# [Leetcode]: 62:unique Paths

Source: Internet
Author: User

Topic:

A robot is located at the Top-left corner of a m x n grid (marked ' Start ' in the diagram below).

The robot can only move either down or right at any point in time. The robot is trying-to-reach the bottom-right corner of the grid (marked ' Finish ' in the diagram below).

How many possible unique paths is there?

Above is a 3 x 7 grid. How many possible unique paths is there?

Note: m and N would be is at most 100.

Idea 1: Recursion

Number of paths to end point = Sum of the number of paths of all points that can reach the end, that is: path [m][n] = path [m-1][n] + path [m][n-1]

Recursively, until start point (0,0)

Code:

`     Public Static intUniquepaths (intMintN) {if(n = = 1 && m==1){            //the initial position            return1; }        Else if(n = = 1 && m>1){            returnUniquepaths (1,m-1); }        Else if(N >1 && m==1){            returnUniquepaths (n-1,1); }        Else{            returnUniquepaths (n-1,m) +uniquepaths (n,m-1); }    }`

The result was timed out at the time of M=17 n=23. So realize the limitations of the topic is the need for speed!

Idea 2: Dynamic planning

Number of paths to a point = Sum of the number of paths of all points that can reach that point: path [m][n] = path [m-1][n] + path [m][n-1]

Unlike recursion, recursion is from large to small, and dynamic planning is

The code is as follows:

`     Public Static intUniquepaths (intMintN) {int[] Arrresult =New int[M][n];  for(inti = 0;i<m;i++){             for(intj = 0;j<n;j++){                if(i = = 0 && j== 0) {arrresult[0][0] = 1; }                Else if(i = = 1 && j== 0) {arrresult[1][0] = 1; }Else if(i = = 0 && j== 1) {arrresult[0][1] = 1; }                //The above is the fill base value                Else{                    intRow = 0; intColumn = 0; if(i> 0) {row= Arrresult[i-1][j]; }                                        if(j> 0) {column= Arrresult[i][j-1]; } Arrresult[i][j]= row +column; }            }        }                        returnArrresult[m-1][n-1]; }`

[Leetcode]: 62:unique Paths

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