It is shown that for any x e d n umb e r o f v ariables, the linear programming problems with n linear inequalities can be solved deterministically by n parallel processors in sub-logarithmic time. The parallel time bound (counting only the arithmetic operations) is O((log log n) d ) where d is the number of variables. In the one-dimensional case this bound is optimal. If we t a k e i n to account the operations needed for processor allocation, the time bound is O((loglog n) d+c ) where c is an

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... absolute constant. Proof: It follows from Corollary 2.2 that jN r=2 (A G 0 )j > minf(1 + ) r=2 jAj n = 2g = m i n fn n=2g = n=2 and similarly, N r=2 (B G 0 ) > n = 2. This implies that N r=2 (A G 0 ) \ N r=2 (B G 0 ) 6 = and hence in G 0 there is a path of length less than or equal to r between A and B or, equivalently, a n e d g e o f ( G 0 ) r . Let c = 2 log 1+ 7, so that (1 + ) c=2 = 7 . Proposition 2.4. For every su ciently large n and for every positive integer r, there exists a graph G of degree less than d = 7 r with the following property: if t c = d, then every two disjoint sets A and B of vertices of G, such that jAj = jBj = n=t, a r e c onnected by an edge. Proof: The proof follows directly from Corollary 2.3 since t = d 1=c = 7 r=c = ( 1 + ) r=2 . Corollary 2.5. In the expander of Proposition 2.4, for every two disjoint sets of vertices A and B of vertices such that jAj = jBj = 2 n=t, there exist more than n=t edges between A and B.