added dp documentation (#106)

* added documentation for dp path

* added link to dp_path

docs/source/dp_path.rst

@@ -0,0 +1,45 @@
+============================
+The Dynamic Programming Path
+============================
+
+The dynamic programming (DP) approach described in reference [1] provides an efficient
+way to find an asymptotically optimal contraction path by running the following steps:
+
+  1. Compute all traces, i.e. summations over indices occurring exactly in one
+     input.
+  2. Decompose the contraction graph of inputs into disconnected subgraphs. Two
+     inputs are connected if they share at least one summation index.
+  3. Find the contraction path for each of the disconnected subgraphs using a
+     DP approach: The optimal contraction path for all sets of ``n`` (ranging
+     from 1 to the number of inputs) connected tensors is found by combining
+     sets of ``m`` and ``n-m`` tensors.
+
+Note that computing all the traces in the very beginning can never lead to a
+non-optimal contraction path.
+
+Contractions of disconnected subgraphs can be optimized independently, which
+still results in an optimal contraction path. However, the computational
+complexity of finding the contraction path is drastically reduced: If the
+subgraphs consist of ``n1``, ``n2``, ... inputs, the computational complexity
+is reduced from ``O(exp(n1 + n2 + ...))`` to ``O(exp(n1) + exp(n2) + ...)``.
+
+The DP approach will only perform pair contractions and will never compute
+intermediate outer products as in reference [1] it is shown that this always
+results in an asymptotically optimal contraction path.
+
+A major optimization for DP is the cost capping strategy: The DP optimization
+only memorizes contractions for a subset of inputs, if the total cost for this
+contraction is smaller than the cost cap. The cost cap is initialized with
+the minimal possible cost, i.e. the product of all output dimensions, and is
+iteratively increased by multiplying it with the smallest dimension
+until a contraction path including all inputs is found.
+
+Note that the worst case scaling of DP is exponential in the number
+of inputs. Nevertheless, if the contraction graph is not completely random,
+but exhibits a certain kind of structure, it can be used for large
+contraction graphs and is guaranteed to find an asymptotically optimal
+contraction path. For this reason it is the most frequently used contraction
+path optimizer in the field of tensor network states.
+
+
+[1] Robert N. C. Pfeifer, Jutho Haegeman, and Frank Verstraete Phys. Rev. E 90, 033315 (2014). https://arxiv.org/abs/1304.6112

docs/source/index.rst

@@ -165,6 +165,7 @@ Table of Contents
    branching_path
    greedy_path
    random_greedy_path
+   dp_path
    custom_paths
 
 .. toctree::