**Category:**Dynamic Programming

**URL:**http://www.topcoder.com/stat?c=problem_statement&pm=10709

We are given a number which is composed of a sequence of digits. Each digit is represented by a combination of matches represented in a certain position to represent the digit. We are given the option of using up to K different match moves and are asked to calculate the number of different integers we can construct without discarding or adding any new matches to our given number.

First, we decompose the problem into several parts. The first involves somehow representing each digit in our program. A simple way to do this is to use a 2D integer array in which index (i,j) is equal to 1 if the digit i has a match in position j. An alternative way is to use a bitmask with the bit representation of the j-th bit set to 1 if there is a match in position j.

Next, we need to somehow compute the transitional costs between two different digits. For example, we need to quantify how many moves we would use up if we convert say a 6 to a 3. Since we are only allowed a limited number of moves its obvious we need a way to keep track of how many we have used up. One possible way is to determine how many places the two numbers differ. Then the number of moves we need to make is equivalent to exactly half of this.

There is another issue we need to deal with, that is the situation where the number of matches used to construct two digits differ. For example, the digit 6 uses 6 matches whereas digit 3 only uses 5 matches. We need to keep track of how much surplus (or deficit) matches we get from transitioning from one digit to another. This is important as we cannot add or remove matches from the final number - i.e. the number of matches used in constructing a unique integer must be the same as the original integer. Now we need to keep track of two things when we consider transforming one digit to another: the number of moves it takes and the difference in matches we have used so far.

We can define the cost function as follows:

To calculate the surplus/deficit it's quite simple, just calculate the number of matches used for each digit and subtract one from the other depending on which direction the transformation is taken. The above cost function ensures that we don't double count the number of moves required from moving a match from a surplus digit to a deficit digit. A way to visualise this is we "precharge" the surplus with an additional move but we are also granted a free move since this surplus is already counted once so the actual number of moves decreases if we need to import more matches from another transformation.

Lastly, we need to somehow compute all the ways we can construct unique integers given that we keep the number of matches used the same (i.e. at the end of our decision algorithm the number of surplus matches is exactly 0) as well as using less than or equal to K moves (which we subtract along the way). This suggests a recursive algorithm from potentially changing the first digit to something else, depending on the transformation we are left with either more/less/equal number of matches and the number of available moves to be less or equal to what we started with. We must also handle the case where we "borrow" matches from the latter part of the integer, so we must also handle cases where our match balance is negative. This can be done by computing an offset to keep the total number positive (to use for our caching purposes).

To efficiently compute the number of ways we need to memoise or apply bottom up DP to our algorithm. Note that the number of unique states is rather small so computational time isn't an issue. An implementation of the above ideas is shown below (using memoisation):

class NumbersAndMatches { public: long long differentNumbers(long long, int); }; int mat[10][7] = { {1,1,1,0,1,1,1}, // 0 {0,0,1,0,0,1,1}, // 1 {1,0,1,1,1,0,1}, // 2 {1,0,1,1,0,1,1}, // 3 {0,1,1,1,0,1,0}, // 4 {1,1,0,1,0,1,1}, // 5 {1,1,0,1,1,1,1}, // 6 {1,0,1,0,0,1,0}, // 7 {1,1,1,1,1,1,1}, // 8 {1,1,1,1,0,1,1} // 9 }; // defines the total number of matches used for digit i int tot[10] = {6, 3, 5, 5, 4, 5, 6, 3, 7, 6}; long long dp[19][155][155]; pair<int,int> costTab[10][10]; string num; #define SURPLUS_MOD 72 long long func(int idx, int moves, int surplus) { // offset due to the possibility of negative surplus int surplusMod = surplus + SURPLUS_MOD; if (idx >= num.size()) { // requires the surplus amount of matches to be 0 otherwise its not valid if (moves >= 0 && surplus == 0) return 1; return 0; } if (dp[idx][moves][surplusMod] != -1) return dp[idx][moves][surplusMod]; long long res = 0; // try each digit int curDig = num[idx] - '0'; for (int dig = 0; dig <= 9; dig++) { int c = costTab[curDig][dig].first; int d = costTab[curDig][dig].second; if (moves - c >= 0) { res += func(idx+1, moves - c, surplus + d); } } return dp[idx][moves][surplusMod] = res; } long long NumbersAndMatches::differentNumbers(long long N, int K) { long long res = 0; memset(dp,-1,sizeof(dp)); // compute the cost function for every digit -> digit transition for (int i = 0; i <= 9; i++) { for (int j = 0; j <= 9; j++) { int c = 0; for (int k = 0; k < 7; k++) { if (mat[i][k] != mat[j][k]) c++; } int s = tot[i] - tot[j]; costTab[i][j] = make_pair((c+s)/2, s); } } // decompose the number into a string long long vN = N; while (vN > 0) { num += ((vN % 10) + '0'); vN /= 10; } reverse(num.begin(),num.end()); // calculate the number of ways return res = func(0, K, 0); }

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