Aha An Illuminating Perspective

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Aha! An Illuminating Perspective

• David Ginat, Tel-Aviv
• Dan Garcia, Berkeley
• Bill Gasarch, Maryland
• February 2002

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? ?
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Tennis Tournament

• N players
• R players in a round R
  div 2 matches If R odd then one
  player advances without a match
• Total of matches?
• N317 1st round 158 advance 158
  winners 1 2nd round 79

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Illuminating Perspective

• Every match has one loser
• A loser exits the tournament
• Every player loses at some point except for the
  winner
• Count the of losers rather than of
  matches
• N players ? N-1 losers ? N-1 matches

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Edge Removals

• Connected graph G(V,E)
• Min of edges to remove so there will be no
  cycle?
• Cycle breaking
• DFS, and removal of edges that close cycles

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Illuminating Perspective

• The Max graph with no cycles?
• A tree!
• Thus, regardless of E, the number of remaining
  edges should be V-1.
• Answer E-(V-1)
• Implicit Characteristics

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Implicit Factors

• Implicit, related patterns
• Link to other problems
• Connected graph G(V,E)
• Min of edges to remove so there will be no
  cycle?
• ? Break cycles? Inefficient
• Largest G with no cycles tree therefore
  E-(V-1)

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Beetles

• N?N board, beetle on each square
• Each beetle moves to a diagonally-adjacent
  square
• Max of occupied squares?

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Beetles

• Swap inside 2?2 sub-boards
• Yields cover of 16-18 squares
• Swap in pairs along the diagonals
• Yields cover of 20 squares

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Beetles

• Color the rows White-Gray
• 3?5 beetles leave white squares
• Only 2?5 Beetles enter them
• Odd N max is (N-1)?N

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Infected Squares

• N?N board, N-1 infected squares
• Each square with 2 or more infected neighbors get
  infected
• Can the whole board be infected?

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Infected Squares

• Focus on the growing infected area operational
  reasoning
• Try to identify Maximal Structures
• Can the whole board be infected?

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Infected Squares

• Focus on an implicit factor ? the infected
  circumference
• Invariant the infected circumference does not
  grow assertional reasoning
• Can the whole board be infected?

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Box Sizes

• N box sizes - B1, B2
• Is there some Bi which is at least twice than
  some Bj ?
• 33, 23, 15, 17 ? Yes
• 33, 23, 25, 17 ? No

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Stone Packing 1

• Stones of size S (int)
• N box sizes - B1, B2 (int)
• of stones packed in the boxes B1 div S B2
  div S
• Is there a pair of boxes whose unification will
  enable packing an extra stone?
• 3 33, 13, 16, 5, 19, 22 ? Yes
• 3 33, 13, 4, 19, 10, 25 ? No

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Stone Packing 2

• Stones of size S (int)
• N box sizes - B1, B2 (int)
• of stones packed in the boxes B1 div S B2
  div S
• What is the maximal of extra stones that can be
  added by unifying pairs of boxes?
• 5 33, 11, 5, 16, 9, 21, 26 ? 1
• 5 33, 11, 5, 17, 9, 22, 27 ? 2

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The Extremal Perspective

• Implicit Min/Max factors
• Boxes Min, Max
• Packing-1 Max, Second-Max
• Packing-2 Order (by remainders) Match
  iteratively from ends
• 5 33, 11, 5, 17, 9, 22, 27
• 0 5, 1 11, 2 17, 2 22, 2 27, 3 33, 4 9

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Auxiliary Coloring

• Partitioning of elements
• Applications
• Graph algorithms
• bipartite graph, cycle lengths
• Data structures
• red-black trees
• Tiling problems
• Proofs, invariant properties

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Integer Removals Game

• K, K1, K2, KN (Ngt20)
• Two players, alternating turns, removing one
  integer at a time
• Until only 2 integers X,Y remain
• If gcd(X,Y)1 then player-I wins otherwise
  player-II wins
• Should I be player-I/player-II?
• Strategy?
• 7 8 9 10 11 12 13

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Integer Removals Game

• Two cases Even N, Odd N
• 1. Focus on primes
• 2. Focus on of divisors of each i
• 3. Focus on mutual divisors of each pair of
  integers Represent as a graph
• For odd N, be player-I, and always remove the i
  with the largest of divisors

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Integer Removals Game

• Gcd(X,X1)1 for every X
• For odd N
• 1. Be player-I.
• 2. Remove one end.
• 3. Partition the remaining is to buddies.
• 4. Respond to each move of player-II by removing
  the buddy
• 7 8 9 10 11 12 13

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Integer Removals Game

• Gcd characteristics
• Buddy notion
• Partitioning
• Invariant property
• Gcd in various math computations
• Buddy system, complement
• Partitioning a decomposition
• Invariant core of every algorithm

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Illuminating Perspectives

• Design of algorithms
• Analysis of algorithms
• Implicit factors
• Auxiliary elements
• Step-by-step progress
• Links to a variety of schemes and applications