The code complexity extension for Visual-Studio is now compatible with 2019 version. Thanks to Michael Murphy for the assistance!
The new version is available at the Visual-Studio Gallery and the source code at GitHub.
The code complexity extension for Visual-Studio is now compatible with 2019 version. Thanks to Michael Murphy for the assistance!
The new version is available at the Visual-Studio Gallery and the source code at GitHub.
This post is an introduction to SciPy sparse graphs. It will present a variation of a known problem followed by a simple solution and implementation.
We are given a directed acyclic graph (DAG) with dynamic edge costs. The graph is large in regard to nodes, it is expected to have millions of nodes. Additionally, the graph is expected to have very few edges, so the average degree is very small.
We need to construct a data structure that given two states, a source and a target, can figure our efficiently if there’s a path from the source to the target. In case it can, what would be the minimum cost of such path?
Our goal is to reduce the query time, where we expected the pairs of source and target to be uniformly picked.
For example, this representation could be of a system with many workflows, mostly independent, where each step in a workflow requires different effort to finish.
The following graph is an example of such graph, where the edge labels represent the costs:
A query could be: Can 2 reach 6? The answer, in this case, is yes. The cost of reaching 6 from 2 is 7 (2→6). Please note the cost is the current cost. The path will always exist, but the edges price may change in the future.
Another query could be: Can 1 reach 2? The answer is no. No path from 1 to 2 exists. This answer will stay no forever, regardless any cost changes.
A straightforward approach would aim to reduce the number of shortest-path algorithm executions as most would yield “no path”. Therefore, some sort of method to predict paths existence is needed.
Assuming we want to initialize a cache, we would store in the cache the fact that i is reachable from j rather than the distance itself since the prices are dynamic. In the case where the average degree is very low, we can expect most pairs to be false. Therefore, we can store only the pairs whose value is true. For example, in the case of 1M nodes and an average degree of 2, we expect <<1% to be true.
A common representation of graphs is weighted adjacency matrix. Denote the matrix with A, and we say that the cost of the edge from i to j price is A[i, j] if the edge from i to j exists and 0 otherwise.
0 | 1 | 2 | 3 | |
0 | 0 | 10 | 0 | 11 |
1 | 0 | 0 | 3 | 18 |
2 | 0 | 0 | 0 | 0 |
3 | 0 | 0 | 0 | 0 |
In this example, 0 has an edge to 1, so A[0, 1] = 10. 0 and 2 are not directly connected, so A[0, 2] = 0. The rows of 2 and 3 are all zeros since both are leaves, meaning their out degree is 0.
We expect the majority of cells in the matrix to be 0. In this case, we can take advantage of a sparse matrix representation. The format which we will use is compressed sparse row (CSR), which supports efficient matrices multiplication. It requires O(num of edges) memory. Luckily, SciPy provides an implementation to the CSR matrix.
Given an input in a form of dictionary of edges to weight:
edges = { (0, 1): 2, (0, 2): 3, (2, 3): 10, (3, 4): 1, (3, 5): 7 }
We can transform it to a sparse matrix:
raw_weights = list(edges.values()) sources = [s for s, _ in edges.keys()] targets = [t for _, t in edges.keys()] n = max(sources + targets) + 1 # assume no isolated nodes weights = csr_matrix((raw_weights, (sources, targets)), shape=(n, n))
This will result in our case to:
[[ 0 2 3 0 0 0] [ 0 0 0 0 0 0] [ 0 0 0 10 0 0] [ 0 0 0 0 1 7] [ 0 0 0 0 0 0] [ 0 0 0 0 0 0]]
Remember that the zeros in the matrix are “implicit” so don’t let the current representation mislead you regarding the memory consumption.
After representing the graph as a matrix, we can create a boolean adjacencies matrix. A boolean adjacencies matrix A is one where A[i, j] is true iff there is an edge from i to j or i=j.
