be result of distinct subarrays between current
prefix sum and previous prefix sum "sum - k"
*/

class Solution {
public:
int subarraySum(vector<int>& nums, int k) {
int n = nums.size();
// key=prefix sum, val=appearance
unordered_map<int, int> map;

int cnt = 0;
int sum = 0;

map[0] = 1;
for (int i = 0; i < n; i++) {
sum += nums[i];
if (map.count(sum - k) != 0) {
cnt += map[sum - k];
}

map[sum] += 1;
}

return cnt;
}
};


Warning

Notice you have to initialize the map[0] = 1; this is because for cases such as [1, 1, 1], when i = 1, sum = 2, [1,1] should be counted as one subarray. Without setting map[0] = 1 at first hand, it will give incorrect results.

Subarray Sums Divisible by K¶

• preSum is the basis of continuous subarray profblem.
• One pass solution should explore a property of modulo (preSum[j] - preSum[i]) % K == 0 indicate the when removing K from the larger value preSum[j] n times, we get the smaller value, then preSum[j] % K == preSum[i] % K. This property makes the one pass solution possible.
• Once the modulo property is found, we can use Hash map to assist our counting. The ideas is to keep counting the remainder, once we have seen the same remainder in the map, the new index and all the found indexes can be used to retrive one solution. The count keep in the map show how many of those can be.
class Solution {
public:
int subarraysDivByK(vector<int>& A, int K) {
int n = A.size();
if (n == 0 || K == 0) {
return 0;
}

int preSum = 0;
unordered_map<int, int> mp;
mp[0] = 1;
int count = 0;

for (int i = 0; i < n; i++) {
preSum += A[i];

int reminder = preSum % K;
// deal with negative values
if (reminder < 0)
reminder += K;

if (mp.find(reminder) != mp.end()) {
count += mp[reminder];
}

mp[reminder] += 1;
}

return count;
}
};


==="Naive solution O(n^2)"

class Solution {
public:
int subarraysDivByK(vector<int>& A, int K) {
int n = A.size();
if (n == 0 || K == 0) {
return 0;
}

vector<long> preSum(n + 1, 0);
preSum[0] = 0;
for (int i = 0; i < n; i++) {
preSum[i + 1] = preSum[i] + A[i];
}
int cnt = 0;
for (int i = 0; i < n; i++) {
for (int j = i + 1; j <= n; j++) {
if ((preSum[j] - preSum[i]) % K == 0) {
cnt++;
}
}
}

return cnt;
}
};


Max Sum of Subarry No Larger Than K*¶

1. This problem in geeksforgeeks as "Maximum sum subarray having sum less than or equal to given sum".
2. It has been discussed here.
3. This problem is the basis to solve the problem 363. Max Sum of Rectangle No Larger Than K.

Solution 1 using prefix sum and set

1. calculate prefix and using a set to store individual prefix sum, (vector also works). In each iteration, we lookup the value preSum - k in the set.
2. Notice we can use binary search to find the smallest element that >= preSum - k. We can use lower_bound to achieve that. Notice if it is asking the sum less than k we have to use upper_bound
int maxSumSubarryNoLargerThanK (int A[], int n, int k) {
set<int> preSumSet;
preSumSet.insert(0);

int res = 0, preSum = 0;
for (int i = 0; i < n; ++i) {
preSum += A[i];
set<int>::iterator siter = preSumSet.lower_bound(preSum - k);
if (siter != preSumSet.end() {
res = max(res, preSum - *siter);
}
preSumSet.insert(preSum);
}

return res;
}


Max Sum of Rectangle No Larger Than K¶

Solution 1 iterate the wide of the matrix and using prefix sum and set lower_bound.

1. To optimize it with the brute force solution, you will find this problem is a combination of the problem Maximum Sum Rectangular Submatrix in Matrix and problem Max Sum of Subarry No Larger Than K.
2. From the problem Max Sum of Subarry No Larger Than K, we have to enumerate the width of the sub-matrix and sum up all row elements and get an array of length m, m is the number of rows of the matrix. Then apply the method.
class Solution {
public:
int maxSumSubmatrix(vector<vector<int>>& matrix, int k) {
if (matrix.empty()) return 0;
int m = matrix.size();
int n = m ? matrix[0].size() : 0;
int res = INT_MIN;

for (int l = 0; l < n; ++l) {
vector<int> sums(m, 0);
for (int r = l; r < n; ++r) {
for (int i = 0; i < m; ++i) {
sums[i] += matrix[i][r];
}

set<int> preSumSet;
preSumSet.insert(0);
int preSum = 0, curMax = INT_MIN;
for (int sum : sums) {
preSum += sum;
set<int>::iterator it = preSumSet.lower_bound(preSum - k);
if (it != preSumSet.end()) {
curMax = max(curMax, preSum - *it);
}
preSumSet.insert(preSum);
}

res = max(res, curMax);
}
}

return res;
}
};


Note

1. The complexity is $n⋅n⋅(m+m\log m)=O(n⋅n⋅m\log m)$
2. Notice the use of lower_bound, this function return iterator point to element greater than or equal to the value curSum - k, if use upper_bound, it will return iterator points to element greater than curSum - k, which would miss the equal to K case.

