Algorithms & Data Structures
the bound is a claim, and it is checkable
Concept

Tree shape — where it appears

Which code lengths a wavelet tree gives its symbols, and therefore how deep each leaf sits. It decides the structure’s size and the ranks an access costs, and separately whether the leaves are in the alphabet’s order — which decides what operations the tree can support at all.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

50318223749516674879310811positionthe minimum12 values · depth 3answer 6 (1)

The shape a range question is about

A range minimum is a question about a tree, and the tree is determined by the array. Twelve values, eleven parent links, and the answer to every one of the seventy-eight ranges is a lowest common ancestor — with the values themselves no longer needed.

structures · Range a fixed-length code97,718 bits5.00 ranks · in orderthe best ordered tree77,890 bits3.98 ranks · in orderthe best tree of any shape76,789 bits3.92 ranks · unorderedσ 21 · the ordered tree is 1.43% above the unordered optimum16,384 characters of englishorder costs 1.43%

The tree the operation insists on

The compound walk means "everything that went left is smaller", which is true only if the leaves are in the alphabet's order. Huffman's tree is the smallest and its leaves are in frequency order, so the operation that makes a bidirectional search affordable costs the shape that makes an index small.

structures · Symbol

Named alongside it

The objects these essays reach for when they reach for this one.

AlphabetAlphabetic codeAmortisedCartesian treeCode lengthCompound operationConstant factorDocument listingDynamic programmingEntropyHuffman codeIndex size

All concepts