UNITY is a programming language constructed by K. Mani Chandy and Jayadev Misra for their book Parallel Program Design: A Foundation. It is a theoretical language which focuses on what, instead of where, when or how. The language contains no method of flow control, and program statements run in a nondeterministic way until statements cease to cause changes during execution. This allows for programs to run indefinitely, such as auto-pilot or power plant safety systems, as well as programs that would normally terminate (which here converge to a fixed point).
All statements are assignments, and are separated by #. A statement can consist of multiple assignments, of the form a,b,c := x,y,z, or a := x || b := y || c := z. You can also have a quantified statement list, <# x,y : ''expression'' :: ''statement''>, where x and y are chosen randomly among the values that satisfy expression. A quantified assignment is similar. In <|| x,y : ''expression'' :: ''statement'' >, statement is executed simultaneously for all pairs of x and y that satisfy expression.
Bubble sort the array by comparing adjacent numbers, and swapping them if they are in the wrong order. Using
\Theta(n)
\Theta(n)
\Theta(n2)
\Theta(n)
k is always chosen randomly from \{0,1\}
k manually.Program bubblesort declare n: integer, A: array [0..n-1] of integer initially n = 20 # <|| i : 0 <= i and i < n :: A[i] = rand % 100 > assign <# k : 0 <= k < 2 :: <|| i : i % 2 = k and 0 <= i < n - 1 :: A[i], A[i+1] := A[i+1], A[i] if A[i] > A[i+1] > > end
You can sort in
\Theta(logn)
\Theta(n2)
\Theta(n2)
Program ranksort declare n: integer, A,R: array [0..n-1] of integer initially n = 15 # <|| i : 0 <= i < n :: A[i], R[i] = rand % 100, i > assign <|| i : 0 <= i < n :: R[i] := <+ j : 0 <= j < n and (A[j] < A[i] or (A[j] = A[i] and j < i)) :: 1 > > # <|| i : 0 <= i < n :: A[R[i]] := A[i] > end
Using the Floyd–Warshall algorithm all pairs shortest path algorithm, we include intermediate nodes iteratively, and get
\Theta(n)
\Theta(n2)
\Theta(n3)
Program shortestpath declare n,k: integer, D: array [0..n-1, 0..n-1] of integer initially n = 10 # k = 0 # <|| i,j : 0 <= i < n and 0 <= j < n :: D[i,j] = rand % 100 > assign <|| i,j : 0 <= i < n and 0 <= j < n :: D[i,j] := min(D[i,j], D[i,k] + D[k,j]) > || k := k + 1 if k < n - 1 end
We can do this even faster. The following programs computes all pairs shortest path in
\Theta(log2n)
\Theta(n3)
\Theta(n3logn)
Program shortestpath2 declare n: integer, D: array [0..n-1, 0..n-1] of integer initially n = 10 # <|| i,j : 0 <= i < n and 0 <= j < n :: D[i,j] = rand % 10 > assign <|| i,j : 0 <= i < n and 0 <= j < n :: D[i,j] := min(D[i,j],
After round
r
D[i,j] contains the length of the shortest path from i
j
0...r
0...2r