Transcript Parallel_Algorithms_In_Combinatorial_Optimization_Problems

CS 6260
PARALLEL COMPUTATION
PARALLEL ALGORITHMS IN COMBINATORIAL
OPTIMIZATION PROBLEMS
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Professor:
Elise De Doncker
Presented By:
Lina Hussein
TOPICS COVERED ARE:
Backtracking
 Branch and bound
 Divide and conquer
 Greedy Methods
 Short paths algorithms

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BRANCH AND BOUND

Branch and bound (BB) is a general algorithm for
finding optimal solutions of various optimization
problems, especially in discrete and combinatorial
optimization. It consists of a systematic enumeration
of all candidate solutions, where large subsets of
fruitless candidates are discarded en masse (all
together), by using upper and lower estimated bounds
of the quantity being optimized.
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BRANCH AND BOUND
If we picture the subproblems graphically, then we form
a search tree.
 Each subproblem is linked to its parent and eventually
to its children.
 Eliminating a problem from further consideration is
called pruning or fathoming.
 The act of bounding and then branching is called
processing.
 A subproblem that has not yet been considered is called
a candidate for processing.
 The set of candidates for processing is called the
candidate list.
 Going back on the path from a node to its root is called 4
backtracking.

BACKTRACKING


Backtracking is a general algorithm for finding all (or
some) solutions to some computational problem, that
incrementally builds candidates to the solutions, and
abandons each partial candidate ("backtracks") as
soon as it determines that it cannot possibly be
completed to a valid solution..
The Algorithm systematically searches for a solution
to a problem among all available options. It does so by
assuming that the solutions are represented by vectors
(v1, ..., vi) of values and by traversing in a depth first
manner the domains of the vectors until the solutions
are found.
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BACKTRACKING
A systematic way to iterate through all the possible
configurations of a search space.
 Solution: a vector v = (v1,v2,…,vi)
 At each step, we start from a given partial solution,
say, v=(v1,v2,…,vk), and try to extend it by adding
another element at the end.
 If so, we should count (or print,…) it.
 If not, check whether possible extension exits.



If so, recur and continue
If not, delete vk and try another possibility.
ALGORITHM try(v1,...,vi)
IF (v1,...,vi) is a solution THEN RETURN (v1,...,vi)
FOR each v DO
IF (v1,...,vi,v) is acceptable vector THEN sol = try(v1,...,vi,v)
IF sol != () THEN RETURN sol
END
END
RETURN ()
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PRUNING SEARCH
If Si is the domain of vi, then S1 × ... × Sm is the
solution space of the problem. The validity
criteria used in checking for acceptable vectors
determines what portion of that space needs to be
searched, and so it also determines the resources
required by the algorithm.
 To make a backtracking program efficient enough to
solve interesting problems, we must prune the
search space by terminating for every search path
the instant that is clear not to lead to a solution.

V1
.
.
.
V2
.
.
S1
S2
S2
...........................................................
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BACKTRACKING


The traversal of the solution space can be represented
by a depth-first traversal of a tree. The tree itself is
rarely entirely stored by the algorithm in discourse;
instead just a path toward a root is stored, to enable
the backtracking.
When you move forward on an x =1 branch, add to a
variable that keeps track of the sum of the subset
represented by the node. When you move back on an x
= 1 branch, subtract. Moving in either direction along
an x = 0 branch requires no add/subtract. When you
reach a node with the desired sum, terminate. When
you reach a node whose sum exceeds the desired sum,
backtrack; do not move into this nodes subtrees. When
you make a right child move see if the desired sum is
attainable by adding in all remaining integers; for this
keep another variable that gives you the sum of the
remaining integers.
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BACKTRACKING DEPTH-FIRST SEARCH
x1= 0
x1=1
x2=1
x2= 0
x2=1
x2= 0
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BACKTRACKING DEPTH-FIRST SEARCH
x1= 0
x1=1
x2=1
x2= 0
x2=1
x2= 0
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BACKTRACKING DEPTH-FIRST SEARCH
x1= 0
x1=1
x2=1
x2= 0
x2=1
x2= 0
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BACKTRACKING DEPTH-FIRST SEARCH
x1= 0
x1=1
x2=1
x2= 0
x2=1
x2= 0
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BACKTRACKING DEPTH-FIRST SEARCH
x1= 0
x1=1
x2=1
x2= 0
x2=1
x2= 0
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BACKTRACKING DEPTH-FIRST SEARCH
x1= 0
x1=1
x2=1
x2= 0
x2=1
x2= 0
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EXAMPLE

