Operations

Decrease-Key(H, x, k), where k $\leq$ key(x)

Decrease-Key(H, x, k)
$\;\;\;\;\;$key(x) = k
$\;\;\;\;\;$while parent(x) $\neq$ NIL and key(x) < key(parent(x))
$\;\;\;\;\;$ $\;\;\;\;\;$;; "bubble" new key up
$\;\;\;\;\;$ $\;\;\;\;\;$swap(key(x), key(parent(x)))
$\;\;\;\;\;$ $\;\;\;\;\;$x = parent(x)

Max depth =

Running time = O(lg n)

Delete(H, x)

Delete(H, x)
$\;\;\;\;\;$Decrease-Key(H, x, $-\infty$) $\;\;\;\;\;$ $\;\;\;\;\;$; O(lg n)
$\;\;\;\;\;$Extract-Min(H) $\;\;\;\;\;$ $\;\;\;\;\;$ $\;\;\;\;\;$ $\;\;\;\;\;$; O(lg n)

Running time = O(lg n)

Binomial Heap Example