Editing Iterative deepening depth-first search (section)

==== Time and space complexities ====

The running time of bidirectional IDDFS is given by

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: <math>2\sum_{k = 0}^{n / 2} b^k</math>
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and the space complexity is given by

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: <math>b^{n/2},</math>
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where <math>n</math> is the number of nodes in the shortest <math>s,t</math>-path. Since the running time complexity of iterative deepening depth-first search is <math>\sum_{k=0}^n b^k</math>, the speedup is roughly

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: <math>\frac{\sum_{k=0}^n b^k}{2\sum_{k=0}^{n/2} b^k} =\frac{\frac{1 - b^{n+1}}{1-b}}{2\frac{1 - b^{n/2 + 1}}{1-b}} = \frac{1-b^{n+1}}{2(1 - b^{n/2 + 1})} = \frac{b^{n+1} - 1}{2(b^{n/2+1} - 1)} \approx \frac{b^{n+1}}{2b^{n/2+1}} = \Theta(b^{n/2}).</math>
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