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===Optimizing space complexity===
{{unreferenced section|date=September 2012}}
Let ''n'' be the total number of digits in the two input numbers in [[Radix|base]] ''D''. If the result must be kept in memory then the space complexity is trivially Θ(''n''). However, in certain applications, the entire result need not be kept in memory and instead the digits of the result can be streamed out as they are computed (for example, to system console or file). In these scenarios, long multiplication has the advantage that it can easily be formulated as a [[FL (complexity)|log space]] algorithm; that is, an algorithm that only needs working space proportional to the logarithm of the number of digits in the input ([[Bachmann-Landau notation|Θ]](log ''n'')). This is the ''double'' logarithm of the numbers being multiplied themselves (log log ''N''). Note that operands themselves still need to be kept in memory and their Θ(''n'') space is not considered in this analysis.
The method is based on the observation that each digit of the result can be computed from right to left with only knowing the carry from the previous step. Let ''a''<sub>''i''</sub> and ''b''<sub>''i''</sub> be the ''i''-th digit of the operand, ''r''<sub>''i''</sub> be the ''i''-th digit of the result and ''c''<sub>''i''</sub> be the carry generated for ''r''<sub>''i''</sub> (i=1 is the right most digit) then
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