Given a 2D matrix matrix, handle multiple queries of the following type:
Calculate the sum of the elements of matrix inside the rectangle defined by its upper left corner(row1, col1) and lower right corner(row2, col2).
Implement the NumMatrix class:
NumMatrix(int[][] matrix) Initializes the object with the integer matrix matrix.
int sumRegion(int row1, int col1, int row2, int col2) Returns the sum of the elements of matrix inside the rectangle defined by its upper left corner(row1, col1) and lower right corner(row2, col2).
You must design an algorithm where sumRegion works on O(1) time complexity.
Example 1:
Input
["NumMatrix", "sumRegion", "sumRegion", "sumRegion"]
[[[[3, 0, 1, 4, 2], [5, 6, 3, 2, 1], [1, 2, 0, 1, 5], [4, 1, 0, 1, 7], [1, 0, 3, 0, 5]]], [2, 1, 4, 3], [1, 1, 2, 2], [1, 2, 2, 4]]
Output
[null, 8, 11, 12]
Explanation
NumMatrix numMatrix = new NumMatrix([[3, 0, 1, 4, 2], [5, 6, 3, 2, 1], [1, 2, 0, 1, 5], [4, 1, 0, 1, 7], [1, 0, 3, 0, 5]]);
numMatrix.sumRegion(2, 1, 4, 3); // return 8 (i.e sum of the red rectangle)
numMatrix.sumRegion(1, 1, 2, 2); // return 11 (i.e sum of the green rectangle)
numMatrix.sumRegion(1, 2, 2, 4); // return 12 (i.e sum of the blue rectangle)
Constraints:
m == matrix.length
n == matrix[i].length
1 <= m, n <= 200
-104 <= matrix[i][j] <= 104
0 <= row1 <= row2 < m
0 <= col1 <= col2 < n
At most 104 calls will be made to sumRegion.
Solutions
Solution 1: Two-dimensional Prefix Sum
We use $s[i + 1][j + 1]$ to represent the sum of all elements in the upper left part of the $i$th row and $j$th column, where indices $i$ and $j$ both start from $0$. We can get the following prefix sum formula:
In the initialization method, we preprocess the prefix sum array $s$, and in the query method, we directly return the result of the above formula.
The time complexity for initializing is $O(m \times n)$, and the time complexity for querying is $O(1)$. The space complexity is $O(m \times n)$.
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classNumMatrix:def__init__(self,matrix:List[List[int]]):m,n=len(matrix),len(matrix[0])self.s=[[0]*(n+1)for_inrange(m+1)]fori,rowinenumerate(matrix):forj,vinenumerate(row):self.s[i+1][j+1]=(self.s[i][j+1]+self.s[i+1][j]-self.s[i][j]+v)defsumRegion(self,row1:int,col1:int,row2:int,col2:int)->int:return(self.s[row2+1][col2+1]-self.s[row2+1][col1]-self.s[row1][col2+1]+self.s[row1][col1])# Your NumMatrix object will be instantiated and called as such:# obj = NumMatrix(matrix)# param_1 = obj.sumRegion(row1,col1,row2,col2)
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classNumMatrix{privateint[][]s;publicNumMatrix(int[][]matrix){intm=matrix.length,n=matrix[0].length;s=newint[m+1][n+1];for(inti=0;i<m;++i){for(intj=0;j<n;++j){s[i+1][j+1]=s[i+1][j]+s[i][j+1]-s[i][j]+matrix[i][j];}}}publicintsumRegion(introw1,intcol1,introw2,intcol2){returns[row2+1][col2+1]-s[row2+1][col1]-s[row1][col2+1]+s[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); */
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classNumMatrix{public:vector<vector<int>>s;NumMatrix(vector<vector<int>>&matrix){intm=matrix.size(),n=matrix[0].size();s.resize(m+1,vector<int>(n+1));for(inti=0;i<m;++i){for(intj=0;j<n;++j){s[i+1][j+1]=s[i+1][j]+s[i][j+1]-s[i][j]+matrix[i][j];}}}intsumRegion(introw1,intcol1,introw2,intcol2){returns[row2+1][col2+1]-s[row2+1][col1]-s[row1][col2+1]+s[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); */