We will take advantage of the following property: A^{k}[i, j]=true iff exists a path from i to j of length k-1 or less. Let us look at an example:
It is easy to create the adjacency matrix for this graph, and to follow its powers:
# A [[ True True False False] [False True True False] [False False True True] [False False False True]] # A^{2} [[ True True True False] [False True True True] [False False True True] [False False False True]] # A^{3 }[[ True True True True] [False True True True] [False False True True] [False False False True]]
According to the graph diameter definition, any simple path is as long as the diameter. This means that if we know the graph diameter, we can easily resolve the reachable components by computing A^{diameter}.
Another property which is easy to notice is that the minimal k that satisfies A^{k}=A^{k+1 }is the graph diameter. Also, if k>diameter, then A^{k+m}=A^{k} for any positive m (since no new node will become reachable after walking all paths in the length of the diameter). Looking at A^{2}, A^{4}, A^{8},…, we can see that if two consecutive powers of A equal, we have reached the matrix that represents the reachable components.
All the values in the reachable components are either true or false. Also, for any A^{k}[i, j] that is true for some k, A^{k+m}[i, j] will be true for any positive m. This means that when we compare two powers of A, it is enough to compare the number of true values in the matrix.
Given an initial weights matrix, we can create the adjacencies matrix:
adjacency = (weights + sparse.diags(np.ones(weights.shape[0]), 0, format='csr')).astype(bool)
The code takes the weights matrix and adds 1 to each value over the diagonal to ensure a positive value. Then it converts the matrix to a boolean one, so any cell with non-zero value turns into true.
Now, we can raise the matrix by powers of 2, until the number of true values does not change:
components = adjacency.copy() previous_nnz = 0 while previous_nnz != components.nnz: previous_nnz = components.nnz components **= 2
The code compares the property nnz, which is the number of non-zero values, to the previous iteration count. Oofficially, nnz it is the number of explicit values stored, but in our case, it equals the number of non-zeros). When it reaches a stable matrix, the components matrix contains the reachable components, meaning that components[i, j]=true iff exists a path from i to j.
Given a query with nodes i and j, two actions are required – check if i can reach j, and if so, compute the minimum cost of the path between them.
The format we’ve used for the components matrix is CSR, which is efficient for computing matrix powers, but, it does not allow constant time access for keys. Therefore, we would like to convert it into a more efficient representation, such as a set of pairs.
components = components.tocoo() reachable_pairs = set(zip(components.row, components.col))
Firstly, we convert the CSR matrix into a coordinates matrix, which is more iteration friendly. The row and col attributes are arrays containing the coordinates of the true stored values.
For example, if the matrix is:
[[False True] [True False]]
Then:
row = [0, 1] col = [1, 0]
This results with
reachable_pairs = {(0, 1), (1, 0)}
Now that we have a quick way to check if a path exists, we can execute the dijkstra algorithm on the weights matrix:
if (source, target) in reachable_pairs: distances = dijkstra(weights, indices=[source]) distance = distances[0, target] else: distance = np.inf
The call to dijkstra returns a costs vector from source to all the nodes in the graph. In our case we expect the vast majority to be infinity.
In this post, we’ve seen one approach to efficiently handle shortest path problem on very sparse graphs. We chose a representation based on sparse adjacency matrix and pre-computed all the connected components.
I will mention the benchmarks, ~500 times faster on 100K nodes with an average degree of 2, but please note that this is not enough to generalize. The main reason is that the problem introduced is too naive – no prior knowledge of the components and queries with a uniform distribution over the nodes. This is usually not the case in real life problems. Information about the components, or a distribution with a higher expectation of positive hits, might have changed the approach.
There are other approaches to optimize, yet, it is very encouraging to know how simple can solutions can be, ~20 lines of code, and very lightweight. It can be useful in many cases to translate problems into classic problems in graphs and take advantage of the existing theory and implementations. This is also true even when large amounts of data are involved.
