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

Sparse graph — where it appears

A graph whose edge count is near its vertex count rather than near its square, which is the regime most bounds in the field are quoted for. It is the regime most bounds in the field are quoted for, and the algorithms that win in it are not the ones that win when the edges approach the square of the vertices.

Named by 6 essays across one field — each of them below, with the objects they name alongside it.

10010³10³10⁴10⁵10⁶10⁷Vcounted workBreadth-firstDijkstra, binary heapDijkstra, all V queuedBellman–Ford, all passesV from 64 to 2048, sparse, fixed average degreework = scans + visits + relaxations + queue comparisons

Counting on a graph

An instrumented array counts comparisons, swaps, reads and writes, and none of those is what a graph algorithm spends its time on. Three new primitives are needed — an adjacency scanned, a vertex first reached, an edge relaxed — and once they exist, breadth-first and depth-first search turn out to be the same algorithm by every count kept on arrays.

graphs · Graph sparse, fixed average degree10010³10³10⁴10⁵10⁶VworkDijkstra, binary heap — E log EDijkstra, all V queued — V^2dense, fixed density10010⁴10⁵VworkDijkstra, binary heap — V + EDijkstra, all V queued — Ethe class is a property of the sweep as much as of the algorithmwork = scans + visits + relaxations + queue comparisons

Two parameters, one bound, no order

With one size parameter the candidate classes are ordered — n beats n log n beats n², always, and comparing two bounds is reading them. With two, E log V and V² have no order at all, and which one is smaller is a property of the graph. Sweeping V at fixed degree and at fixed density are different experiments, and the same algorithm fits different classes in the two.

graphs · Graph adjacency scansrelaxationsqueue comparisonsvisitsBreadth-first14,336Dijkstra, all V queued2,118,656Dijkstra, binary heap56,973Bellman–Ford, all passes50,309,120Prim64,884Kruskal74,008V = 2048, E = 6,144, sparse, fixed average degreeevery segment counted exactly

The queue decides the class, and the pseudocode does not name it

Dijkstra's algorithm is eleven lines of pseudocode with a priority queue in the middle of them. Which queue is not stated, and it is the difference between 56,973 units of work and 2,118,656 on the same graph. Two of the three queues here also fail to fit the class they are famous for, in a regime each.

graphs · Graph busiest vertexuniform pairsbusiest 16 · Σd² 43,104 · degeneracy 4attachmentbusiest 114 · Σd² 90,296 · degeneracy 4pairs of neighbours examineduniform pairs — every pair18,480uniform pairs — oriented4,090attachment — every pair42,076attachment — oriented3,339V = 1,024, E = 3,072 on both30 triangles and 249 — the answers, which also differ

Two parameters are not enough either

Two graphs on 1,024 vertices with 3,072 edges each — identical in both numbers every bound in this field is written in. Enumerating every pair of neighbours of every vertex costs 18,480 examinations on one and 42,076 on the other. The quantity that separates them is a third parameter, it is computable in linear time, and it appears in no statement of the problem.

graphs · Graph 10010³10³10⁴10⁵10⁶10⁷Vcounted workBreadth-firstDijkstra, binary heapBellman–FordBellman–Ford, all passesV from 64 to 2048, sparse, fixed average degreework = scans + visits + relaxations + queue comparisons

The bound with a precondition

Bellman–Ford is O(V·E), and on a graph of 2,048 vertices it stops after seven passes of the 2,047 the bound allows — a factor of 289 between the bound and the run. Dijkstra is faster and returns a wrong answer on four vertices if one arc is negative. Both facts are about the same clause: the qualifier at the end of the sentence.

graphs · Graph ordered by degreedegeneracy orderthe degeneracy024681012largest out-degreeuniform pairs, degree 452 above duniform pairs, degree 6136 above duniform pairs, degree 1083 above dpreferential attachment, degree 414 above dpreferential attachment, degree 622 above dpreferential attachment, degree 1058 above d1,024 verticesdashed: the degeneracy

A bound right for the wrong reason

Orient every edge of a graph towards its higher-degree endpoint and count triangles among out-neighbours, and the work is O(E·d), where d is the graph's degeneracy. The usual reason given is that the orientation keeps every out-degree at most d. On a graph of 1,024 vertices with degeneracy four, 136 vertices have more than four out-neighbours and one has seven. The bound survives by a different argument, and the orientation that does keep every out-degree at most d does less work.

graphs · Graph

Named alongside it

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

Dijkstra's algorithmAdjacencyCounted primitiveDensityWorst caseDegeneracyDegree distributionHeapHonest limitMeasured countOrientationPreferential attachment

All concepts