Bipropagation algorithm is still iterative like a "backpropagation", but internal layers are not hidden anymore since their desired values are now calculated in advance before learning. That way can machine learning be conducted layer by layer, what represents great improvement (it could be more than a few ten times faster).
"Border pairs method" is a totally new algorithm which has many advantages over "backpropagation" algorithm. Firstly we look for the pairs of patterns of opposite class which are so near, that no third pattern does lie in between them. This is the Border pairs and only this patterns are significant when we want to draw a borderline between classes. So the majority of learning patterns (sometimes more than 90%) is eliminated even before the learning begins. Then we are trying to separate all border pairs with a minimal number of border lines, which represent neurons of the 1st layer, so we find the minimal structure of the neural network. While we draw one by one borderline, the learning of the first layer is done with only one neuron at the same time. Since these neurons have a hard limiter, they put binary output values and so the next layers could even be replaced with a logical circuit. Border pairs also allow a simple reduction of the noise.
More info here.
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