Solution 2 using merge sort

1. The idea is similar that solution 1. Instead of calculate preSum on the fly, we finish calculation and pass it to a mergeSort routine.
2. The use mergeSort here is to find the A[j] - A[i] <= k efficiently, O(nlogn).
3. The complexity is n⋅n⋅(m+m⋅\log m)=O(n⋅n⋅m⋅\log m)
class Solution {
public:
int maxSumSubmatrix(vector<vector<int>>& matrix, int k) {
int m = matrix.size();
int n = m ? matrix[0].size() : 0;
int res = INT_MIN;
vector<long long> sums(m + 1, 0);

for (int l = 0; l < n; ++l) {
vector<long long>sumInRow(m, 0);
for (int r = l; r < n; ++r) {
for (int i = 0; i < m; ++i) {
sumInRow[i] += matrix[i][r];
sums[i + 1] = sums[i] + sumInRow[i];
}
res = max(res, mergeSort(sums, 0, m + 1, k));
if (res == k) return k;
}
}

return res;
}

int mergeSort(vector<long long>& sums, int start, int end, int k) {
if (end == start + 1) return INT_MIN;

int mid = start + (end - start) / 2;
int res = mergeSort(sums, start, mid, k);
if (res == k) return k;

res = max(res, mergeSort(sums, mid, end, k));
if (res == k) return k;

long long cache[end - start];

int j = mid, c = 0, t = mid;
for (int i = start; i < mid; ++i) {
/* search first time sums[j] - sums[i] > k */
while (j < end && sums[j] - sums[i] <= k) ++j;
/* sums[j - 1] - sums[i] <= k, make sure j - 1 is in right side */
if (j - 1 >= mid) {
res = max(res, (int)(sums[j - 1] - sums[i]));
if (res == k) return k;
}
/* parallel merge */
while (t < end && sums[t] < sums[i]) {
cache[c++] = sums[t++];
}
cache[c++] = sums[i];
}

/* parallel merge */
for (int i = start; i < t; ++i) {
sums[i] = cache[i - start];
}

return res;
}
};


Shortest Unsorted Continuous Subarray (Count inversions)¶

Solution 1 using merge sort

1. This problem is the basic of using merge sort to solve lots of hard problems.
long long merge_and_count(int A[], int start, int end) {
if (end - start <= 1) return 0;

long long count = 0;
int mid = start + (end - start) / 2;
count = merge_and_count(A, start, mid)
+ merge_and_count(A, mid, end);

int j = mid;
int cache[end - start];
for (int i = start, k = 0; i < mid; i++) {
while (j < end && A[i] > A[j]) {
cache[k++] = A[j++];
count += mid - i;
}

cache[k++] = A[i];
}
// copy cache[0, j - start) to A[start, j)
// if j < end, then A[j, end) in final place.
for (int i = start; i < j; i++) {
A[i] = cache[i - start];
}

return count;
}


Count Inversion (course assignment)¶

Solution 1 using merge sort

1. This problem is the basic of using merge sort to solve lots of hard problems.
long long merge_and_count(int A[], int start, int end) {
if (end - start <= 1) return 0;

long long count = 0;
int mid = start + (end - start) / 2;
count = merge_and_count(A, start, mid)
+ merge_and_count(A, mid, end);

int j = mid;
int cache[end - start];
for (int i = start, k = 0; i < mid; i++) {
while (j < end && A[i] > A[j]) {
cache[k++] = A[j++];
count += mid - i;
}

cache[k++] = A[i];
}
// copy cache[0, j - start) to A[start, j)
// if j < end, then A[j, end) in final place.
for (int i = start; i < j; i++) {
A[i] = cache[i - start];
}

return count;
}


Count of Smaller Numbers After Self¶

Solution 1 Merge sort

1. One important point to remember is you have to create pairs out of the array element and its index, because during merge sort, when we count each value, we don't know where to put those count values in the result vector.
2. The second merge solutions run much faster than the first one.
class Solution {
public:
vector<int> countSmaller(vector<int>& nums) {
int n = nums.size();
vector<int> res(n, 0);
vector<pair<int, int> > vp;

for (int i = 0; i < n; i++) {
vp.emplace_back(nums[i], i);
}

merge_sort_count(vp, 0, n, res);

return res;
}

private:
void merge_sort_count(vector<pair<int, int> >& nums, int start, int end,
vector<int>& res) {
if (end - start <= 1)
return;

int mid = start + (end - start) / 2;

merge_sort_count(nums, start, mid, res);
merge_sort_count(nums, mid, end, res);

vector<pair<int, int> > cache(end - start, pair<int, int>(0, 0));

int j = mid, k = 0, t = mid;
for (int i = start; i < mid; i++) {
j = mid;
while (j < end && nums[i].first > nums[j].first) { // found smaller elements
res[nums[i].second]++;
j++;
}

while (t < end && nums[i].first > nums[t].first) {
cache[k++] = nums[t++];
}

cache[k++] = nums[i];
}

for (int i = start; i < j; i++) {
nums[i] = cache[i - start];
}

return;
}
};

class Solution {
public:
vector<int> countSmaller(vector<int>& nums) {
int n = nums.size();
vector<int> res(n, 0);
vector<pair<int, int> > vp;

for (int i = 0; i < n; i++) {
vp.emplace_back(nums[i], i);
}

mergeSort(vp, 0, n, res);

return res;
}

void mergeSort(vector<pair<int, int>>& x, int start,
int end, vector<int>& res) {
if (end - start <= 1) return;

int mid = start + (end - start) / 2;
mergeSort(x, start, mid, res);
mergeSort(x, mid, end, res);

vector<pair<int, int> > cache(end - start, pair<int, int>(0, 0));

int i = start, j = mid, k = 0;
while(i < mid && j < end) {
if (x[i].first <= x[j].first) {
cache[k++] = x[i];
res[x[i].second] += j - mid;
++i;
} else {
cache[k++] = x[j++];
}
}

while(i < mid) {
cache[k++] = x[i];
res[x[i].second] += end - mid;
++i;
}

while(j < end)
cache[k++] = x[j++];

for(i = start, k = 0; i < end; ++i, ++k) {
x[i] = cache[k];
}
}
};

class Solution {
public:
class TreeNode {
public:
int val, smallerCnt;
TreeNode* left, *right;
TreeNode(int v, int s) : left(NULL), right(NULL), val(v), smallerCnt(s){}
};
vector<int> countSmaller(vector<int>& nums) {
int n = nums.size();
if(n == 0) return {};
vector<int> res(n, 0);
TreeNode* root = NULL;

for(int i = n - 1; i >= 0; --i)
root = insert(root, nums[i], i, 0, res);

return res;
}

private:
TreeNode* insert( TreeNode* node, int val, int idx, int preSum, vector<int>& res) {
if(node == NULL) {
node = new TreeNode(val, 0);
res[idx] = preSum;
} else if(node->val > val) {
node->smallerCnt++;
node->left = insert(node->left, val, idx, preSum, res);
} else {
node->right = insert(node->right, val, idx,
preSum + node->smallerCnt + ((node->val < val)? 1: 0), res);
}

return node;
}
};


Continuous Subarray Sum¶

Hash solution

Once see a multiple of K, you should consider the modulor operation %

The values put into to the hash only for the first time, this is similar to the case in the problem Maximum Size Subarray Sum Equals k.

class Solution {
public:
bool checkSubarraySum(vector<int>& nums, int k) {
int n = nums.size();
if (n == 0)
return false;

unordered_map<int, int> map;
int sum = 0;

map[0] = -1; // test case [0, 0], 0
for (int i = 0; i < n; i++) {
sum += nums[i];

if (k != 0)
sum = sum % k;

if (map.count(sum) != 0) {
if (i - map[sum] > 1) {
return true;
}
} else {
map[sum] = i;
}
}

return false;
}
};


Contiguous Array¶

Similar problems:

Hash solution

This problem is very similar to the problem Continuous Subarray Sum. However, there is a trick to calculate the cummulative sum, treat 0 as -1.

class Solution {
public:
int findMaxLength(vector<int>& nums) {
int n = nums.size();
int res = 0;
int cnt = 0;
unordered_map<int, int> map;

map[0] = -1; // test case: [0, 1]
for (int i = 0; i < n; i++) {
cnt += nums[i] == 0 ? -1 : 1;
if (map.count(cnt) != 0) {
res = max(res, i - map[cnt]);
} else {
map[cnt] = i;
}
}

return res;
}
};


Split Array with Equal Sum¶

Cummulative sum soluiton

Because of the symetric property of the head subarray and trailing subarray, we can calculate cumulative sum from both direction. This can help to fix the index i and k. we can enumerate the index j in between.

class Solution {
public:
bool splitArray(vector<int>& nums) {
int n = nums.size();
int sum1[n] = {0};
int sum2[n] = {0};

sum1[0] = nums[0];
for (int i = 1; i < n; i++) {
sum1[i] = sum1[i - 1] + nums[i];
}

sum2[n - 1] = nums[n - 1];
for (int i = n - 2; i >= 0; i--) {
sum2[i] = sum2[i + 1] + nums[i];
}

// notice the index bounds
for (int i = 1; i < n - 5; i++) {
for (int k = n - 2; k > i + 3; k--) {
if (sum1[i] - nums[i] == sum2[k] - nums[k]) {
for (int j = i + 2; j < k - 1; j++) {
int sumij = sum1[j] - nums[j] - sum1[i];
int sumjk = sum2[j] - nums[j] - sum2[k];
if (sumij == sumjk) {
return true;
}
}
}
}
}

return false;
}
};


410. Split Array Largest Sum¶

Similar problems:

DP solution

• Notice the edge case: [1, INT_MAX], use double can avoid integer overflow.

Binary Search solution

• This is a greedy search solution that use binary search to accelerate the search speed
• The goal is to "minimize the largest sub-array sum". It is different from Divide Chocolate, which is maximize the smallest sum.
• The bisection condition is not A[m] < target any more. It is a function to check whether the constrain can meet given a guess value mid.
/**
* equivalent to the lintcode copy books problem
*
* last step: mth subarray A[j], ..., A[i - 1].
* State: f[m][n]: minmax sum of m subarrays that include n elements
* Equation: f[m][n] = min_{0<=j<n}(max(f[m - 1][j], sum(A[j], ..., A[n - 1])))
* Init: f[0][n] = INT_MAX;
*       f[0][0] = 0;
* NB: notice a special case: [1, 2147483247], 2
*     the sum will overflow in the state update, You use a double type
*/
class Solution {
public:
int splitArray(vector<int>& nums, int m) {
int n = nums.size();

double f[m + 1][n + 1];
f[0][0] = 0;
for (int i = 1; i <= n; i++) {
f[0][i] = INT_MAX;
}

double sum = 0;
for (int k = 1; k <= m; k++) {
f[k][0] = 0;
for (int i = 1; i <= n; i++) {
sum = 0;
f[k][i] = INT_MAX;
for (int j = i; j >= 0; j--) { //j = i, mean sum = 0.
f[k][i] = min(f[k][i], max(f[k - 1][j], sum));
if (j > 0) {
sum += nums[j - 1];
}
}
}
}

return f[m][n];
}
};

class Solution {
public:
int splitArray(vector<int>& nums, int m) {
int total = 0;
int mx = 0;
for (int num: nums) {
total += num;
mx = max(mx, num);
}

int l = mx, r = total;
while (l < r) {
int mid = l + (r - l) / 2;

if (!canCut(nums, mid, m - 1)) {
l = mid + 1;
} else {
r = mid;
}
}

return l;
}

// whether m cuts are possible, notice the greedy property of this check
// you should notice that if not possible, it is because mid is too small,
// not because it is too large.
bool canCut(vector<int>& nums, int mid, int m) {
int sum = 0;
for (int num: nums) {
if (num > mid) return false;
else if (sum + num <= mid) sum += num; // each cut is greedy
else { // cut is ok so far
m--;
if (m < 0) return false; // more element after all cuts.

sum = num; // init the next group sum
}
}

return true;
}
};

class Solution {
public:
int splitArray(vector<int>& nums, int m) {
int l = *max_element(nums.begin(), nums.end());
int r = accumulate(nums.begin(), nums.end(), 0);

while (l < r) {
int mid = l + (r - l) / 2;
int s = 0;
int c = 0;

