It rained in Sunriver on New Year’s Eve, and it continued to rain for the next couple of days. So, instead of going skiing as planned, to the dismay of my family and friends, I spent the majority of those days working on the third language for homework 3. It was quite the language, too - the homework has three problems, each of which has a solution independent of the others. I invite you to join me in my descent into madness as we construct another language.
Homework 3
Let’s take a look at the three homework problems. The first two are related, but are solved using a different technique:
1. Given an array A of n numbers, a query x, and a number k,
find the k numbers in A that are closest (in value) to x.
For example:
find([4,1,3,2,7,4], 5.2, 2) returns [4,4]
find([4,1,3,2,7,4], 6.5, 3) returns [4,7,4]
find([5,3,4,1,6,3], 3.5, 2) returns [3,4]
Filename: closest_unsorted.py
Must run in O(n) time.
The elements in the returned list must be in the original order.
In case two numbers are equally close to x, choose the earlier one.
This problem requires us to find the k
numbers closest to some
query (which I will call n
) from a list xs
. The list isn’t sorted, and the
problem must run in linear time. Sorting the list would require
the standard
[note:
The $n$ in this expression is not the same as the query n
,
but rather the length of the list. In fact, I have not yet assigned
the length of the input xs
to any variable. If we say that
$m$ is a number that denotes that length, the proper expression
for the complexity is $O(m \log m)$.
]
Thus, we have to take another route, which should
already be familiar: quickselect. Using quickselect, we can find the k
th
closest number, and then collect all the numbers that are closer than the kth
closest number. So, we need a language that:
- Supports quickselect (and thus, list partitioning and recursion).
- Supports iteration,
[note:
Why would we need to iterate multiple times? Note that we could have a list
of numbers that are all the same,
[1,1,1,1,1]
. Then, we'll need to know how many of the numbers equally close as thek
th element we need to include, which will require another pass through the list. ]
That’s a good start. Let’s take a look at the second problem:
2. [WILL BE GRADED]
Now what if the input array is sorted? Can you do it faster?
find([1,2,3,4,4,7], 5.2, 2) returns [4,4]
find([1,2,3,4,4,7], 6.5, 3) returns [4,4,7]
Filename: closest_sorted.py
Must run in O(logn + k) time.
The elements in the returned list must be in the original order.
Note: in case two numbers are equally close to x, choose the smaller one:
find([1,2,3,4,4,6,6], 5, 3) returns [4,4,6]
find([1,2,3,4,4,5,6], 4, 5) returns [2,3,4,4,5]
Hint: you can use Python's bisect.bisect for binary search.
This problem really is easier. We have to find the position of the closest element, and then try expand towards either the left or right, depending on which end is better. This expansion will take several steps, and will likely require a way to “look” at a given part of the list. So let’s add two more rules. We need a language that also:
- Supports looping control flow, such as
while
. - [note: We could, of course, simply use list indexing. But then, we'd just be making a simple imperative language, and that's boring. So let's play around with our design a little, and experimentally add such a "list view" component. ] (like an abstraction over indexing).
This is shaping up to be a fun language. Let’s take a look at the last problem:
3. For a given array A of n *distinct* numbers, find all triples (x,y,z)
s.t. x + y = z. (x, y, z are distinct numbers)
e.g.,
find([1, 4, 2, 3, 5]) returns [(1,3,4), (1,2,3), (1,4,5), (2,3,5)]
Note that:
1) no duplicates in the input array
2) you can choose any arbitrary order for triples in the returned list.
Filename: xyz.py
Must run in O(n^2) time.
Hint: you can use any built-in sort in Python.
This problem requires more iterations of a list. We have several [note: I always make the language before I write the post, since a lot of design decisions change mid-implementation. I realize now that "cursors" would've been a better name for this language feature, but alas, it is too late. ] looking into the list, and depending if the values at each of the cursors add up, we do or do not add a new tuple to a list. So, two more requirements:
- The “cursors” must be able to interact.
- The language can represent [note: We could, of course, hack some other way to return a list of tuples, but it turns out tuples are pretty simple to implement, and help make for nicer programming in our language. ]
I think we’ve gathered what we want from the homework. Let’s move on to the language!
A Language
As is now usual, let’s envision a solution to the problems in our language. There
are actually quite a lot of functions to look at, so let’s see them one by one.
