The following program example can be used to sum the array values of size N elements in parallel on a system containing N computing cores $($there is a separate processor for each array element$)$:

for j $=1$ to log $2($N$)\{$

for k $=1$ to N $\{$

if $(($k $+1)\%$ pow$(2,$j$)==0)\{$

values$[$k$]+=$ values$[$k $-$ pow$(2,($j$-1))]$

$\}\}\}$

This has the effect of summing the elements in the array as a series of partial sums, as shown in Figure $6\mathrm{.}16.$ After the code has been executed, the sum of all elements in the array is stored in the last array location.

Are there any race conditions in the above code example? If so$,$ identify

where they occur and illustrate with an example. If not, demonstrate

why this algorithm is free from race conditions.