public override long CalculateTotalCoins(int valueToCalculate, long[] arr, CalcState parentState = null, int depth = 1)
{
#if DEBUG
if (valueToCalculate <= 0)
{
throw new Exception($"Unexpected valueToCalculate - {valueToCalculate}");
}
if (coin.Next != null)
{
throw new Exception("Root node is supposed to be singleton");
}
if (arr.Length <= valueToCalculate)
{
throw new Exception("passed in array is supposed to be >= max valueToCalculate");
}
if (parentState != null)
{
throw new Exception("Expected parentState to be null on calling root node");
}
Grid.StartDebug(this, valueToCalculate);
#endif
// we are singleton - e.g., '9' if we get hit by '18' then
// there are 2 coins we can make ...
if (valueToCalculate % Head == 0)
{
int myCoins = valueToCalculate / Head;
// keep tally in global record;
arr[myCoins]++;
#if DEBUG
Grid.Debug(this, myCoins, 1);
Grid.EndDebug(this);
#endif
return(1 + GetChildren().Sum(x =>
x.CalculateTotalCoins(valueToCalculate, arr, new CalcState(myCoins, myCoins - 1, 1), depth + 1)));
}
return(0);
}
/// <summary>
/// Given an array from our parent indicating how many Total coins the parent
/// has produced - we have a relatively simple way of calculating how many
/// total coins we will produce.
///
/// We need three items :
///
/// A) the parent array of total coins produced for each combo
/// B) the number of coins to subtract for every combination
/// C) the number of combinations to subtract every time we add one of our coins
///
/// B) and C) are the same for every input value.
///
/// To derive A -> Every time we ADD our unit to the previous (parent) combination
/// we are adding one coin = +1 but subtracting "UNIT coins/bottom UNIT".
/// so, e.g., if we are 'head' in the combination (4,1) then we subtract 3 coins
/// from each entry in the input array to reflect the fact we are reducing by 3 overall.
/// if we are 'head' in the combination (8,4) then we subtract 1 coin.
///
/// so UNIT/HEAD_UNIT - 1 is the number of coins to remove from the input array
/// when we add one of our UNITS (B)
///
/// Number of combinations we need to generate is derived by how many combinations
/// does our parent produce for one of our Units.
///
/// E.g., say we add ONE of our units to the set of parent combinations
/// we will be adding just one combination to the total, but, as the PARENT
/// combinations are already factored into the output - we need to make sure we do not
/// duplicate them - so so we subtract however many there are.
///
/// e.g., if we are '4' in 1,2,4 ... then when we are triggered first time
/// when value is '7' we will be triggered with 3 # combinations from 1,2
/// - (5,1)=6,(3,2)=5,(1,3)=4 ... we are adding a new combination
/// (1,1,1)=3 so the total set will contain 4 combinations - but we
/// have already output 3 - so, in this example, the difference in combo numbers for adding a new Unit (C)
/// will be 3-1 = 2.
///
/// We calculate the 'C' the first time we are triggered based on the
/// number of combos from our parent - easy.
///
/// Our parent provides us with #of combinations, the max-number of coins and the 'step' between each coin
///
/// We will be transforming that into an array where each row provides our corresponding number of combinations.
///
/// Because each new coin -> 1 to (Value/Units) -> reduces the combination set and reduces the maximum number
/// we get a triangular output, where each row is number of coins
///
/// E.g., if provided with following array indicating (45-26)=19 combinations and top one has 45 coins.
///
/// 45,
/// 44
/// 43,
/// 42,
/// 41
/// ...
/// 26
///
/// we can convert this to following downward sloping triangle
/// 45
/// 44
/// 43 43
/// 42 42
/// 41 41
/// 40 40 40
/// 39 39 39
/// 38 38 38
/// 37 37 37 37
/// 36 36 36 36
/// 35 35 35 35
/// 34 34 34 34 34
/// 33 33 33 33 33
/// 32 32 32 32 32
/// 31 31 31 31 31 31
/// 30 30 30 30 30 30
/// 29 29 29 29 29 29
/// 28 28 28 28 28 28 28
/// 27 27 27 27 27 27 27
/// 26 26 26 26 26 26 26
/// 25 25 25 25 25 25 25 25
/// 24 24 24 24 24 24 24
/// 23 23 23 23 23 23
/// 22 22 22 22 22 22
/// 21 21 21 21 21
/// 20 20 20 20
/// 19 19 19 19
/// 18 18 18
/// 17 17
/// 16 16
/// 15
///
/// where each new column is a new combination,
/// number of new columns is given by Value/Units,
/// the top downward slope is given by (B) - no coins to subtract for each combination
/// and length of each successive column is (C) minus previous length
///
/// and then we can convert this to e.g,.
///
/// 45 = 1
/// 44 = 1
/// 43 = 2
/// 42 = 2
/// etc.,
///
/// And add that to our state - indicating that, e.g., there are 2 combinations where total coins is '43' etc.,
///
/// We could iterate over each column to do this - but this would be inefficient.
///
/// so instead, we do it directly - by calculating the height of the whole triangle.