// count the possible cuts
for (int n : nums) {
if ((s += n) > mid) {
s = n;
if (++c > m - 1) {
break;
}
}
}

if (c > m - 1) {
l = mid + 1;
} else {
// the "> mid" above guarantee the "no greater than"
// the guess value, if c == m - 1, mid could be the result
r = mid;
}
}

return l;
}
};

class Solution {
public:
int splitArray(vector<int>& nums, int m) {
int l = *max_element(nums.begin(), nums.end());
int r = accumulate(nums.begin(), nums.end(), 0);

while (l < r) {
int mid = l + (r - l) / 2;
int s = 0;
int c = 1;

// count the possible cuts
for (int n : nums) {
if (s + n > mid) { // voilate the constrains
s = 0;
c++;
}
s += n;
}

if (c > m) {
l = mid + 1;
} else {
// the "> mid" above guarantee the "no greater than"
// the guess value, if c == m, mid could be the result
r = mid;
}
}

return l;
}
};


Copy books (lintcode)¶

Description

Given n books and the ith book has A[i] pages. You are given k people to copy the n books. the n books list in a row and each person can claim a continuous range of the n books. For example, one copier can copy the books from ith to jth continuously, but he can not copy the 1st book, 2nd book and 4th book (without the 3rd book).

They start copying books at the same time and they all cost 1 minute to copy 1 page of a book. What's the best strategy to assign books so that the slowest copier can finish at the earliest time?

Example

Given array A = [3,2,4], k = 2. Return 5 (First person spends 5 minutes to copy book 1 and book 2 and the second person spends 4 minutes to copy book 3.)

Solution 1 Binary search

See the solution for 410. Split Array Largest Sum

Solution 2 DP solution

There are i books, consider the last copier, he can copy A[j], ..., A[i-1]. The first k-1 copier copy A[0], ..., A[j - 1].

• Define state: f[k][i], meaning the k-th copier copy i books.
• State transition equation: $f[k][i] = \min_{0 \le j \le i} \max(f[k - 1][j], A[j] + ... + A[i - 1])$
class Solution {
public:
/**
* last step: last copier copy A[j], ... A[i-1]
* first k-1 copier --> A[0], ... A[j - 1].
* f[k][i]: k copier copy i books.
* f[k][i] = \min_{0 \le j \le i} \max(f[k - 1][j], A[j] + ... + A[i - 1])
*/
int copyBooks(vector<int> &pages, int K) {
int n = pages.size();
if (n == 0) {
return 0;
}

if (K > n) {
K = n;
}

int f[K + 1][n + 1];

/* init */
f[0][0] = 0;
for (int j = 1; j <= n; j++) {
f[0][j] = INT_MAX;
}

int sum = 0;
for (int k = 1; k <= K; k++) {
f[k][0] = 0;
for (int i = 1; i <= n; i++) {
sum = 0;
f[k][i] = INT_MAX;
for (int j = i; j >= 0; j--) {
f[k][i] = min(f[k][i], max(f[k - 1][j], sum));
if (j > 0) {
sum += pages[j - 1];
}
}
}
}

return f[K][n];
}
};


Note

We have to enumerate the index j, the highlighted code used a clever technique to optimize this task. It enumerate j backwards. While this seems impossible at the first glance, how can you calculate the states from right to left in DP? Notice the index j is in the upper row (row k-1). Once we are in the k-th row, the values in the k-1-th row are all given.

Maximum Average Subarray I¶

Prefix sum solution

class Solution {
public:
double findMaxAverage(vector<int>& nums, int k) {
int n = nums.size();
double sums[n] = {0};
double max_avg = INT_MIN;

sums[0] = nums[0];
for (int i = 1; i < n; i++) {
sums[i] = sums[i - 1] + nums[i];
}

for (int i = k - 1; i < n; i++) {
double avg = (sums[i] - sums[i - k + 1] + nums[i - k + 1]) / k;
max_avg = max(max_avg, avg);
}

return max_avg;
}
};


Maximum Average Subarray II¶

Prefix sum solution

This is still a brute force solution.

• time complexity: $O(n^2)$
• space complexity: $O(n)$

Space optimized solution

• We could avoid using the prefix sum array and only use two variables to record the prefix sum at any particular instance. One for record prefix sum of exact k elements. Another for the inner loop to check whether removing an element from the beginning will make a new maximum value or not.

Space optimized solution

• We could avoid using the prefix sum array and only use two variables to record the prefix sum at any particular instance. One for record prefix sum of exact k elements. Another for the inner loop to check whether removing an element from the beginning will make a new maximum value or not.

Deque solution

class Solution {
public:
double findMaxAverage(vector<int>& nums, int k) {
int n = nums.size();
vector<int> sums = nums;

for (int i = 1; i < n; ++i) {
sums[i] = sums[i - 1] + nums[i];
}

double res = (double) sums[k - 1] / k;
for (int i = k; i < n; ++i) {
double t = sums[i];
if (t > res * (i + 1)) res = t / (i + 1);
for (int j = 0; j < i - k + 1; ++j) {
t = sums[i] - sums[j];
if (t > res * (i - j)) res = t / (i - j);
}
}

return res;
}
};

class Solution {
public:
double findMaxAverage(vector<int>& nums, int k) {
int n = nums.size();
/* range is half open */
double sumsAll = accumulate(nums.begin(), nums.begin() + k, 0);
double sums = sumsAll, res = sumsAll / k;

for (int i = k; i < n; ++i) {
sumsAll += nums[i];
sums = sumsAll;
if (sums > res * (i + 1)) res = sums / (i + 1);
for (int j = 0; j < i - k + 1; ++j) {
sums -= nums[j];
if (sums > res * (i - j)) res = sums / (i - j);
}
}

return res;
}
};

class Solution {
public:
double findMaxAverage(vector<int>& nums, int k) {
int n = nums.size();
double upper = INT_MIN, lower = INT_MAX;