Example of the use Branch and Bound & backtracking
is Puzzles!
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14 1
13 2 3 12
6 11 5 10
9 8 7 15

1 2 3 4
5 6 7 8
9 10 11 12
13 14 15
For such problems, solutions are at different levels of
the tree
http://www.hbmeyer.de/backtrack/backtren.htm
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TOPICS COVERED ARE:
Branch and bound
 Backtracking
 Divide and conquer
 Greedy Methods
 Short paths algorithms

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DIVIDE AND CONQUER
divide and conquer (D&C) is an important algorithm
design paradigm based on multi-branched recursion.
The algorithm works by recursively breaking down a
problem into two or more sub-problems of the same (or
related) type, until these become simple enough to be
solved directly. The solutions to the sub-problems are
then combined to give a solution to the original
problem.
 This technique is the basis of efficient algorithms for
all kinds of problems, such as sorting (e.g., quick sort,
merge sort).

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ADVANTAGES


Solving difficult problems:
 Divide and conquer is a powerful tool for solving
conceptually difficult problems, such as the classic Tower of
Hanoi puzzle: it break the problem into sub-problems, then
solve the trivial cases and combine sub-problems to the
original problem.
Roundoff control
 In computations with rounded arithmetic, e.g. with floating
point numbers, a D&C algorithm may yield more accurate
results than any equivalent iterative method.
 Example, one can add N numbers either by a simple loop
that adds each datum to a single variable, or by a D&C
algorithm that breaks the data set into two halves,
recursively computes the sum of each half, and then adds
the two sums. While the second method performs the same
number of additions as the first, and pays the overhead of
the recursive calls, it is usually more accurate.
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IN PARALLELISM...

Divide and conquer algorithms are naturally
adapted for execution in multi-processor
machines, especially shared-memory systems
where the communication of data between
processors does not need to be planned in
advance, because distinct sub-problems can be
executed on different processors.
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TOPICS COVERED ARE:
Branch and bound
 Backtracking
 Divide and conquer
 Greedy Methods
 Short paths algorithms

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GREEDY METHODS
A greedy algorithm:

is any algorithm that follows the problem solving
metaheuristic of making the locally optimal choice at each
stage with the hope of finding the global optimum.
A metaheuristic method:



Is method for solving a very general class of computational
problems that aims on obtaining a more efficient or more
robust procedure for the problem.
Generally it is applied to problems for which there is no
satisfactory problem-specific algorithm designed to solve it.
It targeted to the combinatorial optimization (problems
that’s are a problems in which has an optimization function
to( minimize or maximize) subject to some constraints and its
goal is to find the best possible solution
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EXAMPLES

The vehicle routing problem (VRP)


Travelling salesman problem


A number of goods need to be moved from certain
pickup locations to other delivery locations. The goal
is to find optimal routes for a fleet of vehicles to visit
the pickup and drop-off locations.
Given a list of cities and their pair wise distances, the
task is to find a shortest possible tour that visits each
city exactly once.
Coin Change

(making change for n $ using minimum number of coins)
The knapsack problem
 The Shortest Path Problem

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KNAPSACK

The knapsack problem or rucksack problem
is a problem in combinatorial optimization. It
derives its name from the following maximization
problem of the best choice of essentials that can
fit into one bag to be carried on a trip. Given a set
of items, each with a weight and a value,
determine the number of each item to include in
a collection so that the total weight is less than a
given limit and the total value is as large as
possible.
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THE ORIGINAL KNAPSACK PROBLEM (1)
 Problem