/// incrementing the no_of_combos until we reach the mid-point (25) in example above
/// and then decrementing the no_of_combos after that until we reach the end point (15).
///
/// It is grim - but efficient ...
///
///
/// </summary>
/// <param name="valueToCalculate"></param>
/// <param name="arr"></param>
/// <param name="parentState"></param>
/// <param name="depth"></param>
/// <returns></returns>
/// <exception cref="Exception"></exception>
public override long CalculateTotalCoins(int valueToCalculate, long[] arr, CalcState parentState = null, int depth = 1)
{
#if DEBUG
if (depth != this.Depth)
{
throw new Exception("Calc tree out of synch somehow ");
}
if (valueToCalculate <= 0)
{
throw new Exception($"Unexpected valueToCalculate - {valueToCalculate}");
}
#endif
if (differenceInParentComboNumberForEachCoin == -1)
{
// every time we add a new coin at this level we have same as previous level -
// _minus_ the number of parent combinations it takes to make one of our coins.
// (given by GetMaxCombinationsForValue - where SumOfUnits is our minimum trigger value).
// plus one - because this is a new combination.
differenceInParentComboNumberForEachCoin = parent.GetMaxCombinationsForValue(SumOfUnits) - 1;
}
if (noCoinsToSubtractForEachUnit == -1)
{
noCoinsToSubtractForEachUnit = (Head / GetRootUnit()) - 1;
if (noCoinsToSubtractForEachUnit < 1)
{
throw new ArgumentException("noCoinsToSubtractForEachUnit expected to be >=1");
}
}
// every time we have a new coin we reduce our number of coins by
// noCoinsToSubtractForEachUnit - (moving our triangle coin columns down by X)
// but we also have less combinations to add ... this represents our net shift).
if (decrementInTotalCoinsForEachCoin == -1)
{
decrementInTotalCoinsForEachCoin = noCoinsToSubtractForEachUnit - differenceInParentComboNumberForEachCoin;
}
#if DEBUG
Grid.StartDebug(this, valueToCalculate);
#endif
if (valueToCalculate - SumOfUnits >= 0)
{
var thisNumberCombinations = ((valueToCalculate - SumOfUnits) / Head) + 1;
int cnt = 0;
int max = -1;
int min = int.MaxValue;
int ourStartingNoCoins = parentState.MaxParentCoins - noCoinsToSubtractForEachUnit;
// our first column - to the right-angled part of triangle height = same as height of parentState
// minus the decrementInTotalCoinsForEachCoin
int ourHeight = (parentState.NumberCombinations - differenceInParentComboNumberForEachCoin);
if (ourHeight <= 0)
{
throw new Exception($"Invalid height of triange - {ourHeight}");
}
foreach (var(triangleRow, number) in new CoinTriangle(ourStartingNoCoins,
ourHeight, thisNumberCombinations,
noCoinsToSubtractForEachUnit, decrementInTotalCoinsForEachCoin))
{
if (max == -1)
{
max = triangleRow;
}
min = triangleRow;
arr[triangleRow] += number;
#if DEBUG
Grid.Debug(this, triangleRow, number);
#endif
if (number > 0)
{
cnt++;
}
}
#if DEBUG
Grid.EndDebug(this);
#endif
var ourState = new CalcState(max, min, cnt);
return(cnt + GetChildren().Sum(x => x.CalculateTotalCoins(valueToCalculate, arr, ourState, depth + 1)));
}
return(0);
}
public override long CalculateTotalCoins(int valueToCalculate, long[] arr, CalcState parentState = null,
int depth = 1)
{
#if DEBUG
if (depth != 2)
{
throw new Exception("CalculationNodeVector should only be called when depth == 2");
}
if (parentState.NumberCombinations != 1)
{
throw new Exception("Expecting scalar from root node");
}
if (noCoinsToSubtractForEachUnit == -1)
{
noCoinsToSubtractForEachUnit = (Head / GetRootUnit()) - 1;
if (noCoinsToSubtractForEachUnit < 1)
{
throw new ArgumentException("noCoinsToSubtractForEachUnit expected to be >=1");
}
}
Grid.StartDebug(this, valueToCalculate);
#endif
long cnt = 0;
if (valueToCalculate - SumOfUnits >= 0)
{
var max = -1;
var min = -1;
var startValue = parentState.MaxParentCoins;
for (int i = 0; i < ((valueToCalculate - SumOfUnits) / Head) + 1; i++)
{
startValue -= noCoinsToSubtractForEachUnit;
if (max == -1)
{
max = startValue;
}
min = startValue;
arr[startValue]++;
#if DEBUG
Grid.Debug(this, startValue, 1);
#endif
cnt++;
}
#if DEBUG
Grid.EndDebug(this);
#endif
var ps = new CalcState(max, min, (int)cnt);
foreach (var c in GetChildren())
{
cnt += c.CalculateTotalCoins(valueToCalculate, arr, ps, depth + 1);
}
}
return(cnt);
}
public abstract Int64 CalculateTotalCoins(int valuetoCalculate, Int64[] arr, CalcState parentState = null, int depth = 1);