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Tournament Trees PowerPoint PPT Presentation


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Tournament Trees. Winner trees. Loser Trees. Winner Tree – Definition. Complete binary tree with n external nodes and n – 1 internal nodes. External nodes represent tournament players. - PowerPoint PPT Presentation

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Tournament Trees

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Tournament trees l.jpg

Tournament Trees

Winner trees.

Loser Trees.


Winner tree definition l.jpg

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 at the internal node.

Root has overall winner.


Winner tree for 16 players l.jpg

player

match node

Winner Tree For 16 Players


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Winner Tree For 16 Players

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Smaller element wins =>min winner tree.


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Winner Tree For 16 Players

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height is log2 n (excludes player level)


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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.


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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).


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Replace Winner And Replay

Replace winner with 6.


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Replay matches on path to root.


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Replay matches on path to root.


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


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Loser Tree

Each match node stores the match loser rather than the match winner.


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Min Loser Tree For 16 Players

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Min Loser Tree For 16 Players

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Min Loser Tree For 16 Players

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Winner

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Complexity of loser tree initialize l.jpg

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 left child winner.

  • Total time is O(n).

  • More preciselyTheta(n).


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Winner

Replace winner with 9 and replay matches.


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Complexity Of Replay

  • One match at each level that has a match node.

  • O(log n)

  • More preciselyTheta(log n).


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Tournament Tree Applications

  • Run generation.

  • k-way merging of runs during an external merge sort.

  • Truck loading.


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Truck Loading

  • n packages to be loaded into trucks

  • each package has a weight

  • each truck has a capacity of c tons

  • minimize number of trucks


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Bin Packing

  • n items to be packed into bins

  • each item has a size

  • each bin has a capacity of c tons

  • minimize number of bins


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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.


Bin packing heuristics l.jpg

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 it fits.

    • If there is no bin into which current item fits, start a new bin.


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Bin Packing Heuristics

  • First Fit Decreasing.

    • Items are sorted into decreasing order.

    • Then first fit is applied.


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Bin Packing Heuristics

  • Best Fit.

    • Items are packed one at a time in given order.

    • To determine the bin for an item, first determine setSofbins into which the item fits.

    • If S is empty, then start a new bin and put item into this new bin.

    • Otherwise, pack into bin ofSthat has least available capacity.


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Bin Packing Heuristics

  • Best Fit Decreasing.

    • Items are sorted into decreasing order.

    • Then best fit is applied.


Performance l.jpg

Performance

  • For first fit and best fit:

    Heuristic Bins <= (17/10)(Minimum Bins) + 2

  • For first fit decreasing and best fit decreasing:

    Heuristic Bins <= (11/9)(Minimum Bins) + 4


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Max Winner-Tree For 16 Bins

Item size = 7


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Max Winner-Tree For 16 Bins

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Complexity of first fit l.jpg

Complexity Of First Fit

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