First, let’s look at qselect
.
function qselect(xs, k, c) {
if xs == [] {
return 0;
}
traverser bisector(list: xs, span: (0,len(xs)));
traverser pivot(list: xs, random: true);
let pivotE = pop!(pivot);
let (leftList, rightList) = bisect!(bisector, (x) -> c(x) < c(pivotE));
if k > len(leftList) + 1 {
return qselect(rightList, k - len(leftList) - 1, c);
} elsif k == len(leftList) + 1 {
return pivotE;
} else {
return qselect(leftList, k, c);
}
}
After the early return, the first interesting part of the language is the use of what I have decided to call a list traverser. The list traverser is a generalization of a list index. Whenever we use a list index variable, we generally use the following operations:
- Initialize: we set the list index to some initial value, such as 0.
- Step: If we’re walking the list from left to right, we increment the index. If we’re walking the list from right to left, we decrement the index.
- Validity Check: We check if the index is still valid (that is, we haven’t gone past the edge of the list).
- Access: Get the element the cursor is pointing to.
A
[note:
A fun fact is that we've just rediscovered C++
iterators. C++
containers and their iterators provide us with the operations I described:
We can initialize an iterator like auto it = list.begin()
. We
can step the iterator using it++
. We can check its validity
using it != list.end()
, and access what it's pointing to using
*it
. While C++ uses templates and inheritance for this,
we define a language feature specifically for lists.
]
describes these operations. The declartion for the bisector
traverser creates a “cursor” over the list xs
, that goes between the 0th
and last elements of xs
. The declaration for the pivot
traverser creates
a “cursor” over the list xs
that jumps around random locations in the list.
The next interesting part of the language is a traverser macro. This thing,
that looks like a function call (but isn’t), performs an operation on the
cursor. For instance, pop!
removes the element at the cursor from the list,
whereas bisect!
categorizes the remaining elements in the cursor’s list
into two lists, using a boolean-returning lambda (written in Java syntax).
Note that this implementation of qselect
takes a function c
, which it
uses to judge the actual value of the number. This is because our qselect
won’t be finding the smallest number, but the number with the smallest difference
with n
. n
will be factored in via the function.
Next up, let’s take a look at the function that uses qselect
, closestUnsorted
:
function closestUnsorted(xs, k, n) {
let min = qselect(list(xs), k, (x) -> abs(x - n));
let out = [];
let countEqual = k;
traverser iter(list: xs, span: (0, len(xs)));
while valid!(iter) {
if abs(at!(iter)-n) < abs(min-n) {
let countEqual = countEqual - 1;
}
step!(iter);
}
traverser iter(list: xs, span: (0, len(xs)));
while valid!(iter) {
if abs(at!(iter)-n) == abs(min-n) and countEqual > 0 {
let countEqual = countEqual - 1;
let out = out + [at!(iter)];
} elsif abs(at!(iter)-n) < abs(min-n) {
let out = out + [at!(iter)];
}
step!(iter);
}
return out;
}
Like we discussed, it finds the k
th closest element (calling it min
),
and counts how many elements that are equal need to be included,
by setting the number to k
at first, and subtracting 1 for every number
it encounters that’s closer than min
. Notice that we use the valid!
and
step!
macros, which implement the operations we described above. Notice
that the user doesn’t deal with adding and subtracting numbers, and doing
comparisons. All they have to do is ask “am I still good to iterate?”
Next, let’s take a look at closestSorted
, which will require more
traverser macros.
function closestSorted(xs, k, n) {
let start = bisect(xs, n);
let counter = 0;
traverser left(list: xs, span: (0, start), reverse: true);
traverser right(list: xs, span: (start, len(xs)));
while counter != k and canstep!(left) and valid!(right) {
if abs(at!(left, 1) - n) < abs(at!(right) - n) {
step!(left);
} else {
step!(right);
}
let counter = counter + 1;
}
while counter != k and (canstep!(left) or valid!(right)) {
if canstep!(left) { step!(left); }
else { step!(right); }
let counter = counter + 1;
}
return subset!(left, right);
}
The first new macro is canstep!
. This macro just verifies that
the traverser can make another step. We need this for the “reverse” iterator,
which indicates the lower bound of the range of numbers we want to return,
because subset!
(which itself is just Python’s slice, like xs[a:b]
), uses an inclusive bottom
index, and thus, we can’t afford to step it before knowing that we can, and that
it’s a better choice after the step.
