Advanced Algorithm Design and Analysis

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Advanced Algorithm Design and Analysis
CSE 4080 Computer Science Project

• Student Gertruda Grolinger
• Supervisor Prof. Jeff Edmonds

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Topics

• Divide and Conquer Fast Fourier Transformations

• Recursion Parsing

• Greedy Algorithms Matroids

• Dynamic Programming Parsing CFG

• Network Flow Steepest Assent, Edmonds-Karp,
• Matching

• Classifying problems

• NP-completeness Reductions

• Approximation Algorithms Knapsack

• Linear Programming What to put in a hotdog?

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The Pebble Game
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The Pebble Game

• One player game, played on a DAG

• Used for studying time-space trade-off

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Formalization

• Directed acyclic graph

• Bounded in-degree

Output nodes
Nodes
Input nodes
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Three main rules
1. A pebble can be placed on any input node
2. A pebble can be placed on a node v if
all predecessors of the node v are marked
with pebbles
3. A pebble can be removed from a node at
any time
Note a pebble removed from the graph can be
reused
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Strategysequence of legal moves which ends in
pebbling the distinguished node f
The Goalto place a pebble on some previously
distinguished node f while minimizing the
number of pebbles used
A moveplacing or removing one of the
pebbles according to the three given rules
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Example 1
7 moves and 7 pebbles
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Example 2
11 moves and 3 pebbles
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Interpretation
1. A pebble can be placed on any input node
LOAD
2. A pebble can be placed on a node v if
all predecessors of the node v are marked
with pebbles COMPUTE
3. A pebble can be removed form a node at
any time DELETE

• Use as few pebbles as possible REGISTERS

• Use as few moves as possible TIME

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In general How many pebbles are required to
pebble a graph with n nodes?
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Pyramid graph Pk
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Pyramid graph Pk
Fact 1 Every pebbling strategy for Pk (k gt
1) must use AT LEAST k 1 pebbles.
That is O( ) pebbles expressed in number of
edges n.
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Pyramid graph Pk
k 5
We need at least k 1 6
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Pyramid graph Pk
Lets consider having k 5 pebbles
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Arbitrary graph with restricted in-degree (d 2)
Fact 2 Every graph of in-degree 2 can be
pebbled with O(n/log n) pebbles (n nodes in
the graph).
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Arbitrary graph with restricted in-degree (d 2)
O(n/log n)
Proof

• Recursive pebbling strategy

• Cases

• Recursions for each case

• Solutions
• P(n) O(n/log n)

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A Pebble problem
Pebble a DAG that has a rectangle of nodes that
is w wide and h high (h ltltw).
For each node v (not on the bottom row), there
will be d nodes In(v) u1, u2,ud on the
previous row with edges to v.
The goal is to put a pebble on any single node
on the top row.
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Dynamic Programming

• Algorithm that uses as many pebbles
• (memory) as needed, but does not redo
• any work

1. Loop invariant

1. How much time and how many pebbles?

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Dynamic Programming algorithm

• Place a pebble on each of the nodes on the
• bottom row

• After the i th iteration there is a pebble on
  each
• node of the i th row from the bottom

• Next iteration place a pebble on each node
• on the (i 1)st row and remove pebbles from the
• i th row so they can be reused

• Exit when i h

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Time and pebbles (space) needed

• Time O(h w)

• Pebbles 2 w O(w)

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Recursive Backtracking

• Task to place a pebble on some node v which
• is r rows from the bottom

• Algorithm (recursive) to accomplish this task
• that re-does work but uses as few pebbles as
• possible

• Time(r ) and Pebbles(r) are time and pebbles
• used to place a pebble on one node r rows
• from the bottom

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Recursive Backtracking algorithm

• My task is to place a pebble on some node v
• which is r rows from the bottom

• I ask a friend, for each of the d nodes ui Î
  In(v)
• to place a pebble on ui

• Once there is a pebble on all nodes ui Î In(v),
• I place a pebble on node v

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Time(r ) and Pebbles(r)

• Time(r) d Time(r-1) 1 d Time(r-1)
• d(r-1)

• Pebbles(r ) Pebbles(r-1) (d 1)
• (d - 1) (r 1) 1

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Conclusion time-space trade-off

• Time comparison
• DP Q(h w)
• RB Q(d (h-1))

• Space comparison
• DP Q(w)
• RB Q(d h) where h ltlt w

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References

• Gems of theoretical computer science
• U. Schöning, R. J. Pruim

2. Asymptotically Tight Bounds on Time-Space
Trade-offs in a Pebble Game T. Lengauer, R.
E. Tarjan
3. Theoretical Models 2002/03 P. van Emde
Boas
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Thank you for your attention ?

• Questions ?