graph theory began in 1736 when euler noticed that a walking-tour puzzle about seven bridges in königsberg was really a question about dots and lines — and that the geometry (lengths, positions, the river) was irrelevant. 𐃏 that act of abstraction — keep the incidence structure, discard everything else — is the whole subject. a graph is the minimal mathematical model of “things, some pairs of which are related”, which is why it keeps turning up: networks, molecules, dependencies, state machines, social graphs (Epp, Susanna S., 2019).
Trees
An inspiration for the structure of these notes is my computer vision notes.
Divide-and-Conquer Recurrences
Many divide-and-conquer algorithms follow the same pattern: split the input into smaller pieces, solve each piece recursively, and combine the results. This shared structure means their running times satisfy recurrences of the same shape.
Definition
(Divide-and-Conquer Recurrence)
A divide-and-conquer recurrence has the form:
\[T(n) = aT\!\left(\left\lceil n/b \right\rceil\right) + \Theta(n^d)\]
where:
- \(a \geq 1\) is the number of subproblems (the branching factor),
- \(b > 1\) is the factor by which the input shrinks at each level,
- \(n^d\) is the cost of the divide and combine step.
Three forces compete: the branching factor \(a\) creates more work at each level, the shrinkage factor \(b\) makes each subproblem cheaper, and the non-recursive cost \(n^d\) sets the price of splitting and merging. The Master Theorem tells us which force wins.
Backlinks (2)
1. Wiki /wiki/
Knowledge is a paradox. The more one understand, the more one realises the vastness of his ignorance.