Dynamization of Static Data Structures

Most of us in software engineering have been there. You design a static data structure that supports blazing-fast queries for counting elements, searching patterns, or similar tasks. However, the moment you need to support insertions, performance collapses because rebuilding from scratch after every update is too slow.

If you're lucky, you might find (or invent!) a different data structure which:

1. Supports cheap runtime insertion
2. Offers comparable query performance
3. Has reasonable implementation/maintenance complexity

Otherwise, you may benefit from dynamization — a powerful technique that, by just adding a thin boilerplate orchestration layer, allows you to:

1. Reuse your static algorithm (almost) verbatim
2. Get solid query/update performance 
3. Keep implementation effort low

There are, of course, some limitations on its applicability, as well as performance trade-offs to consider. We'll cover these in a later section.

Overview of the Dynamization Algorithm

Core Idea

The core idea behind the Dynamization algorithm is pretty simple: instead of maintaining one monolithic structure, you keep several smaller, disjoint static substructures. 

1. Queries are supported by independently probing each of these substructures and aggregating the results.
2. Insertions are supported by occasionally rebuilding (or merging) a few substructures, amortizing the cost over many updates

By using an exponential merging strategy, dynamization ensures that the number of substructures remains small: on the order of O(log N). This allows us to support both queries and updates with low overhead.

Applicability

This technique can only be used if the query operation is decomposable, i.e, the query on a union of sets is a combination of queries on each set. 

Some examples of decomposable operations where Dynamization can be applied are:

• sum → add partial sums 
• count_if → add partial counts
• min/max→ take the smallest/largest of the partial minimums

Some examples of non-decomposable functions that won't work with Dynamization are: 

• median, 90th percentile, mode → you can’t get the overall answer from the partial answers alone

Performance Considerations

1. Storage overhead: Every element is part of exactly one substructure at a time, and thus, dynamization does not add any storage overhead.
2. Query overhead: Querying requires probing O(log N) substructures independently and merging the results, thus adding an overhead of O(log N) on top of the static structure. 
   • If the static structure answered in O(1), dynamization can answer in O(log N)
   • If the static structure answered in O(log N), dynamization can answer in O(log²N)
3. Insertion overhead: Insertions may trigger merges, but by always at least doubling the size of the merged structures, you can achieve a low amortized complexity for insertions.
   • If merging costs O(k), insertions cost O(log N) amortized, where k is the number of elements being merged in that operation.

Real-World Applications

1. Scheduling Workloads

In scheduling systems, new tasks are often ingested daily into partitions keyed by their arrival date, but queries are typically keyed by the execution date. A careless implementation can lead to scanning thousands of arrival date partitions every day just to filter out most rows. By applying dynamization, you can efficiently maintain substructures indexed by execution date, so queries like “which tasks run tomorrow?” become efficient without wasted scans. This kind of pattern appears in many domains, from job schedulers that enqueue tasks for future deletion or data cleanup, to video platforms that manage scheduled publication or expiration dates for content.

2. Dynamic Keyword Enforcement

Imagine you’re running a content moderation system where new keywords or banned phrases are added every day, and you need to scan large volumes of text for matches. Aho–Corasick is the go-to algorithm for matching text against a fixed set of patterns, but it is built for static keyword lists, and rebuilding is required to support adding new elements to the list, which can be prohibitively expensive for frequent insertions. With dynamization, you maintain smaller automatons that grow and merge over time, letting queries run across them efficiently even as the keyword set expands. The same idea applies beyond text as well in video moderation, where new fingerprints or signature hashes are continuously added to detect copyrighted or restricted content.

3. Video Tagging Pipelines

Imagine you are running a video platform where every new upload is analyzed to generate embeddings, which are later used for search and recommendation. Over time, the embedding library grows continuously as more videos are uploaded. Queries often need to find the nearest neighbors to a given embedding, to identify similar videos, or to compute aggregates like the average embedding of a collection of videos. Specialized vector databases and indexes exist for these operations, but they are often optimized for batch construction and can be expensive to update online. With dynamization, you can maintain and query multiple smaller embedding indexes, letting you scale similarity search and aggregate embedding statistics efficiently while avoiding the complexity of maintaining a fully dynamic vector index.

Limitations

Dynamization isn’t a magic wand. It’s a clever technique for extending static data structures, but it comes with caveats and should be applied thoughtfully. In the wrong setting, the overheads may outweigh the benefits. Here are some limitations of dynamization: 

1. Adds extra query overhead. Every query must probe multiple substructures, which adds a logarithmic factor compared to the static version. Depending on your use case and latency requirements, building an alternate data structure that supports insertions natively may be beneficial. 
2. Does not support universal querying. Dynamization requires decomposable queries as discussed in the "Applicability" section.
3. Does not natively support the removal of elements. Dynamization is built for insertions. Supporting removal typically requires tombstones, lazy cleanup, or periodic full rebuilds.

Further Reading

This article has focused on the high-level concepts and practical applications of dynamization. In part 2, I’ll explain the full algorithm and share a sample implementation. 

Readers may learn more from Jon Bentley and James B. Saxe’s 1980 paper Decomposable Searching Problems.