for (auto num : nums) {
upper = max(upper, (double)num);
lower = min(lower, (double)num);
}

while (lower + 0.00001 < upper) {
double mid = lower + (upper - lower) / 2;
if (isLarger(nums, mid, k)) { // is average value >= mid?
lower = mid;
} else {
upper = mid;
}
}

return lower;
}

/* return true if a greater average value is possible */
bool isLarger(vector<int>& nums, double mid, int k) {
int n = nums.size();
double sums = 0, prev = 0, prev_min = 0;
for (int i = 0; i < k; i++) {
sums += nums[i] - mid;
}

if (sums >= 0) {
return true;
}
/*
* we keep looking for whether a subarray sum of length >= k in array
* "sums" is possible to be greater than zero. If such a subarray exist,
* it means that the target average value is greater than the "mid"
* value. We look at the front part of sums that at least k element
* apart from i. If we can find the minimum of the sums[0, 1, ..., i - k]
* and check if sums[i] - min(sum[0, 1, ..., i - k]) >= 0. If this is the
* case, it indicate there exist a subarray of length >= k with sum
* greater than 0 in sums, we can return ture, otherwise, false.
*/
for (int i = k; i < n; i++) {
sums += nums[i] - mid;
prev += nums[i - k] - mid;
prev_min = min(prev_min, prev);
if (sums >= prev_min)
return true;
}

return false;
}
};

class Solution {
public:
double findMaxAverage(vector<int>& nums, int k) {
int n = nums.size();
vector<double> sums(n, 0);
deque<int> q;

sums[0] = nums[0];
for (int i = 1; i < n; ++i)
sums[i] = sums[i - 1] + nums[i];

double res = sums[n - 1] / n;
for (int j = k - 1; j < n; ++j)
{
while(q.size() >= 2 &&
density(sums, q[q.size() - 2], q.back() - 1) >=
density(sums, q.back(), j - k)) {

q.pop_back();
}

q.push_back(j - k + 1);

while(q.size() >= 2 &&
density(sums, q[0], j) <= density(sums, q[1], j)) {

q.pop_front();
}

res = max(res, density(sums, q.front(), j));
}

return res;
}

private:
double density(vector<double>& sums, int l, int r) {
if (l == 0)
return sums[r] / (r + 1);

return (sums[r] - sums[l - 1]) / (r - l + 1);
}
};


Note

Notice the initial value of prev_min is set to 0 not INT_MAX; Try to understand why set the initial value of prev_min to INT_MAX cannot pass the test case: [8,9,3,1,8,3,0,6,9,2], 8.

Range Sum Query - Immutable¶

Prefix sum solution

Use prefix sum to record the accumulative sum of the array in the constructor. The algorithm is $O(n)$ in space and $O(1)$ in time.

class NumArray {
private:
vector<int> sums;
int n;
public:
NumArray(vector<int> nums) {
n = nums.size();
sums.resize(n + 1, 0);

sums[0] = 0;
for (int i = 1; i <= n; ++i) {
sums[i] = sums[i - 1] + nums[i - 1];
}
}

int sumRange(int i, int j) {
return sums[j + 1] - sums[i];
}
};
/**
* Your NumArray object will be instantiated and called as such:
* NumArray obj = new NumArray(nums);
* int param_1 = obj.sumRange(i,j);
*/


Range Sum Query - Mutable¶

Segment tree solution

Using segment tree, the solution is given at Leetcode Solution.

class NumArray {
private:
vector<int> tree;
int n;
public:
NumArray(vector<int> nums) {
n = nums.size();
tree.resize(2 * n, 0);

for (int i = n, j = 0; i < 2 * n; ++i, ++j) {
tree[i] = nums[j];
}

for (int i = n - 1; i > 0; --i) {
tree[i] = tree[2 * i] + tree[2 * i + 1];
}
}

void update(int i, int val) {
int pos = n + i;
int left = 0;
int right = 0;

tree[pos] = val;
while (pos > 0) {
left = pos;
right = pos;
if (pos % 2 == 0) {
right = pos + 1;
}

if (pos % 2 == 1) {
left = pos - 1;
}

tree[pos / 2] = tree[left] + tree[right];
pos /= 2;
}
}

int sumRange(int i, int j) {
int left = i + n;
int right = j + n;
int sum = 0;

while (left <= right) {
if (left % 2 == 1) {
sum += tree[left];
left++;
}

if (right % 2 == 0) {
sum += tree[right];
right--;
}

left /= 2;
right /= 2;
}

return sum;
}
};

/**
* Your NumArray object will be instantiated and called as such:
* NumArray obj = new NumArray(nums);
* obj.update(i,val);
* int param_2 = obj.sumRange(i,j);
*/


Binary Indexed Tree solution I

Using Binary indexed tree, we are able to solve it optimally in $O(\log n)$ . The solution originally from here

class NumArray {
private:
vector<int> arr;
vector<int> BIT;
int n;
public:
NumArray(vector<int> nums) {
arr = nums;
n = nums.size();
BIT.resize(n + 1, 0);

for (int i = 0; i < n; i++) {
init(i, arr[i]);
}
}

void init(int i, int val) {
i++;
while(i <= n) {
BIT[i] += val; // BIT[i]  = nums[0, i - 1];
i += i & (-i);
}
}

void update(int i, int val) {
int diff = val - arr[i];
arr[i] = val;
init(i, diff);
}

int sumRange(int i, int j) {
return getSum(j) - getSum(i - 1);
}

int getSum(int i) {
i++;
int sum = 0;
while(i > 0) {
sum += BIT[i];
i -= i & (-i);
}

return sum;
}
};

/**
* Your NumArray object will be instantiated and called as such:
* NumArray obj = new NumArray(nums);
* obj.update(i,val);
* int param_2 = obj.sumRange(i,j);
*/


