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Invariant

As explained in Advanced Algorithms and Data Structures

An invariant is an internal property of a data structure or mathematical object that remains true throughout a process or across permitted transformations.

An invariant is an internal property that remains true throughout a data structure’s lifetime or across permitted transformations. It is the implementation-side counterpart to an API guarantee: algorithms update the stored data while preserving the conditions required by external clients. section 2.4

For a sorted list, the invariant is that each element is no greater than its successor. A priority queue may promise that top() returns the highest-priority element; its concrete implementation must preserve internal conditions that make this guarantee reliable. section 2.4

Use invariants as implementation checkpoints:

  • State the property precisely before writing update operations. section 2.4
  • Update the data model without violating the property; splitting, merging, or rearranging may be necessary. section 2.4 section 10.2.2
  • Check the invariant after each structural change, because violating it can break the external API or make searches inefficient. section 2.4 section 10.2.2

R-tree maintenance

An R-tree requires every non-root leaf to contain between m and M points, every non-root internal node to contain between m and M children, and all leaves to share a level. Each node also stores the smallest axis-aligned rectangle containing its contents. Insertion splits an overflowing node; removal merges an underfull node with a sibling. These repairs restore the occupancy and bounding-rectangle invariants, while the shared leaf level keeps the tree balanced. section 10.2.2

Table 2.1 API and contract for priority queue (view table figure)

Abstract data structure: Priority queue

API

 
class PriorityQueue {
  top() → element
  peek() → element
  insert(element, priority)
  remove(element)
  update(element, newPriority)
  size() → int
}

Contract with client

The top element returned by the queue is always the element with highest priority currently stored in the queue

Use the priority-queue API table to compare external guarantees with the internal invariants that implementations must preserve. section 2.4

Figure 10.3 Cartesian plane representation of a (possible) R-tree for our city maps as presented in the example of figure 9.4 (the names of the cities are omitted to avoid confusion). This R-tree contains 12 bounding rectangles, from R1 to R12, organized in a hierarchical structure. Notice that rectangles can and do overlap, as shown in the bottom half.

Use the R-tree diagram to connect bounding-rectangle invariants with hierarchical spatial organization and overlapping regions. section 10.2.2

How Advanced Algorithms and Data Structures covers it

How deep it goes
Explains definition, API relationship, and structural maintenance through sorted lists, priority queues, and R-trees.
Examples
Uses sorted-list ordering, priority-queue guarantees, and R-tree occupancy and bounding rectangles as concrete examples.
What it assumes
Assumes familiarity with data structures, APIs, nodes, and algorithms while defining invariants from basic examples.
What you can do after
Build it from scratch by stating invariants and designing update operations that preserve them.
How it connects
Connects invariants with APIs, data models, algorithms, priority queues, heaps, B-trees, and R-trees.
Cover of Advanced Algorithms and Data Structures
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