class Solution {

int minPatches(vector<int>& nums, int n) {

return minPatches_02(nums, n);

return minPatches_01(nums, n);

}

// Greedy Algorithm

// (Assume the array is sorted already)

//

// Let do some observation at first,

//

// 1) if we have [1,2] then we can cover 1, 2, 3

// 2) if we have [1,2,3] then we can cover 1,2,3,4,5,6

// So, it looks we can simply add all of nums together, then we can find out max number we can reach.

//

// 3) if we have [1,2,5], we can see

// 3.1) [1,2] can cover 1,2,3, but we cannot reach 4,

// 3.2) then we patch 4, then we have [1,2,4] which can cover 1,2,3(1+2),4,5(1+4),6(2+4), 7(1+2+4)

// 3.3) we can see [1,2,4] can reach to 7 - sum all of them

// 3.4) then [1,2,4,5], we can reach to 12 - 1,2,3,4,5,6,7,8(1+2+5),9(4+5),10(1+4+5), 11(2+4+5), 12(1+2+4+5)

//

// So, we can have figure out our solution

//

// 0) considering the `n` we need to cover.

// 1) maintain a variable we are trying to patch, suppose named `try_patch`

// 2) if `try_patch` >= nums[i] then, we just keep add the current array item,

// and set the `try_patch` to the next patch candidate number - `sum+1`

// 3) if `try_patch` < nums[i], which means we need to patch.

//

int minPatches_01(vector<int>& nums, int n) {

long covered = 0; //avoid integer overflow

int patch_cnt = 0;

int i = 0;

while (i<nums.size() ){

// set the `try_patch` is the next number which we cannot cover

int try_patch = covered + 1;

// if the `try_patch` can cover the current item, then just sum it,

// then we can have the max number we can cover so far

if ( try_patch >= nums[i]) {

covered += nums[i];

i++;

} else { // if the `try_patch` cannot cover the current item, then we find the number we need to patch

patch_cnt++;

//cout << "patch " << try_patch << endl;

covered = covered + try_patch;

}

if (covered >=n) break;

}

//for the case - [1], 7

//the above while-loop just process all of the numbers in the array,

//but we might not reach the goal, so, we need keep patching.

while (covered < n) {

int try_patch = covered + 1;

patch_cnt++;

//cout << "patch " << try_patch << endl;

covered = covered + try_patch;

}

return patch_cnt;

}

//The following solution just re-organizes the solution above, and make it shorter

int minPatches_02(vector<int>& nums, int n) {

long covered = 0;

int patch_cnt = 0;

int i = 0;

while ( covered < n){

if (i<nums.size() && nums[i] <= covered + 1) {

covered += nums[i++];

}else{

//cout << "patch " << covered + 1 << endl;

covered = 2 * covered + 1;

patch_cnt++;

}

}

return patch_cnt;

}

};