In this post, I'll describe a neat trick for maintaining a summary quantity (e.g., sum, product, max, log-sum-exp, concatenation, cross-product) under changes to its inputs. The trick and it's implementation are inspired by the well-known max-heap datastructure. I'll also describe a really elegant application to fast sampling under an evolving categorical distribution.

Setup: Suppose we'd like to efficiently compute a summary quantity under changes to its \(n\)-dimensional input vector \(\boldsymbol{w}\). The particular form of the quantity we're going to compute is \(z = \bigoplus_{i=1}^n w_i\), where \(\oplus\) is some associative binary operator with identity element \(\boldsymbol{0}\).

more formally...

\(\boldsymbol{w} \in \boldsymbol{K}^n\)

\(\oplus: \boldsymbol{K} \times \boldsymbol{K} \mapsto \boldsymbol{K}\).

Associative: \((a \oplus b) \oplus c = a \oplus (b \oplus c)\) for all \(a,b,c \in \boldsymbol{K}\).

Identity element: \(\boldsymbol{0} \in \boldsymbol{K}\) such that \(k \oplus \boldsymbol{0} = \boldsymbol{0} \oplus k = k\), for all \(k \in \boldsymbol{K}\).

The trick: Essentially, the trick boils down to parenthesis placement in the expression which computes \(z\). A freedom we assumed via the associative property.

I'll demonstrate by example with \(n=8\).

Linear structure: We generally compute something like \(z\) with a simple loop. This looks like a right-branching binary tree when we think about the order of operations,

$$ z = (((((((w_1 \oplus w_2) \oplus w_3) \oplus w_4) \oplus w_5) \oplus w_6) \oplus w_7) \oplus w_8). $$



Heap structure: Here the parentheses form a balanced tree, which looks much more like a recursive implementation that computes the left and right halves and \(\oplus\)s the results (divide-and-conquer style),

$$ z = (((w_1 \oplus w_2) \oplus (w_3 \oplus w_4)) \oplus ((w_5 \oplus w_6) \oplus (w_7 \oplus w_8))). $$



The benefit of the heap structure is that there are \(\mathcal{O}(\log n)\) intermediate quantities that depend on any input, whereas the linear structure has \(\mathcal{O}(n)\). The intermediate quantities correspond to the values of each of the parenthesized expressions.

Since fewer intermediate quantities depend on a given input, fewer intermediates need to be adjusted upon a change to the input. Therefore, we get faster algorithms for maintaining the output quantity \(z\) as the inputs change.

Heap datastructure (aka binary index tree or Fenwick tree): We're going to store the values of the intermediates quantities and inputs in a heap datastructure, which is a complete binary tree. In our case, the tree has depth \(1 + \lceil \log_2 n \rceil\), with the values of \(\boldsymbol{w}\) at it's leaves (aligned left) and padding with \(\boldsymbol{0}\) for remaining leaves. Thus, the array's length is \(< 4 n\).

This structure makes our implementation really nice and efficient because we don't need pointers to find the parent or children of a node (i.e., no need to wrap elements into a "node" class like in a general tree data structure). So, we can pack everything into an array, which means our implementation has great memory/cache locality and low storage overhead.

Traversing the tree is pretty simple: Let \(d\) be the number of internal nodes, nodes \(1 \le i \le d\) are internal. For node \(i\), left child \(\rightarrow {2 \cdot i},\) right child \(\rightarrow {2 \cdot i + 1},\) parent \(\rightarrow \lfloor i / 2 \rfloor.\) (Note that these operations assume the array's indices start at \(1\). We generally fake this by adding a dummy node at position \(0\), which makes implementation simpler.)

Initializing the heap: Here's code that initializes the heap structure we just described.

def sumheap ( w ): "Create sumheap from weights `w` in O(n) time." n = w . shape [ 0 ] d = int ( 2 ** np . ceil ( np . log2 ( n ))) # number of intermediates S = np . zeros ( 2 * d ) # intermediates + leaves S [ d : d + n ] = w # store `w` at leaves. for i in reversed ( range ( 1 , d )): S [ i ] = S [ 2 * i ] + S [ 2 * i + 1 ] return S

Updating \(w_k\) boils down to fixing intermediate sums that (transitively) depend on \(w_k.\) I won't go into all of the details here, instead I'll give code (below). I'd like to quickly point out that the term "parents" is not great for our purposes because they are actually the dependents: when an input changes the value the parents, grand parents, great grand parents, etc, become stale and need to be recomputed bottom up (from the leaves). The code below implements the update method for changing the value of \(w_k\) and runs in \(\mathcal{O}(\log n)\) time.

def update ( S , k , v ): "Update w[k] = v` in time O(log n)." d = S . shape [ 0 ] i = d // 2 + k S [ i ] = v while i > 0 : # fix parents in the tree. i //= 2 S [ i ] = S [ 2 * i ] + S [ 2 * i + 1 ]

Remarks

Numerical stability : If the operations are noisy (e.g., floating point operator), then the heap version may be better behaved. For example, if operations have an independent, additive noise rate \(\varepsilon\) then noise of \(z_{\text{heap}}\) is \(\mathcal{O}(\varepsilon \cdot \log n)\), whereas \(z_{\text{linear}}\) is \(\mathcal{O}(\varepsilon \cdot n)\). (Without further assumptions about the underlying operator, I don't believe you can do better than that.)