Similarly, we have the at!(t, i)
macro, which looks at the
traverser t
, with offset i
.
We have two loops. The first loop runs as long as we can expand the range in both directions, and picks the better direction at each iteration. The second loop runs as long as we still want more numbers, but have already hit the edge of the list on the left or on the right.
Finally, let’s look at the solution to xyz
:
sorted function xyz(xs, k) {
traverser x(list: xs, span: (0,len(xs)));
let dest = [];
while valid!(x) {
traverser z(list: xs, span: (pos!(x)+2,len(xs)));
traverser y(list: xs, span: (pos!(x)+1,pos!(z)));
while valid!(y) and valid!(z) {
if at!(x) + at!(y) == at!(z) {
let dest = dest + [(at!(x), at!(y), at!(z))];
step!(z);
} elsif at!(x) + at!(y) > at!(z) {
step!(z);
} else {
step!(y);
}
}
step!(x);
}
return dest;
}
I won’t go in depth, but notice that the expression in the span
part
of the traverser
declaration can access another traverser. We treat
as a feature the fact that this expression isn’t immediately evaluated at the place
of the traverser declaration. Rather, every time that a comparison for a traverser
operation is performed, this expression is re-evaluated. This allows us to put
dynamic bounds on traversers y
and z
, one of which must not exceed the other.
Note also a new keyword that was just used: sorted
. This is a harmless little
language feature that automatically calls .sort()
on the first argument of
the function.
This is more than enough to work with. Let’s move on to the implementation.
Implementation
Again, let’s not go too far into the details of implementing the language from scratch. Instead, let’s take a look into specific parts of the language that deserve attention.
Revenge of the State Monad
Our previous language was, indeed, a respite from complexity. Translation was straightforward, and the resulting expressions and statements were plugged straight into a handwritten AST. We cannot get away with this here; the language is powerful enough to implement three list-based problems, which comes at the cost of increased complexity.
We need, once again, to generate temporary variables. We also need to keep track of
which variables are traversers, and the properties of these traversers, throughout
each function of the language. We thus fall back to using Control.Monad.State
:
type Translator = State (Map.Map String ValidTraverserData, [Py.PyStmt], Int)
There’s one part of the state tuple that we haven’t yet explained: the list of statements.
Generating Statements
Recall that our translation function for expressions in the first homework had the type:
translateExpr :: Expr -> Translator ([Py.PyStmt], Py.PyExpr)
We then had to use do
-notation, and explicitly concatenate lists
of emitted statements. In this language, I took an alternative route: I made
the statements part of the state. They are thus implicitly generated and
stored in the monad, and expression generators don’t have to worry about
concatenating them. When the program is ready to use the generated statements
(say, when an if
-statement needs to use the statements emitted by the condition
expression), we retrieve them from the monad:
collectStatements :: Translator a -> Translator ([Py.PyStmt], a)
collectStatements t = do
modify (first $ const [])
a t
ss gets $ \(_, ss, _) -> ss
modify (first $ const [])
return (ss, a)
I should note, for transparency, that there’s a bug in my use of this function.
When I compile if
-statements, I accidentally place statements generated by
the condition into the body of the if
. This bug doesn’t manifest
in the solutions to the homework problems, and so I decided not to spend any more
time on fixing it.
Validating Traverser Declarations
We declare two separate types that hold traverser data. The first is a kind of “draft”
type, TraverserData
:
data TraverserBounds = Range Py.PyExpr Py.PyExpr | Random
data TraverserData = TraverserData
{ list :: Maybe String
, bounds :: Maybe TraverserBounds
, rev :: Bool
}
This record holds all possible configurations of a traverser
that occur as the program is iterating through the various key: value
pairs in
the declaration. For instance, at the very beginning of processing a traverser declaration,
our program will use a “default” TraverserData
, with all fields set to Nothing
or
their default value. This value will then be modified by the first key/value pair,
changing, for instance, the list that the traverser operates on. This new modified
TraverserData
will then be modified by the next key/value pair, and so on. Doing
this with every key/value pair (called an option in the below snippet)
is effectively a foldl operation.
