10.4: Switch Count
- Page ID
- 48359
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Another goal in designing a communication network is to use as few switches as possible. The number of switches in a complete binary tree is \(1 + 2 + 4 + 8 + \cdots + N\), since there is 1 switch at the top (the “root switch”), 2 below it, 4 below those, and so forth. By the formula for geometric sums from Problem 5.4,
\[\nonumber \sum_{i = 0}^{n} r^i = \dfrac{r^{n+1} - 1}{r - 1},\]
the total number of switches is \(2N - 1\), which is nearly the best possible with \(3 \times 3\) switches.