Whenever a leaf splits

Whenever the number of keys doubles

Whenever the key goes into a leaf that has only one key

Only when the split reaches the root, so the root itself splits

Half the height of the red-black tree

The height of the corresponding 2-3 tree

The number of red nodes

⌊log₂ n⌋, where n is the number of keys

8

4

7

15

Both guarantee O(log n) search, insertion and deletion in the worst case. AVL trees enforce a stricter balance condition, which gives a smaller worst-case height; left-leaning red-black trees instead enforce black balance and left-leaning red links.

Red-black trees are O(log n) only on average; AVL trees guarantee it in the worst case.

Red-black trees never need rotations; colour flips are enough.

An AVL tree with n keys always has height at most log₂ n.

2b + 1

b

b + 1

2b

1

One for every ancestor on the path from the new leaf to the root

2

Θ(log n)

A shortest path then uses at most |V| − 1 edges, and after k passes every distance whose shortest path uses at most k edges is correct.

After |V| − 1 passes every edge has been relaxed |V| − 1 times, which is enough for any graph at all.

It is a safety margin; half as many passes would always do.

|V| − 1 is the largest number of negative edges a graph can have without a negative cycle.

Each pass changes at least one distance, so |V| − 1 passes change |V| − 1 of them.

Θ(M + N)

Θ(M·N)

Θ(M) — it does not depend on N at all.

Θ(M·log N)

Θ(N)

Every arc carries between 0 and its capacity, and the total leaving s equals the total arriving at t.

Every arc is either empty or saturated.

Every arc carries between 0 and its capacity, and no cycle carries flow.

At every vertex, including s and t, the flow in equals the flow out.

Every arc carries between 0 and its capacity, and at every vertex other than s and t the flow in equals the flow out.

Θ(|V|) on the sparse graph and Θ(|V|²) on the dense one.

Θ(|V| log |V|) on the sparse graph and Θ(|V|³) on the dense one.

Θ(|V|²) on both.

Θ(|V|²) on the sparse graph and Θ(|V|³) on the dense one.

Θ(|V|²) on the sparse graph and Θ(|V|⁴) on the dense one.