The code complexity extension for Visual-Studio is now compatible with 2017 version.
The new version is available at the Visual-Studio Gallery and the source code at GitHub.
In the previous post we started modifying functions body for AOP purposes. So far we’ve transformed a function to an AST node with extended code. In this post we will transform it to bytecode and replace the original method code with the new one.
As to this moment, we have these two:
Given the node, we can use Python built-in compile. Compile takes either string or AST node. The node is transformed into a module which we can later execute.
def compile_node(modified_node): compiled_method = compile(modified_node, '<string>', 'exec') return compiled_method
The two arguments we send to compile are the source file name to which the method will be linked. In our case, we’re not going to keep the method compiled method, and only use its bytecode. Therefore, we can leave a default value ‘<string>’ which is commonly used. The second argument is ‘exec’, which signals the compiler that multiple statements may appear in the input code and that we don’t care about the returned value. This is a simplified explanation, but sufficient for this context.
Python code object does not allow setting the bytecode. Therefore we need to create a new code object, which is identical to the original code, other than the bytecode and stacksize:
def create_replacement_code(original_function, compiled_function): # these are the code object field names, ordered by the expected order of code object constructor code_arg_names = ['co_argcount', 'co_nlocals', 'co_stacksize', 'co_flags', 'co_code', 'co_consts', 'co_names', 'co_varnames', 'co_filename', 'co_name', 'co_firstlineno', 'co_lnotab'] # fill all the args based on the original code code_args = [getattr(original_function.func_code, key, None) for key in code_arg_names] # replace the bytecode and stacksize with the compiled bytecode and stacksize from the modified function code = extract_code(compiled_function) # replace bytecode and stacksize with the compiled method values code_args[code_arg_names.index('co_code')] = code.co_code code_args[code_arg_names.index('co_stacksize')] = code.co_stacksize return CodeType(*code_args)
The instantiation of the code object is undocumented. Yet, some examples are available online. Please note that this signature fits python 2.7 and was changed in 3.0. Now we finished transforming a python code (which is the original function) into a new code object, containing the modified bytecode.
The last two methods calls can be extracted to:
def create_func_code(original_function, modified_node): compiled_method = compile_node(modified_node) return create_replacement_code(original_function, compiled_method)
The last step required is to switch the function’s code into the modified one. Luckily, this step is very simple:
def replace_code(original_function, code): original_function.func_code = code
Now, when the decorated method is executed, it’ll contain the modified bytecode, which checks if variables used in comparisons are not None.
from _ast import BoolOp, And, Name, Compare, IsNot, Load, Num from ast import NodeTransformer, fix_missing_locations import ast import inspect from types import FunctionType, CodeType class ComparisonTransformer(NodeTransformer): def visit_Compare(self, node): parts = [node.left] + node.comparators # check if any constant number is involved in the comparison if not any(isinstance(part, Num) for part in parts): return node # get all the "variables" involved in the comparison names = [element for element in parts if isinstance(element, Name)] if len(names) == 0: return node # create a reference to None none = Name(id='None', ctx=Load()) # create for each variable a node that represents 'var is not None' node_verifiers = [Compare(left=name, ops=[IsNot()], comparators=[none]) for name in names] # combine the None checks with the original comparison # e.g. 'a < b < 1' --> 'a is not None and b is not None and a < b < 1 return BoolOp(op=And(), values=node_verifiers + [node]) def rewrite_comparisons(original_function): assert isinstance(original_function, FunctionType) node = parse_method(original_function) rewrite_method(node) code = create_func_code(original_function, node) replace_code(original_function, code) return original_function def replace_code(original_function, code): original_function.func_code = code def parse_method(original_function): return ast.parse(inspect.getsource(original_function)) def rewrite_method(node): # assuming the method has single decorator (which is the rewriter) - remove it node.body[0].decorator_list.pop() # we rename the method to ensure separation from the original one. # this step has no real meaning and not really