Binary Indexed Tree solution II

Similar to the above solution, We have combined the init and update. To make it consistant with the solution with problem Range Sum Query 2D - Mutable

class NumArray {
private:
vector<int> arr;
vector<int> BIT;
int n;
public:
NumArray(vector<int> nums) {
n = nums.size();
BIT.resize(n + 1, 0);
arr.resize(n, 0);

for (int i = 0; i < n; i++) {
update(i, nums[i]);
}
}

/* We can combine the init and update like this */
void update(int i, int val) {
int diff = val - arr[i];
arr[i] = val; // here we initialize arr[i]

i++;
while(i <= n) {
BIT[i] += diff; // BIT[i]  = nums[0, i - 1];
i += i & (-i);
}
}

int sumRange(int i, int j) {
return getSum(j) - getSum(i - 1);
}

int getSum(int i) {
i++;
int sum = 0;
while(i > 0) {
sum += BIT[i];
i -= i & (-i);
}

return sum;
}
};

/**
* Your NumArray object will be instantiated and called as such:
* NumArray obj = new NumArray(nums);
* obj.update(i,val);
* int param_2 = obj.sumRange(i,j);
*/


Range Sum Query 2D - Immutable¶

Prefix sum solution

Extended from the 1d array, we can use the prefix sum of the 2d matrix. we use extra space to store the accumulative sum of the submatrix with upper left coordinate (0, 0) and lower right coordinate (i, j).

class NumMatrix {
private:
vector<vector<int> > dp;
public:
NumMatrix(vector<vector<int>> matrix) {
int m = matrix.size();
if (m == 0) return;

int n = matrix[0].size();

//dp = vector<vector<int> (m + 1, vector<int>(n + 1, 0));
dp.resize(m + 1, vector<int>(n + 1, 0));

for (int i = 0; i < m; i++) {
for (int j = 0; j < n; j++) {
dp[i + 1][j + 1] = dp[i][j + 1] + dp[i + 1][j] + matrix[i][j] - dp[i][j];
}
}
}

int sumRegion(int row1, int col1, int row2, int col2) {
return dp[row2 + 1][col2 + 1] - dp[row2 + 1][col1] - dp[row1][col2 + 1] + dp[row1][col1];
}
};

/**
* Your NumMatrix object will be instantiated and called as such:
* NumMatrix obj = new NumMatrix(matrix);
* int param_1 = obj.sumRegion(row1,col1,row2,col2);
*/


Note

This version create the m + 1 by n + 1 dp array to record the prefix sum. The code is clean and elegant.

Alternative prefix sum solution

The idea is the same, in the following solution, we have a m by n 2d array to record the accumulative sum. See how complex the code is.

class NumMatrix {
private:
vector<vector<int>> dp;
int m;
int n;
public:
NumMatrix(vector<vector<int>> matrix) {
m = matrix.size();
if (m == 0)
return;
n = matrix[0].size();
dp.resize(m, vector<int>(n, 0));

for (int i = 0; i < m; i++) {
for (int j = 0; j < n; j++) {
dp[i][j] = matrix[i][j];
if (i > 0) {
dp[i][j] += dp[i - 1][j];
}

if (j > 0) {
dp[i][j] += dp[i][j - 1];
}

if (i > 0 && j > 0) {
dp[i][j] -= dp[i - 1][j - 1];
}
}
}
}

int sumRegion(int row1, int col1, int row2, int col2) {
int res = 0;
res = dp[row2][col2];

if (row1 > 0) {
res -= dp[row1 - 1][col2];
}

if (col1 > 0) {
res -= dp[row2][col1 - 1];
}

if (row1 > 0 && col1 > 0 ) {
res += dp[row1 - 1][col1 - 1];
}

return res;
}
};

/**
* Your NumMatrix object will be instantiated and called as such:
* NumMatrix obj = new NumMatrix(matrix);
* int param_1 = obj.sumRegion(row1,col1,row2,col2);
*/


Note

This version is a little complex. However, the way it was written reflects a very important practice when operating on a 2d array, that is: to check the validation of the array.

Range Sum Query 2D - Mutable¶

Binary Indexed Tree solution

We use 2D version of Binary Index Tree. Some of the explaination can be found at Topcoder tutorial

class NumMatrix {
private:
vector<vector<int> > nums;
vector<vector<int> > tree;
int m;
int n;

public:
NumMatrix(vector<vector<int>> matrix) {
if (matrix.size() == 0 || matrix[0].size() == 0)
return;
m = matrix.size();
n = matrix[0].size();

tree.resize(m + 1, vector<int>(n + 1, 0));
nums.resize(m, vector<int>(n, 0));

for (int i = 0; i < m; i++) {
for (int j = 0; j < n; j++) {
update(i, j, matrix[i][j]);
}
}
}

void update(int row, int col, int val) {
if (m == 0 || n == 0) return;
int diff = val - nums[row][col];
nums[row][col] = val;
for (int i = row + 1; i <= m; i += i & (-i)) {
for (int j = col + 1; j <= n; j += j & (-j)) {
tree[i][j] += diff;
}
}
}

int sumRegion(int row1, int col1, int row2, int col2) {
if (m == 0 || n == 0)
return 0;

return getSum(row2 + 1, col2 + 1) - getSum(row1, col2 + 1) - getSum(row2 + 1, col1) + getSum(row1, col1);
}

int getSum(int row, int col) {
int sum = 0;
for (int i = row; i > 0; i -= i & (-i)) {
for (int j = col; j > 0; j -= j & (-j)) {
sum += tree[i][j];
}
}

return sum;
}
};

/**
* Your NumMatrix object will be instantiated and called as such:
* NumMatrix obj = new NumMatrix(matrix);
* obj.update(row,col,val);
* int param_2 = obj.sumRegion(row1,col1,row2,col2);
*/


Count of Range Sum¶

Solution 1 Merge sort using inplace_merge()

