Similar to Yang Hui's triangle-N cup of water

Problem:

There is a pyramid. From top to bottom, there is a cup on the first layer, there are two cups on the second layer, and so on. Number the cup in the following shape:

1
2 3
4 5 6

The capacity of each cup is C-litre, and the water is raised from the top of the tower. When the first cup is full, it will overflow to the second and third cups. When the numbers 2 and 3 are full, the numbers 2 overflows to the numbers 4 and 5, and the numbers 3 overflows to the numbers 5 and 6. Note that the numbers 5 accept the water from the two cups. And so on. Given C and L, how much water is in cup n.

From @ Chen liren

Analysis:

This question does not come up with a good method, but just a simulation (or DP ),

F [leveli, Index] indicates the amount of water received by the index glass at the leveli layer. So:

F [leveli, Index] = (F [leveli-1, index-1]-C)/2 + (F [leveli-1, Index]-C)/2.

Here f [leveli-1, index-1]-C, F [leveli-1, Index]-C is not less than 0, otherwise the corresponding item is 0.

Implementation: Use hashmap as a memorandum for DP.

Code: time complexity O (N), space complexity O (N)

Double getflow (int c, int leveli, int index, unordered_map <int, double> & hashmap) {If (index <0 | index> leveli) return 0; int n = (leveli + 1) * leveli/2 + index; // The total index auto P = hashmap. find (n); If (P! = Hashmap. end () return p-> second; double leftflow = getflow (C, leveli-1, index-1, hashmap); double rightflow = getflow (C, leveli-1, index, hashmap); Return hashmap [N] = (leftflow> = C? (Leftflow-C)/2: 0) + (rightflow> = C? (Rightflow-C)/2: 0) ;}int main () {int L = 10; int c = 5; int n = 1; // The N value starts from 0, int leveli = 0, sum = 0; while (sum + (leveli + 1) <n + 1) // The initialization level: levelisum + = leveli ++ 1; int Index = N-(leveli + 1) * leveli/2; // calculate the row index unordered_map <int, double> hashmap; hashmap [0] = L; // initialize mapdouble flow = getflow (C, leveli, index, hashmap); cout <(flow> C? C: flow) <Endl ;}

Supplement: Yang Hui triangle knowledge

The quadratic coefficient of n times corresponds to n + 1 row of the Yang Hui triangle.
For example, in (a + B) ^ 2 = a ^ 2 + 2AB + B ^ 2, the two-time binary form corresponds to the 3rd row coefficient 1 2 1 of the Yang Hui triangle.

Nature: