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Algorithms Strategies - Objected Oriented Program II - Notes | CMSC 132, Study notes of Computer Science

University of Maryland Computer Science

Prof. Nelson Padua-Perez

Material Type: Notes; Professor: Padua-Perez; Class: OBJECT-ORIENTED PROG II; Subject: Computer Science; University: University of Maryland; Term: Unknown 1989;

Typology: Study notes

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Uploaded on 02/13/2009

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CMSC 132:

Object-Oriented Programming II

Algorithm Strategies

Department of Computer Science

University of Maryland, College Park

General Concepts

Algorithm strategy

Approach to solving a problem

May combine several approaches

Algorithm structure

Iterative ⇒execute action in loop

Recursive ⇒reapply action to subproblem(s)

Problem type

Satisfying ⇒find any satisfactory solution

Optimization ⇒find best solutions (vs. cost metric)

Discover Study notes of Computer Science University of Maryland

Partial preview of the text

Download Algorithms Strategies - Objected Oriented Program II - Notes | CMSC 132 and more Study notes Computer Science in PDF only on Docsity!

CMSC 132:

Object-Oriented Programming II

Algorithm Strategies

Department of Computer Science

University of Maryland, College Park

General Concepts

Algorithm strategy

Approach to solving a problem May combine several approaches

Algorithm structure

Iterative ⇒ execute action in loop Recursive ⇒ reapply action to subproblem(s)

Problem type

Satisfying ⇒ find any satisfactory solution Optimization ⇒ find best solutions (vs. cost metric)

Some Algorithm Strategies

Recursive algorithms

Backtracking algorithms

Divide and conquer algorithms

Dynamic programming algorithms

Greedy algorithms

Brute force algorithms

Branch and bound algorithms

Heuristic algorithms

Recursive Algorithm

Based on reapplying algorithm to subproblem

Approach

Solves base case(s) directly
Recurs with a simpler subproblem
May need to convert solution(s) to subproblems

Backtracking Algorithm – Example

Color a map with no more than four colors

If all countries have been colored return success Else for each color c of four colors and country n If country n is not adjacent to a country that has been colored c Color country n with color c Recursively color country n+ If successful, return success Return failure

Divide and Conquer

Based on dividing problem into subproblems

Approach

Divide problem into smaller subproblems Subproblems must be of same type Subproblems do not need to overlap
Solve each subproblem recursively
Combine solutions to solve original problem

Usually contains two or more recursive calls

Divide and Conquer – Examples

Quicksort

Partition array into two parts around pivot Recursively quicksort each part of array Concatenate solutions

Mergesort

Partition array into two parts Recursively mergesort each half Merge two sorted arrays into single sorted array

Dynamic Programming Algorithm

Based on remembering past results

Approach

Divide problem into smaller subproblems Subproblems must be of same type Subproblems must overlap
Solve each subproblem recursively May simply look up solution (if previously solved)
Combine solutions into to solve original problem
Store solution to problem

Generally applied to optimization problems

Dynamic Programming – Example

Djikstra’s Shortest Path Algorithm

S = ∅

C[X] = 0

C[Y] = ∞ for all other nodes

while ( not all nodes in S )

find node K not in S with smallest C[K]

add K to S

for each node M not in S adjacent to K

C[M] = min ( C[M] , C[K] + cost of (K,M) )

Stores results of

smaller subproblems

Greedy Algorithm

Based on trying best current (local) choice

Approach

At each step of algorithm Choose best local solution

Avoid backtracking, exponential time O(2 n^ )

Hope local optimum lead to global optimum

Greedy Algorithm – Example

Kruskal’s Minimal Spanning Tree Algorithm

sort edges by weight (from least to most)

tree = ∅

for each edge (X,Y) in order

if it does not create a cycle

add (X,Y) to tree

stop when tree has N–1 edges

Picks best

local solution

at each step

Brute Force Algorithm

Based on trying all possible solutions

Approach

Generate and evaluate possible solutions until Satisfactory solution is found Best solution is found (if can be determined) All possible solutions found Return best solution Return failure if no satisfactory solution

Generally most expensive approach

Branch and Bound – Example

Branch and bound algorithm for TSP

Find possible paths using recursive backtracking Track cost of best current solution found Stop searching path if cost > best current solution Return lowest cost path

If good solution found early, can reduce search

May still require exponential time O(2 n^ )

Heuristic Algorithm

Based on trying to guide search for solution

Heuristic ⇒ “rule of thumb”

Approach

Generate and evaluate possible solutions Using “rule of thumb” Stop if satisfactory solution is found

Can reduce complexity

Not guaranteed to yield best solution

Heuristic Algorithm – Example

Heuristic algorithm for TSP

Find possible paths using recursive backtracking Search 2 lowest cost edges at each node first Calculate cost of each path Return lowest cost path from first 100 solutions

Algorithms Strategies - Objected Oriented Program II - Notes | CMSC 132, Study notes of Computer Science

Related documents

Partial preview of the text

Download Algorithms Strategies - Objected Oriented Program II - Notes | CMSC 132 and more Study notes Computer Science in PDF only on Docsity!

CMSC 132:

Object-Oriented Programming II

Algorithm Strategies

Department of Computer Science

University of Maryland, College Park

General Concepts

Algorithm strategy

Algorithm structure

Problem type

Recursive algorithms

Backtracking algorithms

Divide and conquer algorithms

Dynamic programming algorithms

Greedy algorithms

Brute force algorithms

Branch and bound algorithms

Heuristic algorithms

Based on reapplying algorithm to subproblem

Approach

Color a map with no more than four colors

Based on dividing problem into subproblems

Approach

Usually contains two or more recursive calls

Quicksort

Mergesort

Based on remembering past results

Approach

Generally applied to optimization problems

C[X] = 0

C[Y] = ∞ for all other nodes

find node K not in S with smallest C[K]

add K to S

for each node M not in S adjacent to K

C[M] = min ( C[M] , C[K] + cost of (K,M) )

Stores results of

smaller subproblems

Based on trying best current (local) choice

Approach

Avoid backtracking, exponential time O(2 n^ )

Hope local optimum lead to global optimum

sort edges by weight (from least to most)

tree = ∅

for each edge (X,Y) in order

if it does not create a cycle

add (X,Y) to tree

stop when tree has N–1 edges

Picks best

local solution

at each step

Based on trying all possible solutions

Approach

Generally most expensive approach

Branch and bound algorithm for TSP

If good solution found early, can reduce search

May still require exponential time O(2 n^ )

Based on trying to guide search for solution

Heuristic ⇒ “rule of thumb”

Approach

Can reduce complexity

Not guaranteed to yield best solution

Heuristic algorithm for TSP

Not guaranteed to find best solution

Heuristics used frequently in real applications

Wide range of strategies

Choice depends on