Definition
Want to carry essential items in one bag
Given a set of items, each has
A cost (i.e., 12kg)
 A value (i.e., 4$)

 Goal

To determine the # of each item to include in a collection
so that
The total cost is less than some given cost
 And the total value is as large as possible

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THE ORIGINAL KNAPSACK PROBLEM (2)

Three Types




0/1 Knapsack Problem
 restricts the number of each kind of item to zero or one
Bounded Knapsack Problem
 restricts the number of each item to a specific value
Unbounded Knapsack Problem
 places no bounds on the number of each item
Complexity Analysis

The general knapsack problem is known to be NP-hard


No polynomial-time algorithm is known for this problem
Here, we use greedy heuristics which cannot guarantee the
optimal solution
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0/1 KNAPSACK PROBLEM (1)

Problem: John wishes to take n items on a trip



The weight of item i is wi & items are all different (0/1 Knapsack
Problem)
The items are to be carried in a knapsack whose weight capacity is c
 When sum of item weights ≤ c, all n items can be carried in the
knapsack
 When sum of item weights > c, some items must be left behind
Which items should be taken/left?
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0/1 KNAPSACK PROBLEM (2)

John assigns a profit pi to item i



All weights and profits are positive numbers
John wants to select a subset of the n items to take
 The weight of the subset should not exceed the capacity of the
knapsack (constraint)
 Cannot select a fraction of an item (constraint)
 The profit of the subset is the sum of the profits of the selected
items (optimization function)
 The profit of the selected subset should be maximum (optimization
criterion)
Let xi = 1 when item i is selected and xi = 0 when item i is not selected
 Because this is a 0/1 Knapsack Problem, you can choose the item
or not choose it.
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GREEDY ATTEMPTS FOR 0/1 KNAPSACK
 Apply

Greedy attempt on capacity utilization



greedy method:
Greedy criterion: select items in increasing order of weight
When n = 2, c = 7, w = [3, 6], p = [2, 10],
if only item 1 is selected  profit of selection is 2  not best
selection!
Greedy attempt on profit earned


Greedy criterion: select items in decreasing order of profit
When n = 3, c = 7, w = [7, 3, 2], p = [10, 8, 6],
if only item 1 is selected  profit of selection is 10  not best
selection!
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THE SHORTEST PATH PROBLEM


Path length is sum of weights of edges on path in directed
weighted graph
 The vertex at which the path begins is the source vertex
 The vertex at which the path ends is the destination vertex
Goal
 To find a path between two vertices such that the sum of the
weights of its edges is minimized
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TYPES OF THE SHORTEST PATH PROBLEM

Three types
 Single-source single-destination shortest path
 Single-source all-destinations shortest path
 All pairs (every vertex is a source and destination)
shortest path
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SINGLE-SOURCE SINGLE-DESTINATION SHORTED
PATH


Possible greedy algorithm
 Leave the source vertex using the cheapest edge
 Leave the current vertex using the cheapest edge to the next vertex
 Continue until destination is reached
Try Shortest 1 to 7 Path by this Greedy Algorithm
 the algorithm does not guarantee the optimal solution
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GREEDY SINGLE-SOURCE ALL-DESTINATIONS SHORTEST
PATH (1)

Problem: Generating the shortest paths in increasing order of length from one
source to multiple destinations
Greedy Solution
 Given n vertices, First shortest path is from the source vertex to itself
 The length of this path is 0
 Generate up to n paths (including path from source to itself) by the greedy
criteria
 from the vertices to which a shortest path has not been generated,
select one that results in the least path length
 Construct up to n paths in order of increasing length
Note:
The solution to the problem consists of up to n paths.
The greedy method suggests building these n paths in order of increasing length.
First build the shortest of the up to n paths (I.e., the path to the nearest
destination).
Then build the second shortest path, and so on.

