CSCE350 Algorithms and Data Structure
Lecture 7
Jianjun Hu
Department of Computer Science and Engineering
University of South Carolina
2009.9.
Outline
Learning how to understand Pseudocodes of Mysterious
programs
Brute Force Strategy for Algorithm Design
 The art of lazy algorithm is to count/estimate its running time
 Is it doable within given timeframe?
Sequential Search – Smallest Distance of
Two numbers
Find smallest distance between any 2 numbers
// Search key K in A[0..n - 1]
ALGORITHM MinDis tan ce( A[0..n  1])
Output:minimum distance between two of its numbers
d min  
for i  0 to n  1
for j  0 to n  1
if i  j and A[i ]  A[ j ]  d min
d min | A[i ]  A[ j ] |
return dmin
What is time efficiency of this algorithm?
Sequential Search – Brute Force
Find whether a search key is present in an array
// S e archk e yK in A[0..n-1]
ALGO RITHMSequentialSearch( A[0..n ], K )
A[n ]  K
i0
wh i leA[i ]  K do
i  i 1
if i  n re tu rni
e l sere tu rn-1
What is time efficiency of this algorithm?
Brute-Force String Matching
Find a pattern in the text: Pattern – ‘NOT’, text –
‘NOBODY_NOTICED_HIM’
Typical Applications – ‘find’ function in the text editor, e.g., MS-Word,
Google search
ALGO RITHMBFStringMatch(T [0..n  1], P[0..m  1])
for i  0 to n-m do
j0
while j  m and P[ j ]  T [i  j ]
j  j 1
if j  m re turni
re turn-1
What is the time efficiency of this algorithm?
Closest-Pair and Convex Hull Problems by
Brute Force
Closest-Pair problem
• Given n points in a plane, find the closest pair
• How to solve this problem and what is the time efficiency of
this algorithm?
Convex-Hull problem
• Convex hull is the tightest convex polygon that bounds a set of
n points in a plane
• Convex polygon – any two points in this polygon results in the
inclusion of the segment that links these two points also in this
polygon
Closest-Pair problem
Given n points in a plane, find the closest pair
Convex/NonConvex Polygons
Convex Hull
Imagine a rubber band around a set of nails
Nails touched by the band  extreme points
P6
P7
P2
P8
P3
P4
P5
P1
Solve Convex-Hull Problem
Connect any pair of points by a line segment.
Each line segment partitions the plane to the two half planes
If all the n points are on the same side of this line segment
ax+by-c >0 or <0
Such a line segment is an edge of the convex-hull polygon
What is the time efficiency of this Brute-Force algorithm?
For each possible pair of points, we need to check whether all
the remaining n-2 points are on the same side of the line
segment that connects these pair of points.
For Sorting, String Matching, and Convex-Hull problems, we
will revisit them by designing more efficient algorithms.
Exhaustive Search
A brute-force approach to combinatorial problem
• Generate each and every element of the problem’s domain
• Then compare and select the desirable element that satisfies
the set constraints
• Involve combinatorial objects such as permutations,
combinations, and subsets of a given set
• The time efficiency is usually bad – usually the complexity
grows exponentially with the input size
Three examples
• Traveling salesman problem
• Knapsack problem
• Assignment problem
Traveling Salesman Problem
Find the shortest tour through a given n cities that visits each
city exactly once before returning to the starting city
Using graph model: city  vertex, road  edge, length of the
road  edge weight.
TSP  shortest Hamiltonian Circuit – a cycle that passes
through all the vertices of the graph exactly once
Exhaustive search:
• List all the possible Hamiltonian circuits (starting from any
vertex)
• Ignore the direction
• How many candidate circuits do we have?  (n-1)!/2
• Very high complexity
TSP Example
2
a
5
3
7
8
c
1
Tour
b
d
Length
a ---> b ---> c --->d ---> a
l = 2 + 8 + 1 + 7 = 18
a ---> b---> d ---> c ---> a
l = 2 + 3 + 1 + 5 = 11
a ---> c ---> b ---> d --->a
l = 5 + 8 + 3 + 7 = 23
a ---> c ---> d ---> b ---> a
l = 5 + 1 + 3 + 2 = 11
a---> d ---> b ---> c ---> a
l = 7 + 3 + 8 + 5 = 23
a ---> d ---> c ---> b ---> a
l = 7 + 1 + 8 + 2 = 18
optimal
optimal
Knapsack Problem
Given n items of known weight w1, w2, …wn with values v1,
v2, ..vn,
And a knapsack of capacity W
How to select the items to fill the knapsack such that you get
max value?
Steal money from bank?