1. §3 Worst-Case vs. Amortized
Motivating example: Repeated binary increment,
#bit flips when counting to n?
Inc & decrement ? 1 10 11
100
101 ……
111…111
One increment of value <n: O(log n)
n increments: j<n log j = Θ(n·log n)
Bit #0 flips n/2 times,
bit #1 incurs n/4 flips, bit #2: n/4 flips,
 ≤ 2n = Θ(n): constant (2) per inc
Potential method of analysis:
Let cj denote cost of j-th operation,
Φj:=#1s in counter after j-th op.
 before (j+1)-st op.
1jn cj/n  maxj (cj+Φj−Φj-1) + (Φ0−Φn)/n
Φ0 = 0, Φj  0
= 2 - Φn/n

2. §3 Amortized Analysis 1jn cj/n  maxj (cj+Φj−Φj-1) + (Φ0−Φn)/n
Motivating example: Repeated binary increment,
#bit flips when counting to n?
Inc & decrement ?
Definition: Fix an implementation A of abstract data type D with methods M1, M2,…,Mk.
Let T(n) denote the worst-case cost of any n-element sequence C of calls C1,C2,…,Cn  {M1,…,Mk}.
Then the amortized cost of A is defined as T(n)/n.
Don't confuse:
amortized cost
expected cost
Potential method of analysis:
Let cj denote cost of j-th operation,
Conceive Φj such that right hand side is „small“
average-case cost
1jn cj/n  maxj (cj+Φj−Φj-1) + (Φ0−Φn)/n

3. Worst, Average-Case, Expected, Amortized
Algorithm A, fixed input size n:
worst case
runtime
average case
(eg uniform distribution)
best case
Inputs: A B C D E F G H I J K
asymptotically as n→∞

4. Worst, Average-Case, Expected, Amortized
Randomized algorithm A, fixed input X:
worst case
runtime
expected case
best case
Trial #
worst-case over all X of length n
asympt. as n→∞

5. Worst, Average-Case, Expected, Amortized
Data type D, method M
worst case
runtime
amortized cost
best case
Call # … n
Methods M1,…Mk: worst-case over seq. of length n

6. §3 Relaxed Binomial Trees
A relaxed Binomial Tree
of order k1 consists of a root with k children, jth (j=1…k) being a binom.tree of order j-1
relaxed Bk B0 B0 Bk
Bk-1 B1 B2 B1
relaxed
"Mark" indicates child #j may have order =j-2
 j-2
Lemma: A relaxed Binomial Tree of order k has Fk+2Ω(1.6k) nodes
Binom.Tree of deg. k: =2k nodes
AVL Tree of deg. k:  Fk+3 nodes
Merge
Prune
Proof by induction + homework:
1 + F1 + F2 + … + Fk = Fk+2  φk
φ := (1+√5)/2 > 1.6
Fibonacci no.s Fk 1 1

7. §3 Fibonacci Heaps Bk H A relaxed Binomial Tree
of order k1 consists of a root with k children, jth (j=1…k) being a relaxed binom.tree of order j-2 Bk B0 B0
Bk-1 B1 H
min
: child j may have order j-2
A Fibonacci Heap H is a list of t heap-ordered relaxed binomial trees with pointer to the min.
Lemma: A relaxed Binomial Tree of order k has Fk+2Ω(1.6k) nodes
Lemma: A relaxed Binomial Tree of n nodes has order k  O(log n)
Extract min.key: O(log n)
Decrease key: O(1)
Merge two Fib.heaps: O(1)
Insert element: O(1)
Create 1-elem.Fib.heap: O(1)
amor- tized cost

8. §3 Extract Minimum t cj + Φj–Φj-1  O(log n)
A relaxed Binomial Tree
of order k1 consists of a root with k children, jth (j=1…k) being a relaxed binom.tree of order j-2 Bk B0 B0
Bk-1 B1 H
min
: child j may have order j-2
A Fibonacci Heap H is a list of t heap-ordered relaxed binomial trees with pointer to the min.
Lemma: A relaxed Binomial Tree of n nodes has order k  O(log n) t
Extract min.key: O(log n)
amor- tized cost
Delete target of min.pointr
Merge two Fibonacci heaps.
Consolidate s.t. each tree order k occurs only once!
bucket sort
new? ! !
cj + Φj–Φj-1  O(log n)
jn cj/n  maxj (cj + Φj – Φj-1) + (Φ0−Φn)/n
Potential Φ := Θ( t )
 O(log n)
Φ0=0, Φn≥0

9. §3 Decrease Key cj + Φj–Φj-1  O(1) Φ0=0, Φn≥0 Bk H
A relaxed Binomial Tree
of order k1 consists of a root with k children, jth (j=1…k) being a relaxed binom.tree of order j-2 Bk B0 B0
Bk-1 B1 H
min
: child j may have order j-2
A Fibonacci Heap H is a list of t heap-ordered relaxed binomial trees with pointer to the min.
Lemma: A relaxed Binomial Tree of n nodes has order k  O(log n)
Decrease key: O(1)
amor- tized cost
cut subtree
mark parent
if already marked: reset, cut & cascade up X
cj + Φj–Φj-1  O(1)
Potential Φ := C · t
Potential Φ:=Θ(t+2m)
Φ0=0, Φn≥0

10. §3 Cuts, Marks, and Cascading
Decrease key: O(1)
cut subtree
mark parent
if already marked: reset, cut & cascade up X
cj + Φj – Φj-1  O(1)
Potential Φ:=C·(t+2m)
Φ0=0, Φn≥0