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The lasso can be computed with the LARS or Coordinate Descent algorithm.

What is their computational complexity and when one is quicker than the other?

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I realize it's quite late to give an answer but maybe someone will find it useful.

Here is a nice talk by Trevor Hastie about the coordinate descent. He compares (among others) his two R packages: glmnet (using coordinate descent) and lars (using LARS).

It's shown that coordinate descent is faster in each setting: $p>N, p<N$ and sparse or dense data. There are some examples featuring both simulated and real data.

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