typeNumMatrixstruct{s[][]int}funcConstructor(matrix[][]int)NumMatrix{m,n:=len(matrix),len(matrix[0])s:=make([][]int,m+1)fori:=ranges{s[i]=make([]int,n+1)}fori,row:=rangematrix{forj,v:=rangerow{s[i+1][j+1]=s[i+1][j]+s[i][j+1]-s[i][j]+v}}returnNumMatrix{s}}func(this*NumMatrix)SumRegion(row1int,col1int,row2int,col2int)int{returnthis.s[row2+1][col2+1]-this.s[row2+1][col1]-this.s[row1][col2+1]+this.s[row1][col1]}/** * Your NumMatrix object will be instantiated and called as such: * obj := Constructor(matrix); * param_1 := obj.SumRegion(row1,col1,row2,col2); */
classNumMatrix{privates:number[][];constructor(matrix:number[][]){constm=matrix.length;constn=matrix[0].length;this.s=newArray(m+1).fill(0).map(()=>newArray(n+1).fill(0));for(leti=0;i<m;++i){for(letj=0;j<n;++j){this.s[i+1][j+1]=this.s[i+1][j]+this.s[i][j+1]-this.s[i][j]+matrix[i][j];}}}sumRegion(row1:number,col1:number,row2:number,col2:number):number{return(this.s[row2+1][col2+1]-this.s[row2+1][col1]-this.s[row1][col2+1]+this.s[row1][col1]);}}/** * Your NumMatrix object will be instantiated and called as such: * var obj = new NumMatrix(matrix) * var param_1 = obj.sumRegion(row1,col1,row2,col2) */
/** * Your NumMatrix object will be instantiated and called as such: * let obj = NumMatrix::new(matrix); * let ret_1: i32 = obj.sum_region(row1, col1, row2, col2); */structNumMatrix{// Of size (N + 1) * (M + 1)prefix_vec:Vec<Vec<i32>>,n:usize,m:usize,is_initialized:bool,ref_vec:Vec<Vec<i32>>,}/** * `&self` means the method takes an immutable reference. * If you need a mutable reference, change it to `&mut self` instead. */implNumMatrix{fnnew(matrix:Vec<Vec<i32>>)->Self{NumMatrix{prefix_vec:vec![vec![0;matrix[0].len()+1];matrix.len()+1],n:matrix.len(),m:matrix[0].len(),is_initialized:false,ref_vec:matrix,}}fnsum_region(&mutself,row1:i32,col1:i32,row2:i32,col2:i32)->i32{if!self.is_initialized{self.initialize_prefix_vec();}// Since i32 will let `rustc` complain, just make it happyletrow1:usize=row1asusize;letcol1:usize=col1asusize;letrow2:usize=row2asusize;letcol2:usize=col2asusize;// Return the value in O(1)self.prefix_vec[row2+1][col2+1]-self.prefix_vec[row2+1][col1]-self.prefix_vec[row1][col2+1]+self.prefix_vec[row1][col1]}fninitialize_prefix_vec(&mutself){// Initialize the prefix sum vectorforiin0..self.n{forjin0..self.m{self.prefix_vec[i+1][j+1]=self.prefix_vec[i][j+1]+self.prefix_vec[i+1][j]-self.prefix_vec[i][j]+self.ref_vec[i][j];}}self.is_initialized=true;}}
classNumMatrix(matrix:Array<IntArray>){privatevaln:Intprivatevalm:Intprivatevalmatrix:Array<IntArray>privatevalprefix_sums_matrix:Array<IntArray>privatevarinitialized:Booleaninit{this.n=matrix.sizethis.m=matrix[0].sizethis.matrix=matrixthis.prefix_sums_matrix=Array(n+1){IntArray(m+1)}this.initialized=false}funsumRegion(row1:Int,col1:Int,row2:Int,col2:Int):Int{this.init()returnthis.prefix_sums_matrix[row2+1][col2+1]-this.prefix_sums_matrix[row2+1][col1]-this.prefix_sums_matrix[row1][col2+1]+this.prefix_sums_matrix[row1][col1]}privatefuninit():Boolean{if(!this.initialized){for(iin0..<this.n){for(jin0..<this.m){this.prefix_sums_matrix[i+1][j+1]=this.prefix_sums_matrix[i+1][j]+this.prefix_sums_matrix[i][j+1]-this.prefix_sums_matrix[i][j]+this.matrix[i][j]}}this.initialized=truereturntrue}returnfalse}}/** * Your NumMatrix object will be instantiated and called as such: var obj = NumMatrix(matrix) var * param_1 = obj.sumRegion(row1,col1,row2,col2) */