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### Evaluating Algorithmic Design Paradigms

Sashka Davis

Advised by Russell Impagliazzo

UC San Diego

October 6, 2006

Grace Hopper Celebration of Women in Computing

Suppose you have to solve a problem Π…

No Greedy alg. exists? Or I didn’t think of one?

Is there a Dynamic Programming algorithm that solves Π?

Is there a Backtracking algorithm that solves Π?

Is there a Greedy algorithm that solves Π?

Eureka! I have a DP

Algorithm!

No Backtracking agl. exists? Or I didn’t think of one?

Is my DP algorithm optimal or a better one exists?

Suppose we a have formal model of each algorithmicparadigm

Is there a Dynamic Programming alg. that solves Π?

Is my algorithm optimal, or a better DP algorithm exists?

No Greedy algorithm can solve Π exactly.

Is there a Greedy algorithm that solves Π?

Is there a Backtracking algorithm that solves Π?

No Backtracking algorithm can solve Π exactly.

DP helps!

Yes, it is! Because NO DP alg. can solve Π more efficiently.

The goal

- To build a formal model of each of the basic algorithmic design paradigms which should capture the strengths of the paradigm.
- To develop lower bound technique, for each formal model, that can prove negative results for all algorithms in the class.

Using the framework we can answer the following questions

1. When solving problems exactly:

What algorithmic design paradigm can help?

- No algorithm within a given formal model can solve the problem exactly.
- Wefind an algorithm that fits a given formal model.

2. Is a given algorithm optimal?

- Prove a lower bound matching the upper bound for all algorithms in the class.

3. Solving the problems approximately:

- What algorithmic paradigm can help?
- Is a given approximation scheme optimal within the formal model?

Current hierarchy of formal models [BNR03], [DI04], [ABBO+05],[BODI06]

Dynamic

Programming

Backtracking & Simple DP

Greedy

pBP

prioritized Branching Programs

pBT

prioritized Branching Trees

PRIORITY

PRIORITY

Some of our results

PRIORITY algorithms(formal model of greedy algoritms):

- Dijkstra’s algorithm solves the Single Source Shortest Path (SSSP) in graphs with non-negative edges and cannot be simplified.
- No PRIORITY algorithm can solve the SSSP in graphs with negative weights.
- Proved lower bounds on the approximation ratio for Weighted Vertex Cover, Maximum Independent Set, and Steiner Tree problems.

pBT algorithms(formal model of BT and simple DP algoritms):

- There is no efficient pBT algorithm which finds the shortest path in graphs with negative weights but no negative cycles efficiently.

pBP algorithms ( formal model of Dynamic Programing):

- There is noefficient pBP algorithm which finds the maximum matching in bipartite graphs.

Some of our results

Maximum Matching

in Bipartite graphs

Shortest Path in negative graphs no cycles

Shortest Path in no-negative graphs

Flow Algorithms

Bellman-Ford

pBP

pBT

ADAPTIVE

PRIORITY

Dijkstra’s

FIXED

PRIORITY

Prim’s

Kruskal’s

Minimum Spanning Tree

Dijkstra’s

FIXED

PRIORITY

Prim’s

Kruskal’s

MST

ShortestPath

PRIORITY a formal model of greedy algorithmsConsider Kruskal’s algorithm

- orders edges of the graph ONCEaccording to weight,
- Inspects the next edge according to the order and makes irrevocable decision, to add or not, to the solution (MST)

Consider Prims’s algoritm

- Proceeds in iteration
- each iteration orders edges in the cut in non-descending order according to weight,
- Inspects the next edge according to the order and makes irrevocable decision, to add or not, to the solution MST

Questions:

- Can we canonize all ADAPTIVE algorithms?
- Does there exist a FIXED priority algorithm for SSSP?

Consider one iteration of Dijkstra’s algorithm

N=Not Yet Reached

R=Reached

d(1)

d(2)

u

s

t

v

d(3)

d(4)

Suppose d(3)=min{d(1),d(2),d(3),d(4)}

then (u,v) is added to the solution.

Can Dijkstra’s algorithm be simplified?

ADAPTIVE PRIORITY

- FIXED priority algorithm
- Orders edges ONCE
- Inspects an edge; makes a decision

?

Is there a FIXED priority algorithm that solves SP?

If there is no FIXED priority algorithm for ShortPath problem

then Dijkstra’s algorithm cannot be simplified.

ShortPath problem

ShortPath problem:

Given a graph G=(V,E) and s,t in V.

Find the shortest path from s to t in G.

Instance: set of edges

Solution: a path in G connecting s, t

Theorem: There is no FIXED priority algorithm that solves ShortPath problem exactly.

Corollaries:

- Dijkstra solves the problem exactly and hence cannot be simplified
- The classes of FIXED and ADAPTIVE priority algorithms are distinct.

Lower bound setting

- Lower bound is a game between Adversary and Solver
- Existence of a FIXED priority algorithm is a strategy for Solver
- Existence of a strategy for the Adversary establishes the lower bound

The winning strategy for the Adversary presents a nemesis graph, which can be modified so that the Solver either:

- fails to output a solution
- outputs a path, but not the shortest one

w,k

Modification of the graphIf Solver considers edge (y,1) before edge (z,1)

then the Adversary presents:

u,k

a

y,1

s

t

x,1

z,1

b

a

y,1

s

t

x,1

z,1

b

If Solver considers (y,1) before (z,1)- The Algorithm selects (y,1) first
- Case 1: (y,1) is added to MST
- Case 2:(y,1) is NOT added to MST

The cases when the algorithm selects (u,k) or

(x,1) first reduce to Case 1 and 2.

a

y,1

t

s

z,1

x,1

b

Solver constructs a path {u,y}

Adversary outputs solution {x,z}

Case 1: Solver decides to add (y,1)Case 2: Solver decides (y,1) is NOT part of the path

u,k

a

y,1

t

s

x,1

b

Solver has failed to construct a path.

Adversary outputs a solution {u,y} and wins the game.

ShortPath

Dijkstra’s

FIXED

The outcome of the game:- Solver fails to construct a tree in which t is reachable from s.
- When Solver succeeds, the approximation ratio achieved is (k+1)/2.
- The Adversary can set k arbitrarily large and thus can gain any advantage.
- No FIXED priority algorithm can solve the ShortPath problem.
- Dijkstra solves ShortPath problem, hence it cannot be simplified.
- FIXED priority algorithms are properly contained in ADAPTIVE priority algorithms.

BT

Greedy

ConclusionsBuilding formal models of basic algorithmic design paradigm and developing general lower bound techniques we can answer:

1. What algorithmic design paradigm can help?

(a) Certify the problem as hard: by Proving NO algorithms in the class can solve it.

(b) Or we find an algorithm within a given formal class.

2. If we solved the problem, then we can prove that our algorithm is optimal.

By Proving a matching lower bound for ALL algorithms in the class.

3. If No technique can solve the problem exactly then we use the framework to:

(a) How good an approximation scheme can we get using different algorithmic techniques?

(b) Certify that our approximation algorithm is optimal.

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