update docs

Author: GiggleLiu

docs/src/performancetips.md

@@ -88,6 +88,13 @@ The `TropicalGEMM` pirates the `LinearAlgebra.mul!` interface, hence it takes ef
 The GEMM routine can speed up the computation on CPU for one order, with multi-threading, it can be even faster.
 Benchmark shows the performance of `TropicalGEMM` is close to the theoretical optimal value.
 
+## Sum product representation for configurations
+[`TreeConfigEnumerator`](@ref) can save a lot memory for you to store exponential number of configurations in polynomial space.
+It is a sum-product expression tree to store [`ConfigEnumerator`](@ref) in a lazy style, configurations can be extracted by depth first searching the tree with the `Base.collect` method. Although it is space efficient, it is in general not easy to extract information from it.
+This tree structure supports directed sampling so that one can get some statistic properties from it with an intermediate effort.
+
+(To be written.)
+
 ## Make use of GPUs
 To upload the computing to GPU, you just add need to use CUDA, and offer a new key word argument.
 ```julia

examples/IndependentSet.jl

@@ -157,7 +157,7 @@ loaded_sets = load_configs(filename; format=:binary, bitlength=10)
 #     * [`ConfigsMin`](@ref) for enumerating minimum set size,
 
 
-# ## Weights and open vertices
+# ## Weighted Graphs
 # [`IndependentSet`](@ref) accepts weights as a key word argument.
 # The following code computes the weighted MIS problem.
 problem = IndependentSet(graph; weights=collect(1:10))
@@ -167,6 +167,13 @@ max_config_weighted = solve(problem, SingleConfigMax())[]
 show_graph(graph; locs=locations, vertex_colors=
           [iszero(max_config_weighted.c.data[i]) ? "white" : "red" for i=1:nv(graph)])
 
+# For weighted MIS problem, a property that many people care about is the "energy spectrum", or the largest weights.
+# One just feed a positional argument in the [`SizeMax`](@ref) constructor as the number of largest weights.
+spectrum = solve(problem, SizeMax(10))
+
+# It uses the [`ExtendedTropical`](@ref) as the tensor elements.
+
+# ## Open vertices and MIS tensor analysis
 # The following code computes the MIS tropical tensor (reference to be added) with open vertices 1, 2 and 3.
 problem = IndependentSet(graph; openvertices=[1,2,3])
 

src/arithematics.jl

@@ -176,6 +176,12 @@ end
 Extended Tropical numbers with largest `K` orders keeped,
 or the [`TruncatedPoly`](@ref) without coefficients,
 `TO` is the element type of orders.
+This algebra maps
+
+* `+` to finding largest `K` values of union of two sets.
+* `*` to finding largest `K` values of sum combination of two sets.
+* `0` to set [-Inf, -Inf, ..., -Inf, -Inf]
+* `1` to set [-Inf, -Inf, ..., -Inf, 0]
 
 Example
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