Question

I was trying to solve a Lattice paths problem using dynamic programming method.

Starting in the top left corner of a 22 grid, there are 6 routes (without backtracking) to the bottom right corner.

How many routes are there through a 2020 grid?

Here is the code I have written to solve this question. Where am I going wrong. I seem to get wrong output every time. Am I crossing some boundaries in variable data types?

#include <stdio.h>
int count = 0;
int limita,limitb;
long long int cache[20][20];
unsigned long long int start(int a,int b)
{
    unsigned int long long i = 0;
    if(a == limita && b == limitb)
        return 1;
    if(cache[a][b] != -1)
        return cache[a][b];
    if(a != limita)
        i += start(a+1, b);
    if(b != limitb)
        i += start(a, b+1);
    cache[a][b] = i;
    return i;
}
int main(void)
{
    limita = limitb = 19;
    int i,j;
    for(i = 0; i < 20; i++)
        for(j = 0; j <20;j++)
            cache[i][j] = -1;
    unsigned long long int number = start(0,0);
    printf("The number of ways to reach the end is %llu\n",number);
    return 0;
}

Please help me out

Answer 1

A grid of size 1*1:

    0    1
   0+-----+
    |     |
    |     |
   1+-----+
    |<-2->|

A grid of size 2*2:

    0    1    2
   0+----+----+
    |    |    |
    |    |    |
   1+----+----+
    |    |    |
    |    |    |
   2+----+----+
    |<---3--->|

…

Your algorithm seems to be OK, but you’re counting edges wrong.