applyOption :: TraverserData -> (String, Py.PyExpr) -> Maybe TraverserData
applyOption td ("list", Py.Var s) =
return $ td { list = Just s }
applyOption td ("span", Py.TupleLiteral [f, t]) =
return $ td { bounds = Just $ Range f t }
applyOption td ("random", Py.BoolLiteral True) =
return $ td { bounds = Just Random }
applyOption td ("reverse", Py.BoolLiteral b) =
return $ td { rev = b }
applyOption td _ = Nothing
The data may not have all the required fields until the very end, and its type
reflects that: Maybe String
here, Maybe TraverserBounds
there. We don’t
want to deal with unwrapping the Maybe a
values every time we use the traverser,
especially if we’ve done so before. So, we define a ValidTraverserData
record
that does not have Maybe
arguments, and thus, has all the required data. At the
end of a traverser declaration, we attempt to translate a TraverserData
into
a ValidTraverserData
, invoking fail
if we can’t, and storing the ValidTraverserData
into the state otherwise:
translateStmt (Traverser s os) =
foldlM applyOption defaultTraverser mapM translateOption os >>= saveTraverser
where
saveTraverser :: Maybe TraverserData -> Translator Py.PyStmt
saveTraverser (Just (td@TraverserData { list = Just l, bounds = Just bs})) =
putTraverser s vtd $> translateInitialBounds s vtd
where
vtd = ValidTraverserData
{ validList = l
, validBounds = bs
, validRev = rev td
}
saveTraverser Nothing = fail "Invalid traverser (!)"
Then, every time we retrieve a traverser from the state, define a lookup monadic operation like this:
requireTraverser :: String -> Translator ValidTraverserData
requireTraverser s = gets (\(m, _, _) -> Map.lookup s m) >>= handleMaybe
where
handleMaybe Nothing = fail "Invalid traverser"
handleMaybe (Just vtd) = return vtd
Compiling Macros
I didn’t call them macros for no reason. Clearly, we don’t want to generate
code that
[note:
In fact, there's no easy way to do this at all. Python's integers (if we choose to
represent our traversers using integers), are immutable. Furthermore, unlike C++,
where passing by reference allows a function to change its parameters "outside"
the call, Python offers no way to reassign a different value to a variable given
to a function.
For an example use of C++'s pass-by-reference mechanic, consider std::swap
:
it's a function, but it modifies the two variables given to it. There's no
way to generically implement such a function in Python.
]
We also can’t allow arbitrary expressions to serve as traversers:
our translator keeps some context about which variables are traversers, what their
bounds are, and how they behave. Thus, calls to traverser macros are very much macros:
they operate on AST nodes, and require that their first argument is a variable,
named like the traverser. We use the requireTraverser
monadic operation
to get the traverser associated with the given variable name, and then perform
the operation as intended. The at!(t)
operation is straightforward:
translateExpr (TraverserCall "at" [Var s]) = do
l validList requireTraverser s
return $ Py.Access (Py.Var l) [Py.Var s]
The at!(t,i)
is less so, since it deals with the intricacies of accessing
the list at either a positive of negative offset, depending on the direction
of the traverser. We implement a function to properly generate an expression for the offset:
traverserIncrement :: Bool -> Py.PyExpr -> Py.PyExpr -> Py.PyExpr
traverserIncrement rev by e =
Py.BinOp op e (Py.BinOp Py.Multiply by (Py.IntLiteral 1))
where op = if rev then Py.Subtract else Py.Add
We then implement at!(t,i)
as follows:
translateExpr (TraverserCall "at" [Var s, IntLiteral i]) = do
vtd requireTraverser s
return $ Py.Access (Py.Var $ validList vtd)
[traverserIncrement (validRev vtd) (Py.IntLiteral i) (Py.Var s)]
The most complicated macro is bisect!
. It must be able to step the traverser,
and also return a tuple of two lists that the bisection yields. We also
prefer that it didn’t pollute the environment with extra variables. To
achieve this, we want bisect!
to be a function call. We want this
function to implement the iteration and list construction.
bisect!
, by definition, takes a lambda. This lambda, in our language, is declared
in the lexical scope in which bisect!
is called. Thus, to guarantee correct translation,
we must do one of two things:
- Translate 1-to-1, and create a lambda, passing it to a fixed
bisect
function declared elsewhere. - Translate to a nested function declaration, [note: Inlining, in this case, means replacing a call to a function with the function's body. We do this to prevent the overhead of calling a function, which typically involves pushing on a stack and other extraneous work. If our function is simple, like a simple comparison, it doesn't make sense to spend the effort calling it. ]
Since I quite like the idea of inlining a lambda, let’s settle for that. To do this,
we pull a fresh temporary variable and declare a function, into which we place
the traverser iteration code, as well as the body of the lambda, with the variable
substituted for the list access expression.