required. node.body[0].name = 'internal_method' # transform Compare nodes to fit the 'is not None' requirement ComparisonTransformer().visit(node) # let python try and fill code locations for the new elements fix_missing_locations(node) def create_func_code(original_function, modified_node): compiled_method = compile_node(modified_node) return create_replacement_code(original_function, compiled_method) def compile_node(modified_node): compiled_method = compile(modified_node, '<string>', 'exec') return compiled_method def extract_code(compiled_method): exec compiled_method generated_func = locals()['internal_method'] return generated_func.func_code def create_replacement_code(original_function, compiled_function): # these are the code object field names, ordered by the expected order of code object constructor code_arg_names = ['co_argcount', 'co_nlocals', 'co_stacksize', 'co_flags', 'co_code', 'co_consts', 'co_names', 'co_varnames', 'co_filename', 'co_name', 'co_firstlineno', 'co_lnotab'] # fill all the args based on the original code code_args = [getattr(original_function.func_code, key, None) for key in code_arg_names] # replace the bytecode and stacksize with the compiled bytecode and stacksize from the modified function code = extract_code(compiled_function) # replace bytecode and stacksize with the compiled method values code_args [code_arg_names.index('co_code')] = code.co_code code_args[code_arg_names.index('co_stacksize')] = code.co_stacksize return CodeType(*code_args) @rewrite_comparisons def foo(x): return x < 1 print foo(None)
This code, will output the following, as expected:
False
To prove that this code is works in other context, the following code:
@rewrite_comparisons def bar(x, y, z, w): if x < 1 or y < 2 or z < 3: return 'Default behavior' if w < 1: return 'Expected behavior' return 'Failure' print bar(None, None, None, 0)
Prints the following result:
Expected behavior
Python is amazingly flexible and using ~50 lines of code we can create a micro-framework to manipulate methods behavior. In addition, this flexibility allows code changes in many levels, including instructions level changes.
The vast majority of use cases are on the method level and not the instructions. Yet, this facilities can be useful in some cases an we should be able to take advantage of them.
This post and the next post will address AOP in Python. In general AOP in Python is very simple thanks to Python’s decorators. The aspects which we would like to apply in this post are low-level, meaning they’ll be applied on in-body instructions and not just on method level. The way in which we’re going to implement it will be using code weaving and rewriting.
I previously blogged about similar concept in .Net using Mono Cecil, where we tracked IL instructions.
The topic will be covered by two posts, where the first one will address rewriting code and the second one will deal with replacing the original code.
The general motivation for AOP is to separate the business logic from other functional logic, like logging, security or error handling. Most of the common examples fit the pattern of wrapping the function with new one. Then, perform logic before/after the method is executed. This is very useful, yet, limits our ability to change behavior of specific instructions inside the method which are relevant to the aspect.
During the post we will use a concrete simple example. Let us observe the following example (Python 2.7):
def foo(x): return x < 1 print foo(None)
As you probably know, this will print:
True
This is a common Python (2.7) behavior but might not be intuitive. In general, assuming we had many variables and many comparisons, we’d like to change all to the pattern: VAR is not None and VAR < CONST
The goal of our process will be to transform the method to:
def foo(x): return x is not None and x < 1
Where the aspect we’re applying is Update Comparison of None and Constants.
The steps required by this solution are the following:
Like the common approach, we will use a decorator to modify the function. We will start from this simple decorator and build over it:
def rewrite_comparisons(original_function): assert isinstance(original_function, FunctionType) return original_function @rewrite_comparisons def foo(x): return x < 1
This decorator does nothing, so far.
The first challenge is getting the method code from a function and make it modifiable. Since Python provides bytecode by default for a method, we will use built-in inspect to extract the original source code:
function_source_code = inspect.getsource(original_function)
Inspect uses the code locations linked to the function and read them from the source file. The return value is a string with function. This is different from disassembling code from the method bytecode.
