Definition: A complete binary tree that is stored using the formulabased representation'

1
Chapter 10Tournament Trees

• Definition A complete binary tree that is
  stored using the formula-based representation.
• Basic Operations
• - Find/Replace the min or max element
• 2 Categories
• - Winner Trees
• - Loser Trees

?
?
?
e
d
c
?
a
b
5 players
2
Chapter 10Tournament Trees

• Internal nodes represent a match between the
  children of the node
• 2 Categories
• - Winner Trees - Each internal node records the
  winner of the match played there
• Min Winner Tree Player with the smaller
  value wins
• Max Winner Tree Player with the larger value
  wins
• - Loser Trees Not the same as Min Winner Trees

?
?
?
e
d
c
?
a
b
5 players
3
Chapter 10Tournament Trees

• Winner of each match competes at the next higher
  level

c
c
e
e6
d2
c7
a
a4
b3
5 players
4
Chapter 10Tournament Trees

• Winner at each level?

?
?
?
?
?
?
?
f3
e10
g1
h5
i9
j4
?
?
c7
d2
a4
b3
10 players
5
Chapter 10Tournament Trees

• The champion is
• e

e
i
e
h
c
i
e
f3
e10
g1
h5
i9
j4
a
c
c7
d2
a4
b3
10 players
6
Chapter 10Tournament Trees

• Algorithm for initialize
• 1. Start at lowest level, visit each n-1 nodes
  once O(n)
• or 2. PostOrder traversal O(n)

e
i
e
h
c
i
e
f3
e10
g1
h5
i9
j4
a
c
c7
d2
a4
b3
10 players
7
Chapter 10Tournament Trees

• Replay
• 1. Change es value from 10 to 1
• 2. Complexity of replay is O(log n)

e
i
e
h
c
i
e
f3
e10
g1
h5
i9
j4
a
c
c7
d2
a4
b3
10 players
8
Chapter 10Tournament Trees

• Winner at each level for the Min Winner Tree?

g
g
d
g
d
j
f
f3
e10
g1
h5
i9
j4
b
d
c7
d2
a4
b3
10 players
9
Chapter 10Tournament Trees

• Sort Algorithm
• 1. Initialize Min Winner Tree
• 2. For i 0 to n
• ai min element
• min elements value ? ?
• replay

g
g
d
g
d
j
f
f3
e10
g1
h5
i9
j4
b
d
c7
d2
a4
b3
10 players
10
Chapter 10Tournament Trees

• n 5
• ti gives the winner of the match played at node
  i
• Must be able to determine the parent of node i...

t1
t2
t3
t4
e3
e4
e5
e1
e2
5 players
11
Chapter 10Tournament Trees

• n 5
• (i offset)/2 i ? LowExt
• p
• (i - LowExt n - 1)/2 i gt LowExt

t1
t2
t3
t4
e3
e4
e5
e1
e2
5 players
12
Chapter 10Tournament Trees

• (i offset)/2 i ? LowExt
• p
• (i - LowExt n - 1)/2 i gt LowExt
• s ?log2(n - 1)?
• n 5 External Nodes
• s 2

t1
t2
t3
t4
e3
e4
e5
e1
e2
5 players
13
Chapter 10Tournament Trees

• (i offset)/2 i ? LowExt
• p
• (i - LowExt n - 1)/2 i gt LowExt
• s ?log2(n - 1)?
• s 2
• The number of internal nodes at the lowest
• level is n - 2s
• The number of external nodes at the lowest
• level, LowExt, is 2(n - 2s)
• LowExt 2
• offset 2s1 -1

t1
t2
t3
t4
e3
e4
e5
e1
e2
5 players
14
Chapter 10Tournament Trees

• (i offset)/2 i ? LowExt
• p
• (i - LowExt n - 1)/2 i gt LowExt
• s ?log2(n - 1)?
• The number of internal nodes at the lowest
• level is n - 2s
• LowExt 2(n - 2s)
• offset 2s1 -1

g
g
d
g
d
j
f
f3
e10
g1
h5
i9
j4
b
d
c7
d2
a4
b3
10 players
15
Chapter 10Tournament Trees