1. The core is to figure out how to calculate the result while merging. It is based on the fact that the left half and right half are all sorted. Using the ordering information we are able to locate two points in the right half j and k, between which will fulfill the requirement.
2. Several important points need to be made. 1) calculation of prefix sum of the array. The length is n + 1 not n? 2) the range passed to the merge subroutine are open-end [start, end).
3. The base case of the subrouine. It return zero becuase the case has been counted in the for loop, we don't need to count it again. Not because the base case is 0.
class Solution {
public:
int countRangeSum(vector<int>& nums, int lower, int upper) {
int n = nums.size();
vector<long> sums(n + 1, 0);

for(int i = 0; i< n; i++)
sums[i + 1] = sums[i] + nums[i];

return mergeSort(sums, 0, n + 1, lower, upper);
}

int mergeSort(vector<long>& sums, int start, int end, int lower, int upper) {
if(end - start <= 1) return 0; // note the meaning of this 0.

int mid = start + (end - start) / 2;
count = mergeSort(sums, start, mid, lower, upper)
+ mergeSort(sums, mid, end, lower, upper);

int m = mid, n = mid, count = 0;
for(int i = start; i < mid; i++) {
while(m < end && sums[m] - sums[i] < lower) m++;
while(n < end && sums[n] - sums[i] <= upper) n++;
count += n - m;
}

inplace_merge(sums.begin() + start, sums.begin() + mid, sums.begin() + end);

return count;
}
};


Solution 2 Merge sort using tmp buffer cache

1. Here is how the count is making sense.
       |--------------|-------------------|
sums:  |start         |mid                |end
|---|----------|------|------|-----|
i                 j      k
Because sums[j] - sums[i] >= lower, and sums[k] - sums[i] > upper,
So for the subarray start with i, ending index in [j, k), the range
sum is in [lower, upper]. Notice k should not be included.

class Solution {
public:
int countRangeSum(vector<int>& nums, int lower, int upper) {
int n = nums.size();
long sums[n + 1] = {0};

if (n == 0)
return 0;

for (int i = 0; i < n; i++) {
sums[i + 1] = sums[i] + nums[i];
}

/* n + 1 is the one pass the last element of sums */
return countByMergeSort(sums, 0, n + 1, lower, upper);
}

/* This function will return sorted array sums[start], ... sums[end - 1] */
int countByMergeSort(long sums[], int start, int end, int lower, int upper) {
if (end - start <= 1) return 0;

int mid = start + (end - start) / 2;
int count = countByMergeSort(sums, start, mid, lower, upper)
+ countByMergeSort(sums, mid, end, lower, upper);

long cache[end - start] = {0};

int j = mid, k = mid, t = mid;
for (int i = start, r = 0; i < mid; ++i, ++r) {
while (k < end && sums[k] - sums[i] < lower) k++;
while (j < end && sums[j] - sums[i] <= upper) j++;

count += j - k; /* calculate the result */

/* Merge left and right to get sorted array {sums[start], .. sums[end - 1]}.
* Because left part of sums[start] to sums[mid] are already sorted,
* use cache here to merge prefix of the right part: sum[mid] to sums[t]
* with left part upto sums[i] for all i = {start, mid - 1}. */
while (t < end && sums[t] < sums[i])
cache[r++] = sums[t++];

cache[r] = sums[i];

} /* after this for loop, cache will have partially sorted array
* cache = sums_left = {sums[start], ... sums[t - 1]} element
* of which will be in their final sorted positions.
* array sums_right = {sums[t], sums[end - 1]} is also
* in their final sorted positions. */

/* Since the sums_left is sorted, it have size of t - start,
* here we copy exactly t - start element from cache to sums.
*/
for (int i = start; i < t; i++)
sums[i] = cache[i - start];

return count;
}
};


Solution 3 BST

Solution 4 BIT

Maximum Sum of Two Non-Overlapping Subarrays¶

class Solution {
public:
int maxSumTwoNoOverlap(vector<int>& A, int L, int M) {
int n = A.size();
if (L == 0 || M == 0) {
return 0;
}

vector<int> preSum(n, 0);
preSum[0] = A[0];
for (int i = 1; i < n; i++) {
preSum[i] = preSum[i - 1] + A[i];
}

int res = 0;
// iterate the L using index i
for (int i = 0; i < n - L + 1; i++) {
int Lsum = 0;
if (i == 0) {
Lsum = preSum[i + L - 1];
} else {
Lsum = preSum[i + L - 1] - preSum[i - 1];
}

int Msum = 0;
// iterate the left M array using index j
for (int j = 0; j < i - M; j++) {
int tmp = 0;
if (j == 0) {
tmp = preSum[j + M - 1];
} else {
tmp = preSum[j + M - 1] - preSum[j - 1];
}
Msum = max(Msum, tmp);
}
// iterate the right M array using index j
for (int j = i + L; j < n - M + 1; j++) {
Msum = max(Msum, preSum[j + M - 1] - preSum[j - 1]);
}

res = max(res, Msum + Lsum);
}

return res;
}
};

class Solution {
public:
int maxSumTwoNoOverlap(vector<int>& A, int L, int M) {
int n = A.size();
if (L == 0 || M == 0) {
return 0;
}

vector<int> preSum(n, 0);
preSum[0] = A[0];
for (int i = 1; i < n; i++) {
preSum[i] = preSum[i - 1] + A[i];
}

int res = INT_MIN;
int Lmax = INT_MIN;
int Mmax = INT_MIN;
//
for (int i = L + M; i <= n; i++) {
// L is front, M is back
if (i == L + M) {
Lmax = preSum[L - 1];
} else {
Lmax = max(Lmax, preSum[i - M - 1] - preSum[i - L - M - 1]);
}
// M is front, L is back
if (i == L + M) {
Mmax = preSum[M - 1];
} else {
Mmax = max(Mmax, preSum(i - L - 1) - preSum[i - M - L - 1]);
}

res = max({res,
Lmax + preSum[i - 1] - preSum[i - M - 1],
Mmax + preSum[i - 1] - preSum[i - L - 1]})
}
return res;
}
};

