Tournament Trees

7/27/2019 Tournament Trees

1/35

Tournament Trees

Winner trees.

Loser Trees.


2/35

Winner Tree Definition

Complete binary tree with n external

nodes and n 1 internal nodes.

External nodes represent tournament

players.

Each internal node represents a match

played between its two children;

the winner of the match is stored atthe internal node.

Root has overall winner.


3/35

Winner Tree For 16 Players

player match node


4/35


4 3 6 8 1 5 7 3 2 6 9 4 5 2 5 8

Smaller element wins =>min winner tree.

3 6 1 3 2 4 2 5

3 1 2 2

12

1


5/35


4 3 6 8 1 5 7 3 2 6 9 4 5 2 5 8

height is log2 n (excludes player level)

3 6 1 3 2 4 2 5

3 1 2 2

12

1


6/35

Complexity Of Initialize

O(1)time to play match at each match node.

n 1 match nodes.

O(n) time to initialize n-player winner tree.


7/35

Winner Tree Operations

Initialize

O(n) time

Get winner O(1) time

Replace winner and replay

O(log n)time More preciselyTheta(log n)

Tie breaker (player on left wins in case of a tie).


8/35

Replace Winner And Replay

4 3 6 8 1 5 7 3 2 6 9 4 5 2 5 8

3 6 1 3 2 4 2 5

3 1 2 2

12

1

Replace winner with 6.


9/35


4 3 6 8 6 5 7 3 2 6 9 4 5 2 5 8

3 6 1 3 2 4 2 5

3 1 2 2

12

1

Replay matches on path to root.


10/35


4 3 6 8 6 5 7 3 2 6 9 4 5 2 5 8

3 6 1 3 2 4 2 5

3 1 2 2

12

1

Replay matches on path to root.


11/35


4 3 6 8 6 5 7 3 2 6 9 4 5 2 5 8

3 6 1 3 2 4 2 5

3 1 2 2

12

1

Opponent is player who lost last match played at this node.


12/35

Loser Tree

Each match node stores the match

loser rather than the match winner.


13/35

Min Loser Tree For 16 Players

4 3 6 8 1 5 7 3 2 6 9 4 5 2 5 8

4

3

8


14/35


4 3 6 8 1 5 7 3 2 6 9 4 5 2 5 8

4

6

8

3

5

1

7


15/35


4 3 6 8 1 5 7 3 2 6 9 4 5 2 5 8

4

6

8

3

5

3

7

1

6

2

9


16/35


4 3 6 8 1 5 7 3 2 6 9 4 5 2 5 8

4

6

8

3

5

3

7

2

5

2

8

1

6

4

9


17/35


4 3 6 8 1 5 7 3 2 6 9 4 5 2 5 8

4

6

8

3

5

3

7

2

5

5

8

1

6

4

9


18/35


4 3 6 8 1 5 7 3 2 6 9 4 5 2 5 8

4

6

8

3

5

3

7

2

5

5

8

1

6

4

9


19/35


4 3 6 8 1 5 7 3 2 6 9 4 5 2 5 8

4

6

8

3

5

3

7

2

5

5

8

2

6

4

9


20/35

4 3 6 8 1 5 7 3 2 6 9 4 5 2 5 8

4

6

8

3

5

3

7

2

5

5

8

2

6

4

9

1 Winner


21/35

Complexity Of Loser Tree

Initialize

Start with 2 credits at each node. Use one to pay for the match played at that

node.

Use the other to pay for the store of a leftchild winner.

Total time is O(n).

More preciselyTheta(n).


22/35

Winner

4 3 6 8 1 5 7 3 2 6 9 4 5 2 5 8

4

6

8

3

5

3

7

2

5

5

8

2

6

4

9

1

Replace winner with 9 and replay matches.

9

5

9

3

3

2


23/35

Complexity Of Replay

One match at each level that has a match

node.

O(log n)

More preciselyTheta(log n).


24/35

Tournament Tree Applications

Run generation.

k-way merging of runs during an external

merge sort.

Truck loading.


25/35

Truck Loading

n packages to be loaded into trucks

each package has a weight

each truck has a capacity ofc tons

minimize number of trucks


26/35

Bin Packing

n items to be packed into bins

each item has a size

each bin has a capacity ofc tons

minimize number of bins


27/35

Bin Packing

Truck loading is same as bin packing.

Truck is a bin that is to be packed (loaded).

Package is an item/element.

Bin packing to minimize number of bins is NP-hard.

Several fast heuristics have been proposed.


28/35

Bin Packing Heuristics

First Fit.

Bins are arranged in left to right order.

Items are packed one at a time in given order.

Current item is packed into leftmost bin into which itfits.

If there is no bin into which current item fits, start anew bin.


29/35


First Fit Decreasing.

Items are sorted into decreasing order.

Then first fit is applied.


30/35


Best Fit.

Items are packed one at a time in given order.

To determine the bin for an item, first determine setS

ofbins into which the item fits. IfS is empty, then start a new bin and put item into this

new bin.

Otherwise, pack into bin ofSthat has least available

capacity.


31/35


Best Fit Decreasing.

Items are sorted into decreasing order.

Then best fit is applied.


32/35

Performance

For first fit and best fit:

Heuristic Bins


33/35

Max Winner-Tree For 16 Bins

8

4 3 6 8 1 5 7 3 2 6 9 4 5 2 5 8

4 8 5 7 6 9 5 8

7 9 8

89

9

Item size = 7


34/35

7

1

Max Winner-Tree For 16 Bins

6

4 3 6 1 5 7 3 2 6 9 4 5 2 5 8

4 6 5 7 6 9 5 8

7 9 8

9

9


35/35

Complexity Of First Fit

O(n log n), where n is the number of

items.

Tournament Trees

Documents

Transcript of Tournament Trees