Relationship to max-heap : In the case of a max or min heap, we can avoid allocating extra space for intermediate quantities because all intermediates values are equal to exactly one element of \(\boldsymbol{w}\).

Change propagation: The general idea of adjusting cached intermediate quantities is a neat idea. In fact, we encounter it each time we type make at the command line! The general technique goes by many names—including change propagation, incremental maintenance, and functional reactive programming—and applies to basically any side-effect-free computation. However, it's most effective when the dependency structure of the computation is sparse and requires little overhead to find and refresh stale values. In our example of computing \(z\), these considerations manifest themselves as the heap vs linear structures and our fast array implementation instead of a generic tree datastructure.

Generalizations

No zero? No problem. We don't actually require a zero element. So, it's fair to augment \(\boldsymbol{K} \cup \{ \textsf{null} \}\) where \(\textsf{null}\) is distinguished value (i.e., \(\textsf{null}

otin \boldsymbol{K}\)) that acts just like a zero after we overload \(\oplus\) to satisfy the definition of a zero (e.g., by adding an if-statement).

Generalization to an arbitrary maps instead of fixed vectors is possible with a "locator" map, which a bijective map from elements to indices in a dense array.

Support for growing and shrinking: We support growing by maintaining an underlying array that is always slightly larger than we need—which we're already doing in the heap datastructure. Doubling the size of the underlying array (i.e., rounding up to the next power of two) has the added benefit of allowing us to grow \(\boldsymbol{w}\) at no asymptotic cost! This is because the resize operation, which requires an \(\mathcal{O}(n)\) time to allocate a new array and copying old values, happens so infrequently that they can be completely amortized. We get of effect of shrinking by replacing the old value with \(\textsf{null}\) (or \(\boldsymbol{0}\)). We can shrink the underlying array when the fraction of nonzeros dips below \(25\%\). This prevents "thrashing" between shrinking and growing.

Application

Sampling from an evolving distribution: Suppose that \(\boldsymbol{w}\) corresponds to a categorical distributions over \(\{1, \ldots, n\}\) and that we'd like to sample elements from in proportion to this (unnormalized) distribution.

Other methods like the alias or inverse CDF methods are efficient after a somewhat costly initialization step. But! they are not as efficient as the heap sampler when the distribution is being updated. (I'm not sure about whether variants of alias that support updates exist.)

Method Sample Update Init alias O(1) O(n)? O(n) i-CDF O(log n) O(n) O(n) heap O(log n) O(log n) O(n)

Use cases include

Gibbs sampling, where distributions are constantly modified and sampled from (changes may not be sparse so YMMV). The heap sampler is used in this paper.

EXP3 (mutli-armed bandit algorithm) is an excellent example of an algorithm that samples and modifies a single weight in the distribution.

Stochastic priority queues where we sample proportional to priority and the weights on items in the queue may change, elements are possibly removed after they are sampled (i.e., sampling without replacement), and elements are added.

Again, I won't spell out all of the details of these algorithms. Instead, I'll just give the code.

Inverse CDF sampling

def sample ( w ): "Ordinary sampling method, O(n) init, O(log n) per sample." c = w . cumsum () # build cdf, O(n) p = uniform () * c [ - 1 ] # random probe, p ~ Uniform(0, z) return c . searchsorted ( p ) # binary search, O(log n)

Heap sampling is essentially the same, except the cdf is stored as heap, which is perfect for binary search!

def hsample ( S ): "Sample from sumheap, O(log n) per sample." d = S . shape [ 0 ] // 2 # number of internal nodes. p = uniform () * S [ 1 ] # random probe, p ~ Uniform(0, z) # Use binary search to find the index of the largest CDF (represented as a # heap) value that is less than a random probe. i = 1 while i < d : # Determine if the value is in the left or right subtree. i *= 2 # Point at left child left = S [ i ] # Probability mass under left subtree. if p > left : # Value is in right subtree. p -= left # Subtract mass from left subtree i += 1 # Point at right child return i - d

Code: Complete code and test cases for heap sampling are available in this gist. A fast Cython implementation is available in my Python arsenal.