We can assume that for our functions the source code is available. Otherwise, this first step will fail, and the processing will need to be in bytecode level (which might be covered in other post). In addition, this constrains us to ensure decorator is called before any other decorator. Otherwise, previous decorators might be ignore since their effect is not reflected in the original source code.
After the previous line of code extracted the source, we can parse it to an AST. The motivation for building an abstract syntax tree, is that it’s modifiable and we can compile is back to bytecode.
function_source_code = inspect.getsource(original_function) node = ast.parse(function_source_code)
The node we get is the root one of the parsed code. It links to all the elements in the hierarchy and represents a simplified module code.
Taking for example the foo function, the tree is:
Module # the method declaration (foo) FunctionDef # the arguments list (x) arguments Name Param # return instruction Return # comparison of two elements Compare # load variable (x) Name Load # comparison operator (<) Lt # load constant (1) Num
The AST represents the function, while the decorator is omitted for simplicity. As can easily be seen, the tree represents all the content of the method, including declaration, other methods in context if there are and more. Given the AST, we’d like to modify it a fit the need that our aspect requires.
We will use the AST visitors as an introduction to syntax tree traversal. The node visitors follow a convention where callback names are of pattern visit_NODETYPE(self, node), where node type can be any these. For example, if we want a callback on method calls, we can define one for the Call node and name it visit_Call(self, node).
In our example, we can visit the compare nodes, and print all the operands:
from ast import NodeVisitor class ComparisonVisitor(NodeVisitor): def visit_Compare(self, node): operands = [node.left] + node.comparators print '; '.join(type(operand).__name__ for operand in operands)
For every callback, we are assured the type of the node fits the Compare node type. Given the type, we can investigate it’s members. Comparison in Python is composed of operators (one or more) and operands (two or more). In the case of Compare node, the first operand is called left, and the rest are called comparators. One of the reason for the complicated structure is to support expressions like:
0 < x < 100
Using the visitor we can query the nodes, but not modify them. If we visit the the original foo function:
</pre> node = ast.parse(inspect.getsource(foo)) ComparisonVisitor().visit(node)
The result we expect is:
Name; Num
Since comparison is x < 1, where x is Name load in the context and 1 is a Constant Number in the context.
Python provides transformers, which are a special type of AST visitors. The transformers, in contrast to nodes visitors, modify the nodes they visit. In our example, we’ll look for nodes that represent comparison between variables and numbers, and then extend them to comply with the aspect.
from ast import NodeTransformer from _ast import BoolOp, And, Name, Compare, IsNot, Load, Num class ComparisonTransformer(NodeTransformer): def visit_Compare(self, node): parts = [node.left] + node.comparators # check if any constant number is involved in the comparison if not any(isinstance(part, Num) for part in parts): return node # get all the "variables" involved in the comparison names = [element for element in parts if isinstance(element, Name)] if len(names) == 0: return node # create a reference to None none = Name(id='None', ctx=Load()) # create for each variable a node that represents 'var is not None' node_verifiers = [Compare(left=name, ops=[IsNot()], comparators=[none]) for name in names] # combine the None checks with the original comparison # e.g. 'a < b < 1' --> 'a is not None and b is not None and a < b < 1 return BoolOp(op=And(), values=node_verifiers + [node])
This chunk of code is a simplified (relaxed type input checks no attempts to code location fixes) version of a transformer that visits all nodes of type Compare. The transformer methods names use the same convention as the visitors.
According to the original behavior a new node is being built. This node is a new Boolean expression, which requires all the variables[2] in use to be not None and to satisfy the original comparison.
