Correctness of Constructing Optimal Alphabetic Trees Revisited

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Correctness of Constructing Optimal Alphabetic
Trees Revisited
Theoretical computer science 180 (1997) 309-324

• Marek Karpinski,
• Lawrence L. Larmore,
• Wojciech Rytter

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Outline

• Definitions
• General version of Garsia-Wachs (GW) algorithm
• Proof of GW
• Hu-Tacker (HT) algorithm
• Proof of HT by similarity to GW

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Definitions
Binary tree Every internal node has exactly two
sons
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Definitions
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The Move Operator
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The Move Operator
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The Move Operator
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The Move Operator
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Definitions
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Theorem 1 (correctness of GW)
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Garsia-Wachs Algorithm
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Definitions
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Theorem 2
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Shift Operations
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Shift Operations
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LeftShift Example
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LeftShift Example
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LeftShift Example
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LeftShift Example
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LeftShift Example
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LeftShift Example
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LeftShift Example
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LeftShift Example
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LeftShift Example
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Theorem 2
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Proof of Point 2 in Theorem 2
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Proof of Point 2 in Theorem 2
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Proof of Point 3 Theorem 2
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Definition of Well Shaped Segments
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Definition of Well Shaped Segments
Active Window
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Movability Lemma
If the segment i,,j is left well shaped, then
the active pair (i,i1) can be moved to the other
side of the segment by locally rearranging
sub-trees in the active window without changing
the relative order of the other items and
without changing the level function of the tree.
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Movability Lemma
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Movability Lemma
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Movability Lemma
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Movability Lemma
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Movability Lemma
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Theorem 3
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Point 1 in Theorem 2
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Hu-Tucker Algorithm
Transparent items and opaque items
Compatible pair No opaque items in the middle
Minimal compatible pair (mcp) compatible pair
(i,i1) where Weight(i) weight(i1) is minimal
Tie Breaking Rule
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Hu-Tucker Algorithm
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Hu-Tucker Algorithm
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Hu-Tucker Algorithm
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Hu-Tucker Algorithm
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GW Algorithm
gmp Globaly Minimal Pair
GW - the same as GW but always choose gmp
instead of some other lmp.
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Definitions
Normal sequence sequence of weights
Special sequence sequence of weights, each one
is either transparent or opaque
MoveTransparent operator converts a special
sequence into a normal sequence and moves all
transparent items to their RightPos. (first it
moves the rightmost item, then the one to its
left, etc)
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MoveTransparent
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MoveTransparent
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MoveTransparent
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MoveTransparent
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MoveTransparent
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MoveTransparent
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MoveTransparent
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MoveTransparent
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MoveTransparent
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MoveTransparent
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MoveTransparent
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MoveTransparent
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MoveTransparent
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MoveTransparent
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MoveTransparent
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MoveTransparent
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The Simulation Lemma
Assuming there are no ties
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Proof of the Simulation Lemma
Claim A
Claim B
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Proof of Claim A
By contradiction assume that w,u are not visible
to each other. This means there is an opaque item
q between them. Two cases
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Proof of Claim A
Contradiction no place for q
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Proof of Claim A
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Proof of Claim A
Contradiction no place for q
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Proof of Claim B
Claim B
Proof
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Proof of Claim B
The order of movements is from right to left,
thus w is Processed first then the items between
u and w, and then u. At this point u and w must
be adjacent, thus q must be processed later, and
thus it is to the left of u.
q is not visible from u because (u,w) is mcp
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Proof of Claim B
Let q be the opaque item between q and u visible
from u
Contradiction
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Conclusion
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Claim C
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Proof of claim C
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Proof of Simulation Lemma
The proof of the lemma is by induction
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Tie Braking Rule
Theorem 4 The Tie Breaking Rule (TBR) is correct
Proof
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Proof of TBR
Case 1 All weights are strictly positive
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Proof of TBR
Case 2 Some of the original weights are zero