• (i offset)/2 i ? LowExt
• p
• (i - LowExt n - 1)/2 i gt LowExt
• s ?log2(n - 1)?
• The number of internal nodes at the lowest
• level is n - 2s
• LowExt 2(n - 2s)
• offset 2s1 -1
• Compare with p. 469
• Initialize()
• Play()
• RePlay()

g
g
d
g
d
j
f
f3
e10
g1
h5
i9
j4
b
d
c7
d2
a4
b3
10 players
16
Tournament TreesApplications

• Parallel Sorting Example Requirement is to sort
  16000 records
• - 16 Processors Sort Simultaneously
• - Merge the Results
• 1000 1000 1000
  1000 1000

P1
P2
P3
P16
P15
17
Tournament TreesApplications

• Parallel Sorting Example Requirement is to sort
  16000 records
• - 16 processors sort 1000 records simultaneously
• - Merge the Results
• - Complexity?

. . .
P1
P2
P16
P15
18
Tournament TreesApplications

• Assume a complexity of O(n log n) for the sort
  algorithm on each processor
• Since they sort in parallel across k processors
  its
• O(n/k) log (n/k) O(whatevertimeittakestomerge
  )

. . .
P1
P2
P16
P15
19
Tournament TreesApplications

• Merge Algorithm
• 1. Initialize a min winner tree O(k), where k
  is the number of processors
• 2. Move the winner to the output file O(1)
• 3. Replay O(log k)
• 4. Do this n times O(n log k)

. . .
P1
P2
P16
P15
20
Tournament TreesApplications

• Complexity
• O(n/k) log (n/k) O(n log k)

. . .
P1
P2
P16
P15
21
Tournament TreesLoser Trees

• Reduce the work needed to determine the outcome
  in a RePlay
• Record the LOSER of each match rather than the
  WINNER
• Overall winner recorded at the top

c
c
e
c
e
a
d
e6
e6
d2
d2
c7
c7
a
b
a4
b3
a4
b3
5 players - Max Winner Tree
5 players - Loser Tree
22
Tournament TreesLoser Trees

• Now, when the value of c changes to 2, theres a
  rematch at the parent of c, and c loses. Notice
  the pattern in the replay

e
c
a
c
e
c
e
a
d
c
e6
e6
d2
d2
c7
c7
a
b
c2
a4
b3
a4
b3
5 players - Max Winner Tree
5 players - Loser Tree
23
Tournament TreesLoser Trees

• Now, when the value of c changes to 2, theres a
  rematch at the parent of c, and c loses.

c
c
e
c
e
a
d
c
e6
e6
d2
d2
c7
c7
a
b
c2
a4
b3
a4
b3
5 players - Max Winner Tree
5 players - Loser Tree
24
Tournament TreesApplications

• Bin Packing Problem
• Bins have a capacity c
• n objects need to be packed into the bins
• Object i requires si units of capacity,
• 0 lt si ? c
• A feasible packing is an assignment of objects
  s.t. no bins capacity is exceeded
• A feasible packing that uses the fewest bins is
  optimal

10
10
10
10
10
10
25
Example

• Example The following items have to go in the
  bins.
• 1. Find a feasible solution.
• 2. Find the optimal solution.
• f 5
• e 9
• c 2
• b 3
• d 6
• a 5

10
10
10
10
10
10
26
Algorithms(Approximation Algorithms)

• First Fit Objects are considered for packing in
  the order in which they arrive.
• f 5
• e 9
• c 2
• b 3
• d 6
• a 5

10
10
10
10
10
10
27
Algorithms(Approximation Algorithms)

• Best Fit Objects are packed into the bin with
  the least capacity available into which they will
  fit (cAvailj ? si)
• f 5
• e 9
• c 2
• b 3
• d 6
• a 5

10
10
10
10
10
10
28
Algorithms(Approximation Algorithms)

• First Fit Decreasing Same as First Fit, but
  objects are first reordered s.t. si ? si1
• f 5
• e 9
• c 2
• b 3
• d 6
• a 5

10
10
10
10
10
10
29
Algorithms(Approximation Algorithms)

• Best Fit Decreasing Same as Best Fit, but
  objects are first reordered s.t. si ? si1
• f 5
• e 9
• c 2
• b 3
• d 6
• a 5

10
10
10
10
10
10