[note:
Reading the lexical scope is one thing, but modifying it is another. To prevent
accidental changes to the variables outside a nested function, Python assumes
that variables assigned inside the function body are local to the function. Thus, to make
sure changing our variable (the traverser index) has an effect outside the function
(as it should) we must include the nonlocal
keyword, telling
Python that we're not declaring a new, local variable, but mutating the old one.
]
translateExpr (TraverserCall "bisect" [Var s, Lambda [x] e]) = do
vtd requireTraverser s
newTemp freshTemp
lambdaExpr translateExpr e
let access = Py.Access (Py.Var $ validList vtd) [Py.Var s]
let translated = substituteVariable x access lambdaExpr
let append s = Py.FunctionCall (Py.Member (Py.Var s) "append") [ access ]
let bisectStmt = Py.FunctionDef newTemp []
[ Py.Nonlocal [s]
, Py.Assign (Py.VarPat "l") (Py.ListLiteral [])
, Py.Assign (Py.VarPat "r") (Py.ListLiteral [])
, Py.While (traverserValid (Py.Var s) vtd)
[ Py.IfElse translated
[ Py.Standalone $ append "l" ]
[]
(Just [ Py.Standalone $ append "r" ])
, traverserStep s vtd
]
, Py.Return $ Py.TupleLiteral [Py.Var "l", Py.Var "r"]
]
emitStatement bisectStmt
return $ Py.FunctionCall (Py.Var newTemp) []
The Output
Let’s see what the compiler spits out:
from bisect import bisect
import random
def qselect(xs,k,c):
if xs==[]:
return 0
bisector = 0
pivot = random.randrange(len(xs))
pivotE = xs.pop(pivot)
def temp1():
nonlocal bisector
l = []
r = []
while bisector<len(xs):
if c(xs[bisector])<c(pivotE):
l.append(xs[bisector])
else:
r.append(xs[bisector])
bisector = bisector+1
return (l, r)
(leftList,rightList) = temp1()
if k>len(leftList)+1:
return qselect(rightList, k-len(leftList)-1, c)
elif k==len(leftList)+1:
return pivotE
else:
return qselect(leftList, k, c)
def closestUnsorted(xs,k,n):
min = qselect(list(xs), k, (lambda x: abs(x-n)))
out = []
countEqual = k
iter = 0
while iter<len(xs):
if abs(xs[iter]-n)<abs(min-n):
countEqual = countEqual-1
iter = iter+1
0
iter = 0
while iter<len(xs):
if abs(xs[iter]-n)==abs(min-n) and countEqual>0:
countEqual = countEqual-1
out = out+[xs[iter]]
elif abs(xs[iter]-n)<abs(min-n):
out = out+[xs[iter]]
iter = iter+1
0
return out
def closestSorted(xs,k,n):
start = bisect(xs, n)
counter = 0
left = start
right = start
while counter!=k and left-1*1>=0 and right<len(xs):
if abs(xs[left-1*1]-n)<abs(xs[right]-n):
left = left-1
0
else:
right = right+1
0
counter = counter+1
while counter!=k and (left-1*1>=0 or right<len(xs)):
if left-1*1>=0:
left = left-1
0
else:
right = right+1
0
counter = counter+1
return xs[(left):(right)]
def xyz(xs,k):
xs.sort()
x = 0
dest = []
while x<len(xs):
z = x+2
y = x+1
while y<z and z<len(xs):
if xs[x]+xs[y]==xs[z]:
dest = dest+[(xs[x], xs[y], xs[z])]
z = z+1
0
elif xs[x]+xs[y]>xs[z]:
z = z+1
0
else:
y = y+1
0
x = x+1
0
return dest
Observe that the generated code just uses indices, +
, -
, and various comparison operators.
Our traverser is an example of a zero cost abstraction, a feature that, conceptually,
operates at a higher level, making us no longer worry about adding, subtracting, and
comparing numbers, while, in the final output, not damaging the performance of safety
of the code. Also observe the various 0
standalone statements. This is an issue
with the translator: traverser macros may not always yield an expression, but
the type of translateExpr
and translateStmt
effectively requires one. Thus,
when a macro doesn’t generate anything useful, we give it the placeholder expression 0
.
That concludes this third post in the series. I hope to see you in the next one!