1. Recall Brute Force as a Problem Solving Technique
Exhaustive Search as a Problem Solving Technique
Interesting problems
Selection Sort, Bubble Sort*
String Matching
Closest-Pair
Traveling Salesman
Knapsack
Assignment Problem
Algorithms for generating permutations
Algorithms for generating all subsets

2. Outline Algorithms for generating permutations
Algorithms for generating all subsets
Decrease-and-Conquer

3. Recall from Lecture 7 summation puzzles Two conditions are assumed:
pot + pan = bib
dog + cat = pig
boy + girl = baby
Two conditions are assumed:
the correspondence between letters and decimal digits is one-to-one
the digit zero does not appear as the left-most digit in any of the numbers.
© 2013 Goodrich, Tamassia, Goldwasser
Recursion

4. Recall from Lecture 7 Multiple recursion:
makes potentially many recursive calls
not just one or two
© 2013 Goodrich, Tamassia, Goldwasser
Recursion

5. Algorithm for Multiple Recursion
Algorithm PuzzleSolve(k,S,U):
Input: Integer k, sequence S, and set U (universe of elements to test)
Output: Enumeration of all k-length extensions to S using elements in U without repetitions
for all e in U do
Remove e from U {e is now being used}
Add e to the end of S
if k = 1 then
Test whether S is a configuration that solves the puzzle
if S solves the puzzle then
return “Solution found: ” + S
else
PuzzleSolve(k - 1, S,U)
Add e back to U {e is now unused}
Remove e from the end of S

6. Slide by Matt Stallmann included with permission.
Example
cbb + ba = abc
a,b,c stand for 7,8,9; not necessarily in that order
= 897
[] {a,b,c}
[a] {b,c}
a=7
[b] {a,c}
b=7
[c] {a,b}
c=7
[ab] {c}
a=7,b=8
c=9
[ac] {b}
a=7,c=8
b=9
[ba] {c}
b=7,a=8
[bc] {a}
b=7,c=8
a=9
[ca] {b}
c=7,a=8
[cb] {a}
c=7,b=8
© 2013 Goodrich, Tamassia, Goldwasser
Recursion

7. Visualizing PuzzleSolve( 3 , () ,{ a b c } )
Initial call 2 1 ab ac cb ca bc ba
abc
acb
bac
bca
cab
cba
© 2013 Goodrich, Tamassia, Goldwasser
Recursion

8. Generating Permutations
Algorithm:
If n = 1 return 1; otherwise, generate recursively the list of all permutations of 12…n-1 and then insert n into each of those permutations by starting with inserting n into 12...n-1 by moving right to left and then switching direction for each new permutation
Example: n=3
start 1
insert 2 into 1 right to left 12 21
insert 3 into 12 right to left
insert 3 into 21 left to right
A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 4 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

9. Other permutation generating algorithms
Johnson-Trotter [L] page 145
Lexicographic-order algorithm [L] page 146
Heap’s algorithm (Problem 9 in Assignment 4)
A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 4 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

10. Johnson-Trotter Algorithm
associate a direction with each element k in a permutation
The element k is said to be mobile in such an arrow-marked permutation if its arrow points to a smaller number adjacent to it

11. Johnson-Trotter Algorithm

12. Generating Subsets IDEA: Can we find minimal-change algorithm?
All subsets correspond to the binary string of length n
For n=3
Can we find minimal-change algorithm?
Yes
A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 4 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

13. Generating Subsets
Binary reflected Gray code: minimal-change algorithm for generating 2n bit strings corresponding to all the subsets of an n-element set where n > 0
If n=1 make list L of two bit strings 0 and 1
else
generate recursively list L1 of bit strings of length n-1
copy list L1 in reverse order to get list L2
add 0 in front of each bit string in list L1
add 1 in front of each bit string in list L2
append L2 to L1 to get L
return L
A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 4 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

14. Binary reflected Gray code
A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 4 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.