//TODO


Subarrays with K Different Integers¶

• If you use sliding window to solve this problem, there are lots of corner cases.
• for example: [1, 1, 2, 1, 2, 1, 3, 1], K = 2, how can you ensure you count all the subarray?
• Remember this trick that you can use the atMostKDistinct(A, K) - atMostKDistinct(A, K - 1).
class Solution {
public:
int subarraysWithKDistinct(vector<int>& A, int K) {
return subarrayWithAtMostKDistinct(A, K)
- subarrayWithAtMostKDistinct(A, K - 1);
}
int subarrayWithAtMostKDistinct(vector<int>& A, int K) {
unordered_map<int, int> count;

int i = 0;
int res = 0;
for (int j = 0; j < A.size(); ++j) {
if (!count[A[j]]++) K--;
while (k < 0) {
if (!--count[A[j]]) K++;
i++;
}
res += j - i + 1;
}

return res;
}
};


Category 4 K Sum problems¶

923 3Sum With Multiplicity¶

Solution 1: use a hash table to look up, similar to the classic 2 Sum problem Solution 2: use math trick. From the duplicate case, you realize they can be a nCk problem. 3 cases:

1. choose three same values i == j == k.
2. choose two same values i == j && j != k.
3. choose three distinct values i < j & j < k. notice here it can be others such as i < j & i < k, ..., at least three numbers are not the same.
class Solution:
def threeSumMulti(self, arr: List[int], target: int) -> int:
res = 0
MOD = 10**9 + 7
cnt = collections.Counter()

for i, m in enumerate(arr):
res = (res + cnt[target - m]) % MOD
for n in arr[:i]:
cnt[m + n] += 1

return res


== "Python $O(n * m)$"

class Solution:
def threeSumMulti(self, arr: List[int], target: int) -> int:
res = 0
MOD = 10**9 + 7
c = collections.Counter(arr)

for i, j in itertools.combinations_with_replacement(c, 2):
k = target - i - j
if i == j == k:
res += c[i] * (c[i] - 1) * (c[i] - 2) // 6
elif i == j != k:
res += c[i] * (c[i] - 1) // 2 * c[k]
elif i < k and j < k:
res += c[i] * c[j] * c[k]

return res % MOD


Cagegory 5 2D arry (matrix, grid) problems¶

Max Sum of Rectangle No Larger Than K¶

Most consistent ways of dealing with the series of stock problems

121. Best Time to Buy and Sell Stock¶

Solution 1 O(n) one pass to find the minimum and in the meantime, find the max profit.

class Solution {
public:
int maxProfit(vector<int> &prices) {
int n = prices.size();
if (n == 0) return 0;

int res = 0, low = prices[0];

for (int i = 1; i < n; i++) {
res = max(res, prices[i] - low);
low = min(low, prices[i]);
}

return res;
}
};


122. Best Time to Buy and Sell Stock II¶

Solution 1 Greedy since you can buy as many times as you can

class Solution {
public:
int maxProfit(vector<int> &prices) {
int n = prices.size();
int res = 0;

for (int i = 0; i < n - 1; i++) {
if (prices[i + 1] - prices[i]) {
res += prices[i + 1] - prices[i];
}
}

return res;
}
};


123. Best Time to Buy and Sell Stock III¶

You can now buy at most twice. how to max the profit.

Solution 1 Dynamic programming

• 5 stages: 1. before buy the first <-- optimal solution could be at this stage 2. hold the first 3. sell the first <-- or at this stage, only bought once, 4. hold the second 5. sell the second <-- or at this stage, bought twice.
class Solution {
public:
int maxProfit(vector<int> &A) {
//1, 3, 5:
// f[i][j] = max{f[i - 1][j], f[i - 1][j - 1] + A[i - 1] - A[i - 2]}
// 2, 4:
// f[i][j] = max{f[i - 1][j] + A[i - 1] - A[i - 2],
//               f[i - 1][j - 1], f[i - 1][j - 2] + A[i - 1] - A[i - 2]}
int n = A.size();
if (n == 0) {
return 0;
}

int f[n+1][6];

/* init */
f[0][1] = 0;
f[0][2] = f[0][3] = f[0][4] = f[0][5] = INT_MIN;

for (int i = 1; i <= n; i++) {
for (int j = 1; j <= 5; j += 2) {
f[i][j] = f[i - 1][j];
if (i > 1 && j > 1 && f[i - 1][j - 1] != INT_MIN) {
f[i][j] = max(f[i][j], f[i - 1][j - 1] + A[i - 1] - A[i - 2]);
}

}

for (int j = 2; j < 5; j += 2) {
f[i][j] = f[i - 1][j - 1];
if (i > 1 && f[i - 1][j] != INT_MIN) {
f[i][j] = max(f[i][j], f[i - 1][j] + A[i - 1] - A[i - 2]);
}

if (i > 1 && j > 2 && f[i - 1][j - 2] != INT_MIN) {
f[i][j] = max(f[i][j], f[i - 1][j - 2] + A[i - 1] - A[i - 2]);
}
}
}

return max(f[n][1], max(f[n][3], f[n][5]));
}
};


Solution 2

Use T[i][k][j] to represent the maximum profit of first i days if we allow at most k transactions and the current number of stocks at hand is j (j == 0, 1 because hold two stocks at the same time is not allowed). So we have:

T[i][2][0] = max(T[i - 1][2][0], T[i - 1][2][1] + prices[i - 1]);
T[i][2][1] = max(T[i - 1][2][1], T[i - 1][1][0] - prices[i - 1]);
T[i][1][0] = max(T[i - 1][1][0], T[i - 1][1][1] + prices[i - 1]);
T[i][1][1] = max(T[i - 1][1][1], T[i - 1][0][0] - prices[i - 1]);


Think: How to ensure you fourmular to cover all the possible values?