Graph Traversal Algorithms - Breadth First Search

Analysis and
Design of
Algorithms
BREADTH FIRST SEARCH,
BRANCHING AND BOUNDING

 Instructor
Prof. Amrinder Arora
amrinder@gwu.edu
Please copy TA on emails
Please feel free to call as well

 Available for study sessions
Science and Engineering Hall
GWU
Algorithms Graph Traversal Techniques, Branch & Bound 2
LOGISTICS

Algorithms
Analysis
Asymptotic
NP-
Completeness
Design
D&C
Greedy
DP
Graph
B&B
Applications
WHERE WE ARE
 Done
 Done
 DFS, BFS
 Done
 Done

Knowing that an object exists within a certain
distance from us, in an infinite maze, how can we
find it?
INFINITE MAZE

1. Select an unvisited node s, visit it, have it be the
root in a BFS tree being formed. Its level is called
the current level.
2. From each node x in the current level, in the order
in which the level nodes were visited, visit all the
unvisited neighbors of x. The newly visited nodes
from this level form a new level that becomes the
next current level.
3. Repeat the previous step until no more nodes can
be visited.
4. If there are still unvisited nodes, repeat from Step
1.
BREADTH FIRST SEARCH (BFS)

 http://commons.wikimedia.org/wiki/File:Breadth-
First-Search-Algorithm.gif
 http://www.cs.sunysb.edu/~skiena/combinatorica/a
nimations/anim/bfs.gif
BFS EXAMPLE

 Observations: The first node visited in each level is
the first node from which to proceed to visit new
nodes.
 This suggests that a queue is the proper data
structure to remember the order of the steps.
BFS IMPLEMENTATION

Procedure BFS(input: graph G)
Queue Q; Integer s, x
while (G has an unvisited node) do
s := an unvisited node
visit(s)
Enqueue(s,Q)
While (Q is not empty) do
x := Dequeue(Q)
For (unvisited neighbor y of x) do
visit(y)
Enqueue(y,Q)
BFS PSEUDOCODE

 Robotics
 Optimization problems using Branch and Bound
BFS APPLICATIONS

 Hamiltonian Cycle
 K-clique
 K-coloring
OTHER GRAPH PROBLEMS

Algorithms
Analysis
Asymptotic
NP-
Completeness
Design
D&C
Greedy
DP
Graph
B&B
Applications
WHERE WE ARE
 Done
 Done
 Starting today..
 Done
 Done
 Done

 A systematic method for solving optimization problems
 A general optimization technique that applies where the
greedy method and dynamic programming fail.
 Much slower – often leads to exponential time
complexities in the worst case. If applied carefully, can
lead to algorithms that run reasonably fast on average.
 General idea of B&B is a BFS-like search for the optimal
solution, but not all nodes get expanded.
 Carefully selected criterion determines which node to expand and
when
 Another criterion tells the algorithm when an optimal solution has
been found.
BRANCH AND BOUND

 Given jobs, resources, and cost matrix (cost if resource i
does job j), how to assign jobs to resources, such that
each job is done by a different resource, and the overall
cost is minimum?
 What is the right D&C strategy?
 What is the right BFS strategy?
B&B FOR JOB ASSIGNMENT
J1 J2 J3 J4 J5
R1 2 4 5 3 11
R2 3 5 6 4 8
R3 4 3 5 7 9
R4 3 4 3 8 12
R5 4 2 6 4 9

What is the lower bound?
[In other words, how much will it cost at
minimum to do the job assignment?]
Each job must be done – so if we add minimum cost
per job, then that must be minimum cost
Each person must do a job – so if we add minimum
cost per resource, then that must be minimum cost
Taking the maximum of these two minimums, is a
good “lower bound”.
LOWER BOUND