If we’d look at the output, the the AST will be modified and verify variables are not None before they’re compared to None. The out tree for the modified foo is:
Module # the method declaration (foo) FunctionDef # the arguments list (x) arguments Name Param # return instruction Return # the bool expression that combines with And: # 1. the original comparison # 2. the new check 'VAR is not None' BoolOp And # the 'x is not None' comparison Compare Name Load IsNot Name Load # the original comparison 'x < 1' Compare Name Load Lt Num
In the next phase, we’re going to import the new code as temporary module, which will case the declaration of the new method to be executed again. In order to do so, we’d like to remove the rewriter decorator, since we don’t want it to process the modified function. In addition, we rename the function for safety to avoid collisions between the declared function and other locals. Lastly, we ask python to fix code locations for the new nodes so they can be compiled later on. This is done using fix_missing_locations.
from ast fix_missing_locations def rewrite_method(node): # assuming the method has single decorator (which is the rewriter) - remove it node.body[0].decorator_list.pop() # we rename the method to ensure separation from the original one. # this step has no real meaning and not really required. node.body[0].name = 'internal_method' # transform Compare nodes to fit the 'is not None' requirement ComparisonTransformer().visit(node) # let python try and fill code locations for the new elements fix_missing_locations(node)
During the first phase we got as an input a function (through a decorator), then modified it’s body by visiting it’s body using a syntactic level. Lastly, we modified it’s declaration and source locations so it can be safely imported as a new function.
As you probably notice, the only part in this code which is concerned by the aspect is the transformer. Meaning, if we’d like to apply a different aspect the only part which’ll change is the transformer. In our example the ComparisonTransformer is hard-coded for simplicity, but in real solution we’d provide it as an argument to the decorator.
In the next phase we’ll use the modified function to generate replacement bytecode.
The code complexity extension for Visual-Studio is now compatible with 2015 version.
I’d like to thank Alex Galin for adding support to additional C# language constructs.
The new version is available at the Visual-Studio Gallery and the source code at CodePlex.
QuickSelect is a known and simple algorithm for finding the kth smallest element in an array. The advantages of this algorithm are its linear average performance and the constant memory it requires, in addition to simple implementation.
class QuickSelect
constructor: (@arr) ->
kth: (k) ->
@_select(0, @arr.length - 1, k)
median: ->
@kth Math.floor(@arr.length / 2)
_swap: (i, j) ->
tmp = @arr[i]
@arr[i] = @arr[j]
@arr[j] = tmp
_partition: (left, right, pivotIndex) ->
pivotValue = @arr[pivotIndex]
@_swap pivotIndex, right
storeIndex = left
for i in [left..right]
if @arr[i] < pivotValue
@_swap storeIndex, i
storeIndex++
@_swap right, storeIndex
return storeIndex
_choose_random_pivot: (left, right) ->
left + Math.floor(Math.random() * (right - left + 1))
_select: (left, right, k) ->
if left == right
return @arr[left]
while true
pivotIndex = @_choose_random_pivot left, right
pivotIndex = @_partition(left, right, pivotIndex)
if k == pivotIndex
return @arr[k]
else if k < pivotIndex
right = pivotIndex - 1
else
left = pivotIndex + 1
describe "QuickSelect", ->
describe "1 item", ->
selector = new QuickSelect [1]
it "Should return the item", -> selector.kth(0).should.equal(1)
describe "2 items", ->
selector = new QuickSelect [2,1]
it "Min should be 1", -> selector.kth(0).should.equal(1)
it "Max should be 2", -> selector.kth(1).should.equal(2)
describe "3 items", ->
selector = new QuickSelect [2,1,3]
it "Min should be 1", -> selector.kth(0).should.equal(1)
it "Second should be 2", -> selector.kth(1).should.equal(2)
it "Max should be 3", -> selector.kth(2).should.equal(3)
These simple tests explains how the access to kth element is done. The tests are written also in CoffeeScript.
Array size | 5,000 | 10,000 | 20,000 |
QuickSelect | 1758ms | 3393ms | 6912ms |
Sort | 14302ms | 32136ms | 70566ms |