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GREEDY SINGLE-SOURCE ALL-DESTINATIONS SHORTEST
PATH (2)
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Path
Length
0
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
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Each path (other than first) is a one
edge extension of a previous path
Next shortest path is the shortest one
edge extension of an already generated
shortest path
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Increasing
order
GREEDY SINGLE SOURCE ALL DESTINATIONS:
EXAMPLE (1)
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1
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[1] [2] [3] [4] [5] [6] [7]
d 0
6 2 16 - 14
p 1 1 1 1
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GREEDY SINGLE SOURCE ALL DESTINATIONS :
EXAMPLE (2)
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[1] [2] [3] [4]
d 0
6 2 16
p 1 1 1
[5] [6] [7]
55- 10
- 14
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1
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GREEDY SINGLE SOURCE ALL DESTINATIONS :
EXAMPLE (3)
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[1] [2] [3] [4] [5] [6] [7]
d 0 66 2 16
9 5- 10
- 14
p 1 1 51 33
1
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GREEDY SINGLE SOURCE ALL DESTINATIONS :
EXAMPLE (4)
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[1] [2] [3] [4] [5] [6] [7]
d 0
6 2 9 5- 10
- 14
p 1 1 5 33
1
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GREEDY SINGLE SOURCE ALL DESTINATIONS :
EXAMPLE (5)
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[1] [2] [3] [4] [5] [6] [7]
d 0
6 2 9 5- 1- 14
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p 1 1 5 3- 03
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5
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GREEDY SINGLE SOURCE ALL DESTINATIONS :
EXAMPLE (6)
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[1] [2] [3] [4] [5] [6] [7]
d 0
6 2 9 5- 10
- 14
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p 1 1 5 3- 416
3
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TOPICS COVERED ARE:





Backtracking
Branch and bound
Divide and conquer
Greedy Methods
Short path algorithm
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Parallel Algorithms
Arrays and
Trees
Meshes of
Trees
Packet
Routing
Graph
Algorithms
Greedy
Algorithms
Short paths
Hypercube…&
Networks
Combinatorial
optimization
problems
Knapsack
Problem
.. A branch of optimization. Its domain is optimization
problems where the set of feasible solutions is discrete
or can be reduced to a discrete one, and the goal is to
find the best possible solution
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USE OF ALGORITHMS IN PARALLEL
With Parallelism many Problems appeared ,
some are those of choice of granularity such as
Grouping of tasks or partitioning, scheduling..
And when the physical architecture is to be taken
into account we face the Mapping problem.
 Greedy Methods Packet routing



Routes every packets to its destination through the
shortest path.
Shortest path  Graph algorithms

To compute the least weight directed path between
any two nodes in a weighted graph.
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USE OF ALGORITHMS IN PARALLEL

Branch and Bound Exact Methods
..Based on exploring all possible solutions. In theory it
gives optimal solutions but in practice it can be costly an
unusable for large problems.
 It uses B&B in Mapping Problem:



A mapping, is an application allocation which associate a
processor with a task.
The B&B algorithms will involve mapping a task
progressively between processors by scanning a search
tree that gives all possible combinations. For each
mapping a partial solution is given and for each one a
set of less restricted partial solutions is constructed
similarly by mapping a second task and so on until all
the tasks have been mapped(leaves of the tree are
reached). For each node the cost of mapping is computed
then all branches can be pruned through an estimating
function and he best computed mapping is then choosed.
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Q&A
BRANCH AND BOUND VS. BACKTRACKING?

B&B is An enhancement of backtracking
 Similarity


A state space tree is used to solve a problem.
Difference
The branch-and-bound algorithm does not limit us to any particular
way of traversing the tree and is used only for optimization
problems
 The backtracking algorithm requires traversing the tree and is used
for non-optimization problems as well.

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REFERENCES
Parallel Algorithms and Architectures ,by Michel
Cosnard, Denis Trystram.
 Parallel and sequential algorithms..
 Greedy Method and Compression, Goodrich
Tamassia


http://www.wikipedia.org/
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