 Any random assignment can be an upper bound
 Or, we can use a greedy algorithm for upper bound.
 In the context of a minimization problem, upper
bound means: “Here is a solution that I have already
found. We are not interested in solutions that are
more expensive than this.” So, in other words, this
solution now defines the highest cost (the upper
bound on cost) that we are willing to pay.
 If we find an even better solution, then we will
decrease our upper bound.
UPPER BOUND

 Assigning a job to a resource can be a node in the
BFS tree
 We need to think about the solution space as a tree,
and define what a node in the tree means with
respect to our problem (job assignment)
BRANCHING

 B&B is a general algorithm for finding optimal
solutions of various optimization problems,
especially in discrete and combinatorial
optimization.
 2 main steps
 Branching
 Bounding
 (If bounding allows, then) Node Elimination
MORE.. (FROM WIKIPEDIA)

 We can consider a maximization problem just as
easily in B&B as well.
 In case of a maximization problem, the “lower
bound” comes from a known solution. The “upper
bound” comes from a theoretical/analytical solution
that typically relaxes some constraint.
 Lower bound signifies: A solution we already know exists
 Upper bound signifies: Maximum value we could obtain, even if
some constraints were relaxed.
 An example of a maximization problem is a 0/1
knapsack problem.
B&B FOR MAXIMIZATION PROBLEMS

 Given a set of items, each with a weight and a value,
determine which items 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.
 Different from fractional knapsack, can only take or
not take an item.
 No easy solution – greedy solution may not be
optimal.
0/1 KNAPSACK PROBLEM

 3 Items. Knapsack can hold 50 lbs.
 Item 1 weighs 10 lbs, worth 60$
 Greedy solution: Item 1 + Item 2, worth = $160
 Optimal solution: Item 3 + Item 2, worth = $220
0/1 KNAPSACK EXAMPLE

 Given a complete graph with n vertices, the
salesperson wishes to make a tour, visiting each city
exactly once and finishing at the city he starts from.
Cost of going from city i to city j = c(i,j).
TRAVELING SALESPERSON PROBLEM (TSP)

 Input: n jobs, n employees, and an n x n matrix A
where Aij be the cost if person i performs job j.
 Problem: Find a one-to-one matching of the n
employees to the n jobs so that the total cost is
minimized.
 Formally, find a permutation f such that C(f), where
C(f)=A1f(1) + A2f(2) + ... + Anf(n) is minimized.
JOB ASSIGNMENT PROBLEM

 Do a Breadth First Search
 Evaluate each node for upper and lower bounds
 Branch into the “best” node
 Terminate if solution meets performance parameters
BRANCH AND BOUND TEMPLATE

 Branch and Bound depends upon being able to
“bound” each node in the search tree, so that we can
evaluate whether or not to expand it.
 Lower bound and upper bound depend on the
problem
BRANCH AND BOUND TEMPLATE (CONT.)

 Using a mixture of upper and lower bounds, we can
find when might be a right time to terminate the
search.
TERMINATION CRITERIA

 B&B is an interesting technique that uses BFS and
bounding to eliminate the nodes in the search tree.
 Can solve problems that are intrinsically harder.
 Time complexity may not be polynomial, but may be
practical.
 We can return an approximate, usable solution with
some bound on performance.
 Developing interesting bounds depends upon the
problem and uses problem specific insights.
SUMMARY

Algorithms
Analysis
Asymptotic
NP-
Completeness
Design
D&C
Greedy
DP
Graph
B&B
Applications
WHERE WE ARE
 Done
 Done
 Done
 Done
 Done
 Done

 Job Assignment problem, Branch and Bound
 Review slides again – Be ready to discuss lower
bound and upper bounds
 http://en.wikipedia.org/wiki/Branch_and_bound
 “An Automatic Method of Solving Discrete
Programming Problems”
 http://rjlipton.wordpress.com/2012/12/19/branch-
and-bound-why-does-it-work/
 Preview: NP completeness and lower bound for coins
problem.
READING ASSIGNMENT/PREVIEW

Graph Traversal Algorithms - Breadth First Search

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