A ring buffer (circular array) of fixed capacity C≥ 1 stores a sequence of at most C…

A ring buffer (circular array) of fixed capacity \(C\ge 1\) stores a sequence of at most \(C\) elements using an underlying array \(B[0..C-1]\) together with two integers `head` and `size`, where \(0\le\text{head}<C\) and \(0\le\text{size}\le C\). The sequence in logical order is \[ B[\text{head}],\ B[(\text{head}+1)\bmod C],\ \ldots,\ B[(\text{head}+\text{size}-1)\bmod C]. \] Design procedures `ENQUEUE-BACK(x)`, `DEQUEUE-FRONT()`, and `PEEK(i)` (\(0\le i<\text{size}\)) for this representation. (a) Write pseudocode for the three procedures. `ENQUEUE-BACK` must fail (and leave the structure unchanged) if \(\text{size}=C\). `DEQUEUE-FRONT` must fail if \(\text{size}=0\). `PEEK` must return the element at logical index \(i\) without mutating the buffer. (b) State the representation invariant relating \(B\), `head`, `size`, and \(C\). (c) For each procedure, give a worst-case \(\Theta\) bound in terms of \(C\) and `size`, and argue that the bound does not hide a scan of the buffer.

The live data occupy a wrap-around interval of length `size` starting at `head`; the back of that interval is determined by those two integers.